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丽贝卡-戈尔茨坦的《不完备性:哥德尔的证明和悖论》中的“阅读建议”

已有 1941 次阅读 2021-9-14 16:22 |个人分类:解读哥德尔不完全性定理|系统分类:海外观察

戈尔茨坦的《不完备性:哥德尔的证明和悖论》一书最后给出了关于哥德尔文献的阅读建议:


像我之前和之后的许多人一样,我第一次真正接触到哥德尔不完全性定理,并不是通过研究那篇著名的1931年论文本身,而是在大学期间阅读了Ernest NagelJames R. Newman的著名的《哥德尔证明》( Godel's Proof,纽约大学出版社,1968)。这是一篇通俗的论述,但却能对证明的实质进行一些详细的阐述,我的世界被震撼了,这么多年后重读这本书,我又一次被打动了。这是一本奇妙的小书,以它自己的方式成为经典。


Jaakko Hintikka的《论哥德尔》(On GodelBelmont, CA: Wadsworth Thomson Learning, 2000)这本书非常薄(70页),也是为非专业人士提供的对哥德尔证明的清晰和简洁的介绍。与更著名的《哥德尔证明》一样,Hintikka的证明也是自成一体的,不需要以前的逻辑知识,他也有很好的幽默感。


就逻辑学家的生活而言,《逻辑困境》(Logical DilemmasKurt Godel的生活和工作,Wellesley, MA: A K Peters, 1997),作者John Dawson,是权威的。Dawson不仅是一位逻辑学家,也是哥德尔的档案管理员,他的妻子学会了翻译哥德尔的速记,他在介绍哥德尔的生平方面处于无可匹敌的地位。研究所的数学家Armand Borel告诉我,哥德尔的文学遗迹是由哥德尔的遗孀捐赠给高等研究所的,这些遗迹非常混乱,无序地堆放在腐烂的箱子里;然后 "一个年轻人"John Dawson)提出将这一切整理好。"他做得很好,我听说。" 的确,他做了。


John Dawson也有两篇关于哥德尔的论文,通俗而有趣。"Kurt Godel in Sharper Focus " "The Reception of Godel's Incompleteness Theorems"。这两篇论文都转载于Stuart Shanker编辑的《聚焦哥德尔定理》,以及其他有趣的文章,包括Solomon Feferman "Kurt Godel: 确信和谨慎(Kurt Godel: Conviction and Caution»


王浩采撷了哥德尔的思想,写了三本相当古怪但耐人寻味的书:《从数学到哲学》( From Mathematics to Philosophy,纽约:人文出版社,1974年)、《对库尔特-哥德尔的思考》( Reflections on Kurt Godel,马萨诸塞州剑桥:麻省理工学院出版社,1987年)和《逻辑之旅》( A Logical Journey,马萨诸塞州剑桥:麻省理工学院出版社,1996年)。这些书叙述了王浩与哥德尔的对话,其中穿插了这位逻辑学家的生活史以及王浩自己对他和哥德尔讨论的话题的看法,在内容上弥补了结构上的不足。


有几本哥德尔的回忆录,是由那些最初在维也纳认识他的人写的,它们很吸引人,而且以自己的方式令人感动。首先是Georg Kreisl "Kurt Godel: 1906-1978",《皇家学会会员传记》,第26卷(1980),第148-224页。Kreisl是一位杰出的数学逻辑学家,他的地位很独特,在Kreisl还是学生的时候,他就相当了解维特根斯坦,后来,又在普林斯顿认识了哥德尔。卡尔-门格尔与哥德尔一起被邀请加入维也纳圈,成为汉斯-哈恩的得意门生,他对哥德尔的宝贵的第一手回忆在《库尔特-哥德尔的回忆》中有所叙述,该书收录于《维也纳圈和数学讨论会的回忆》中。Louise Golland, Brian McGuinness, and Abe Sklar (Dordrecht: Kluwer, 1994)。还有Olga Taussky-Todd,她自己也是一位数字理论家,她也是在学生时代第一次认识哥德尔的。她的 "对库尔特-哥德尔的回忆 "载于《哥德尔回忆》(Naples: Bibliopolis, 1987)。


如果读者有兴趣看到一个当代多面手如何将哥德尔定理应用于他自己的创造性科学思维,那么建议他阅读罗杰-彭罗斯的《皇帝的新脑:关于计算机、思想和物理定律》(纽约:企鹅出版社,1989年)和他的《思想的阴影:寻找缺失的意识科学》(牛津:牛津大学出版社,1994年)。像哥德尔一样,彭罗斯是一个公认的数学柏拉图主义者,他对不完全性定理的解释与哥德尔完全一样。他还讨论了许多其他迷人的数学问题包括图灵对哥德尔开始的工作的贡献, Mandelbrot 集,以及彭罗斯自己在平面瓷砖上的工作他认为这些都是指向柏拉图主义的方向。彭罗斯的总体论点是,数学知识,我们拥有这些知识的惊人事实,证明了物理学定律与我们之前的梦想有着根本的不同。


Douglas Hofstadter的普利策奖获奖作品《哥德尔、埃舍尔、巴赫:永恒的金带》(纽约:基础书店,1974年)是一次自反的狂欢。Hofstadter在将逻辑、艺术和音乐中的思想编织在一起方面做得很好,正如书名所说。当有人问我在过去的几年里一直在研究什么时,我说 « 哥德尔 »,往往会得到一个白眼。然后我提到Hofstadter的畅销书的书名,那白眼就会让位给一个微笑和哦,是的


最后,还有哥德尔本人的写作,他发表的几篇论文和许多未发表的作品,载于《作品集》(Collected Works),Solomon Feferman等人(牛津:牛津大学出版社,1986-),到目前为止有三卷。


****

参考文献:

Incompleteness: The Proof and Paradox of Kurt Gödel

https://www.essra.org.cn/upload/202102/Incompleteness%20-%20The%20Proof%20and%20Paradox%20of%20Kurt%20Godel%20by%20Rebecca%20Goldstein2005.pdf


原文:


Suggested Reading


Like many before and after me, my first substantive exposure to Godel's incompleteness theorems came not by way of studying the famous 1931 paper itself but rather by reading, as an undergraduate, the celebrated Godel's Proof by Ernest Nagel and James R. Newman (New York: New York University Press, 1968). This is a popular exposition that yet manages to go into some detail concerning the substance of the proof. My world was rocked. On rereading it after all these years, I was impressed all over again. It's a wonderful little book, in its own way a classic.


Jaakko Hintikka's very slim (70 pages) book, On Godel (Belmont, CA: Wadsworth Thomson Learning, 2000), is also a clear and concise presentation of Godel's proof for the non-expert. Like the more expansive Godel's Proof, Hintikka's proof is self-contained, requiring no previous knowledge of logic. He also has a good sense of humor.


So far as the life of the logician is concerned, Logical Dilemmas: The Life and Work of Kurt Godel (Wellesley, MA: A K Peters, 1997) by John Dawson is definitive. As not only a logician but also GodePs archivist, whose wife learned to translate Godel's shorthand, Dawson was in an unrivaled position for presenting the life of Godel. I was told by Institute mathematician Armand Borel that Godel's literary remains, which had been donated to the Institute for Advanced Study by Godel's widow, were in utter chaos, piled helter-skelter into decaying boxes; and then "a young man" (Dawson) had offered to put it all into order. "He did a good job, I'm told." Indeed he did.


John Dawson also has two papers on Godel that are accessible and interesting: "Kurt Godel in Sharper Focus" and "The Reception of Godel's Incompleteness Theorems." Both are reprinted in Godel's Theorem in Focus, edited by Stuart Shanker, as are other interesting essays, including Solomon Feferman's "Kurt Godel: Conviction and Caution. »


Hao Wang produced three rather eccentric but intriguing books out of the pickings of Godel's mind: From Mathematics to Philosophy (New York: Humanities Press, 1974), Reflections on Kurt Godel (Cambridge, MA: MIT Press, 1987), and A Logical Journey (Cambridge, MA: MIT Press, 1996). The books recount conversations Wang had with Godel, interlaced with history of the logician's life and Wang's own views on the topics he and Godel discussed. What they lack in structure they compensate for in content.


There are several, memoirs of Godel, written by those who had first known him in Vienna, and they are fascinating and in their own way touching. There is first of all Georg Kreisl's "Kurt Godel: 1906-1978," Biographical Memoirs of Fellows of the Royal Society, Vol. 26 (1980), pp. 148-224. Kreisl, an eminent mathematical logician, is in a unique position, having known Wittgenstein quite well when Kreisl was a student, and then, later, having gotten to know Godel in Princeton. Karl Menger had been invited, together with Godel, to join the Vienna Circle as favored students of Hans Hahn and his invaluable first-hand reminiscences of Godel are recounted in "Memories of Kurt Godel," in Reminiscences of the Vienna Circle and the Mathematical Colloquium, ed. Louise Golland, Brian McGuinness, and Abe Sklar (Dordrecht: Kluwer, 1994). And then there is Olga Taussky-Todd, herself a number-theorist, who also had first come to know Godel in their student days. Her "Remembrances of Kurt Godel" is in Godel Remembered (Naples: Bibliopolis, 1987).


If the reader is interested in seeing how a contemporary polymath applies Godel's theorems in his own creative scientific thinking) then he is advised to read Roger Penrose's The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (New York: Penguin, 1989) and his Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford: Oxford University Press, 1994). Like Go'del, Penrose is a confirmed mathematical Platonist; he interprets the incompleteness theorems exactly as Godel did. There's lots of further fascinating mathematics that he discusses—-including Turing's contributions to the work Godel began, the Mandelbrot set, and Penrose's own work on the tiling of the plane—all argued, by him, as pointing in the direction of Platonism. Penrose's overall argument is that mathematical knowledge, the amazing fact that we have it, is evidence that the laws of physics are of a fundamentally different character than we have heretofore dreamt.


Douglas Hofstadter's Pulitzer-prize-winning Godel, Escher, Bach: The Eternal Golden Braid (New York: Basic Books, 1974) is a spirited romp through self-referentiality. Hofstadter does a wonderful job of braiding together ideas in logic, art, and music, just as the title promises. When, upon being asked what I'd been working on these past few years, I'd say "Godel," more often than not I got a blank stare in return. Then I'd mention the title of Hofstadter's bestseller, and the blank stare would give way to a smile and an "oh yes. »


Finally, there is the writing of Godel himself, his few published papers and his many unpublished works, in Collected Works, ed. Solomon Feferman et al. (Oxford: Oxford University Press, 1986-). There are three volumes to date.





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