出版社: Oxford University Press
副标题: The Loss of Certainty
出版年: 1982617
页数: 384
定价: USD 19.95
装帧: Paperback
ISBN: 9780195030853
作者简介 · · · · · ·
M·克莱因，美国纽约大学柯朗数学研究所的荣誉教授，曾任《数学杂志》的副主编，《精确科学史档案的主编，它的著作还有《西方文化中的数学》、《古今数学思想》等。自从欧几里得建立了现代数学的明确模式以来，他是比任何人都更好地理解了数学的思想家。
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理解无限——简明扼要的数学哲学发展史
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魏厚生 (士为悦己者读书)
The Isolation of Mathematics p279 The establishment of existence theorems of differential equations, first undertaken by Cauchy, was intended to guarantee that the mathematical formulations of physical problems do have a solution, so that one could confidently seek that solution. Hence, though this work is totally mathematical, it does have ulterior physical significance. [Later on the author ...20141029 15:31
The Isolation of Mathematics p279The establishment of existence theorems of differential equations, first undertaken by Cauchy, was intended to guarantee that the mathematical formulations of physical problems do have a solution, so that one could confidently seek that solution. Hence, though this work is totally mathematical, it does have ulterior physical significance.[Later on the author said that: ]p282Another factor induced many mathematicians to tackle problems of pure mathematics.Problem of science are rarely solved in toto....There is a fascination to clearcut problems as opposed to problems of neverending complexity and depth.p301To nettle the purists, the applied mathematicians have remarked that the pure mathematicians can find the difficulty in any solution, but the applied men can find the solution to any difficulty.p304Blinded by a century of ever purer mathematics, most mathematicians have lost the skill and the will to read the book of nature. They have turned to fields such as abstract algebra and topology,..., to existence proofs for differential equations that are remote from applications,.... [So it seems that Kline thought that mathematicians should not study equations that are remote from applications. The following is quoted from a PDEer.]总之，如果说“我们为什么要理解流形”的答案是“流形是世界的基本存在和数学的基本研究对象之一”的话，那么“我们为什么要理解PDE”的答案就是“PDE是人类研究世界的基本手段和研究数学的基本方式之一”——而且还是迄今为止被证明最成功的方式之一（尽管我们的成功依然十分渺小）。（3）那我们应该做什么PDE呢？——看上面。——什么？——当然是几何和物理，和我们认识世界中自然会碰到的那些重要的PDE了。嗯……实际上事情没有那么简单。我们今天研究的很多方程，要么没有几何/物理意义或者并不明显（比如分数阶的NLS），要么有几何/物理意义但其背景并不算重要或基本（比如KdV/BenjaminOno，据说是无限长管道的水波的方程，和广相里的方程比起来，在重要性上就呵呵了吧），然后呢，它们有一个共同的名称，叫做Model。这些Model比起真正重要的方程都是要简单很多的。那么为什么还...没错，就是因为它们简单啊。确切的说是因为重要的方程太难了啊。至于有多难，只要想想NavierStokes和Euler，想想被认为50年内看不到解决希望的Cosmic cencorship（正因为如此果断被我写进某篇同人了），想想真空Einstein的局部解从H^(2+) 推进到H^2花了15年和800页以上，你就应该很清楚了。（当然，Poincare conjecture被用Ricci flow解决这件事说明我们还是应该抱有希望的。）按：“一个问题如果太难，可能是我们问的方式不对，”这应该是科研的常识吧。Kline在这章的观点实在武断。p283In fact, most of these papers are devoted to a reformulation in more general or more abstract terms or in new terminology of what had previously existed in more concrete and specific knowledge. And this reformulation provides no gain in power or insight to one who would apply the mathematics.按：Kline这种应用至上的衡量标准是非常惹人怀疑的，不论最后一句在多大程度上正确。These examples of specialization,...do not do justice to the complexity and depth of such problems.按：Kline举的例子都来自数论。然而Gowers在The Importance of Mathematics里举了几个看似毫不相干的数学问题并说明其联系，其中也包含非常容易理解的数论问题。p284Kline为批评专门化（specialization），引用了Bourbaki的一段话（The Architecture of Mathematics, American Mathematical Monthly, 1950, p221, translated by Arnold Dresden）：Many mathematicians take up quarters in a corner of the domain of mathematics, which they do not intend to leave; not only do they ignore almost completely what does not concern their special field, but they are unable to understand the knowledge and the terminology used by colleagues who are working in a corner remote from their own. Even among those who have the widest training, there are none who do not feel lost in certain regions of the immense world of mathematics; those who, like Poincare or Hilbert, put the seal of their genius on almost every domain, constitute a very great exception even among the men of greatest accomplishment.Bourbaki在这里表达了两层意思，一，许多数学家守着自己的一亩三分地不愿越出一步；二，即便数学家有相当广泛的涉猎，也会迷失在数海之中。Poincare和Hilbert这种大神只是少数。尔后Bourbaki就在该文论证公理化方法对于数学的意义（p223）：...the axiomatic method has its cornerstone in the conviction that, bot only is mathematics not a randomly developing concatenation of syllogisms but neither is it a collection of more or less "astute" tricks, arrived at by lucky combinations, in which purely technical cleverness wins the day....the axiomatic method teaches us to look for the deeplying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging, to bring these ideas forward and to put them in their proper light. Kline在紧接着的段落里贬低公理化的价值，但他并没有反驳Bourbaki上面这段话。这并非Kline故意回避，因为Bourbaki的观点想必在Kline看来完全是在数学内部考虑问题。p292295Kline在这里举了几个数学概念的历史起源来批驳纯数学。按：In The Mathematician, published in Works of the Mind Vol. I no. 1 (University of Chicago Press, Chicago, 1947), von Neumann wrote: There are various important parts of modern mathematics in which the empirical origin is untraceable, or, if traceable, so remote that it is clear that the subject has undergone a complete metamorphosis since it was cut off from its empirical roots. The symbolism of algebra was invented for domestic, mathematical use, but it may be reasonably asserted that it had strong empirical ties. However, modem, "abstract" algebra has more and more developed into directions which have even fewer empirical connections. The same may be said about topology. And in all these fields the mathematician's subjective criterion of success, of the worthwhileness of his effort, is very much selfcontained and aesthetical and free (or nearly free) of empirical connections. (I will say more about this further on.) In set theory this is still clearer. The "power" and the "ordering" of an infinite set may be the generalizations of finite numerical concepts, but in their infinite form (especially "power") they have hardly any relation to this world. If I did not wish to avoid technicalities, I could document this with numerous set theoretical examplesthe problem of the "axiom of choice," the "comparability" of infinite "powers," the "continuum problem," etc. The same remarks apply to much of real function theory and real pointset theory. Two strange examples are given by differential geometry and by group theory: they were certainly conceived as abstract, nonapplied disciplines and almost always cultivated in this spirit. After a decade in one case, and a century in the other, they turned out to be very useful in physics. And they are still mostly pursued in the indicated, abstract, nonapplied spirit.有趣的是，这篇文章的倒数第二段为Kline所引用（p291）以支持Kline自己的观点。The Authourity of Naturep338Science is a rationalized fiction, rationalized by mathematics.[Heinrich Hertz:]"One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was orignally put into them."回应 20141029 15:31 
The generalized continuum hypothesis says that the set of all subsets of a set that has the cardinal number N(n) that is 2^(Nn), is N(n+1). Cantor had proved that 2^(Nn) > N(n). ==== rephraze : there's no set whose cardinality is strictly between that of the integers and real numbers.
20140728 15:41
The generalized continuum hypothesis says that the set of all subsets of a set that has the cardinal number N(n) that is 2^(Nn), is N(n+1). Cantor had proved that 2^(Nn) > N(n). ====rephraze : there's no set whose cardinality is strictly between that of the integers and real numbers.回应 20140728 15:41 
Alonzo Church 1. defined recusive function 2. Given a specifc assertion, one cannot always find an algorithm to determine whether it is provable or disprovable. Yuri Matyasevich proved there's no algorithm to determine whether or not integers satisfy the relevant Diophantine problem.
20140728 11:47
Alonzo Church 1. defined recusive function2. Given a specifc assertion, one cannot always find an algorithm to determine whether it is provable or disprovable.Yuri Matyasevich proved there's no algorithm to determine whether or not integers satisfy the relevant Diophantine problem.回应 20140728 11:47 
For any logistic or mathematics system that could be mapped into arithmatical terms, there exist meaningful statements that belongs to these systems but cannot be proved within the systems. Contradicts the view in 19th century thatmathematics is coextensive with the collection of axiomatized branches. Death blow to comprehensive axiomatization.
20140728 10:51
For any logistic or mathematics system that could be mapped into arithmatical terms, there exist meaningful statements that belongs to these systems but cannot be proved within the systems.Contradicts the view in 19th century thatmathematics is coextensive with the collection of axiomatized branches.Death blow to comprehensive axiomatization.回应 20140728 10:51

There are thus, according to Descartes, only two mental acts that enable us to arrive at knowledge without any fear of error: intuition and deduction.
20140623 20:34

Immanuel Kant: <<Prolegomena to Any Future Metaphysics>> We can say with confidence that certain pure a priori synthetical cognitions,pure mathematics and pure physics, are actual and given; for both contain propositions which are thoroughly recognized as absolutely certain...and yet as independent of experience.
20140626 11:30
Immanuel Kant:<<Prolegomena to Any Future Metaphysics>>We can say with confidence that certain pure a priori synthetical cognitions,pure mathematics and pure physics, are actual and given; for both contain propositions which are thoroughly recognized as absolutely certain...and yet as independent of experience.
回应 20140626 11:30

魏厚生 (士为悦己者读书)
The Isolation of Mathematics p279 The establishment of existence theorems of differential equations, first undertaken by Cauchy, was intended to guarantee that the mathematical formulations of physical problems do have a solution, so that one could confidently seek that solution. Hence, though this work is totally mathematical, it does have ulterior physical significance. [Later on the author ...20141029 15:31
The Isolation of Mathematics p279The establishment of existence theorems of differential equations, first undertaken by Cauchy, was intended to guarantee that the mathematical formulations of physical problems do have a solution, so that one could confidently seek that solution. Hence, though this work is totally mathematical, it does have ulterior physical significance.[Later on the author said that: ]p282Another factor induced many mathematicians to tackle problems of pure mathematics.Problem of science are rarely solved in toto....There is a fascination to clearcut problems as opposed to problems of neverending complexity and depth.p301To nettle the purists, the applied mathematicians have remarked that the pure mathematicians can find the difficulty in any solution, but the applied men can find the solution to any difficulty.p304Blinded by a century of ever purer mathematics, most mathematicians have lost the skill and the will to read the book of nature. They have turned to fields such as abstract algebra and topology,..., to existence proofs for differential equations that are remote from applications,.... [So it seems that Kline thought that mathematicians should not study equations that are remote from applications. The following is quoted from a PDEer.]总之，如果说“我们为什么要理解流形”的答案是“流形是世界的基本存在和数学的基本研究对象之一”的话，那么“我们为什么要理解PDE”的答案就是“PDE是人类研究世界的基本手段和研究数学的基本方式之一”——而且还是迄今为止被证明最成功的方式之一（尽管我们的成功依然十分渺小）。（3）那我们应该做什么PDE呢？——看上面。——什么？——当然是几何和物理，和我们认识世界中自然会碰到的那些重要的PDE了。嗯……实际上事情没有那么简单。我们今天研究的很多方程，要么没有几何/物理意义或者并不明显（比如分数阶的NLS），要么有几何/物理意义但其背景并不算重要或基本（比如KdV/BenjaminOno，据说是无限长管道的水波的方程，和广相里的方程比起来，在重要性上就呵呵了吧），然后呢，它们有一个共同的名称，叫做Model。这些Model比起真正重要的方程都是要简单很多的。那么为什么还...没错，就是因为它们简单啊。确切的说是因为重要的方程太难了啊。至于有多难，只要想想NavierStokes和Euler，想想被认为50年内看不到解决希望的Cosmic cencorship（正因为如此果断被我写进某篇同人了），想想真空Einstein的局部解从H^(2+) 推进到H^2花了15年和800页以上，你就应该很清楚了。（当然，Poincare conjecture被用Ricci flow解决这件事说明我们还是应该抱有希望的。）按：“一个问题如果太难，可能是我们问的方式不对，”这应该是科研的常识吧。Kline在这章的观点实在武断。p283In fact, most of these papers are devoted to a reformulation in more general or more abstract terms or in new terminology of what had previously existed in more concrete and specific knowledge. And this reformulation provides no gain in power or insight to one who would apply the mathematics.按：Kline这种应用至上的衡量标准是非常惹人怀疑的，不论最后一句在多大程度上正确。These examples of specialization,...do not do justice to the complexity and depth of such problems.按：Kline举的例子都来自数论。然而Gowers在The Importance of Mathematics里举了几个看似毫不相干的数学问题并说明其联系，其中也包含非常容易理解的数论问题。p284Kline为批评专门化（specialization），引用了Bourbaki的一段话（The Architecture of Mathematics, American Mathematical Monthly, 1950, p221, translated by Arnold Dresden）：Many mathematicians take up quarters in a corner of the domain of mathematics, which they do not intend to leave; not only do they ignore almost completely what does not concern their special field, but they are unable to understand the knowledge and the terminology used by colleagues who are working in a corner remote from their own. Even among those who have the widest training, there are none who do not feel lost in certain regions of the immense world of mathematics; those who, like Poincare or Hilbert, put the seal of their genius on almost every domain, constitute a very great exception even among the men of greatest accomplishment.Bourbaki在这里表达了两层意思，一，许多数学家守着自己的一亩三分地不愿越出一步；二，即便数学家有相当广泛的涉猎，也会迷失在数海之中。Poincare和Hilbert这种大神只是少数。尔后Bourbaki就在该文论证公理化方法对于数学的意义（p223）：...the axiomatic method has its cornerstone in the conviction that, bot only is mathematics not a randomly developing concatenation of syllogisms but neither is it a collection of more or less "astute" tricks, arrived at by lucky combinations, in which purely technical cleverness wins the day....the axiomatic method teaches us to look for the deeplying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging, to bring these ideas forward and to put them in their proper light. Kline在紧接着的段落里贬低公理化的价值，但他并没有反驳Bourbaki上面这段话。这并非Kline故意回避，因为Bourbaki的观点想必在Kline看来完全是在数学内部考虑问题。p292295Kline在这里举了几个数学概念的历史起源来批驳纯数学。按：In The Mathematician, published in Works of the Mind Vol. I no. 1 (University of Chicago Press, Chicago, 1947), von Neumann wrote: There are various important parts of modern mathematics in which the empirical origin is untraceable, or, if traceable, so remote that it is clear that the subject has undergone a complete metamorphosis since it was cut off from its empirical roots. The symbolism of algebra was invented for domestic, mathematical use, but it may be reasonably asserted that it had strong empirical ties. However, modem, "abstract" algebra has more and more developed into directions which have even fewer empirical connections. The same may be said about topology. And in all these fields the mathematician's subjective criterion of success, of the worthwhileness of his effort, is very much selfcontained and aesthetical and free (or nearly free) of empirical connections. (I will say more about this further on.) In set theory this is still clearer. The "power" and the "ordering" of an infinite set may be the generalizations of finite numerical concepts, but in their infinite form (especially "power") they have hardly any relation to this world. If I did not wish to avoid technicalities, I could document this with numerous set theoretical examplesthe problem of the "axiom of choice," the "comparability" of infinite "powers," the "continuum problem," etc. The same remarks apply to much of real function theory and real pointset theory. Two strange examples are given by differential geometry and by group theory: they were certainly conceived as abstract, nonapplied disciplines and almost always cultivated in this spirit. After a decade in one case, and a century in the other, they turned out to be very useful in physics. And they are still mostly pursued in the indicated, abstract, nonapplied spirit.有趣的是，这篇文章的倒数第二段为Kline所引用（p291）以支持Kline自己的观点。The Authourity of Naturep338Science is a rationalized fiction, rationalized by mathematics.[Heinrich Hertz:]"One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was orignally put into them."回应 20141029 15:31 
The generalized continuum hypothesis says that the set of all subsets of a set that has the cardinal number N(n) that is 2^(Nn), is N(n+1). Cantor had proved that 2^(Nn) > N(n). ==== rephraze : there's no set whose cardinality is strictly between that of the integers and real numbers.
20140728 15:41
The generalized continuum hypothesis says that the set of all subsets of a set that has the cardinal number N(n) that is 2^(Nn), is N(n+1). Cantor had proved that 2^(Nn) > N(n). ====rephraze : there's no set whose cardinality is strictly between that of the integers and real numbers.回应 20140728 15:41 
Alonzo Church 1. defined recusive function 2. Given a specifc assertion, one cannot always find an algorithm to determine whether it is provable or disprovable. Yuri Matyasevich proved there's no algorithm to determine whether or not integers satisfy the relevant Diophantine problem.
20140728 11:47
Alonzo Church 1. defined recusive function2. Given a specifc assertion, one cannot always find an algorithm to determine whether it is provable or disprovable.Yuri Matyasevich proved there's no algorithm to determine whether or not integers satisfy the relevant Diophantine problem.回应 20140728 11:47 
For any logistic or mathematics system that could be mapped into arithmatical terms, there exist meaningful statements that belongs to these systems but cannot be proved within the systems. Contradicts the view in 19th century thatmathematics is coextensive with the collection of axiomatized branches. Death blow to comprehensive axiomatization.
20140728 10:51
For any logistic or mathematics system that could be mapped into arithmatical terms, there exist meaningful statements that belongs to these systems but cannot be proved within the systems.Contradicts the view in 19th century thatmathematics is coextensive with the collection of axiomatized branches.Death blow to comprehensive axiomatization.回应 20140728 10:51
这本书的其他版本 · · · · · · ( 全部3 )
 湖南科学技术出版社版 19976 / 2182人读过
 台灣商務版 2004 / 5人读过
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订阅关于Mathematics的评论:
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0 有用 SlowMover 20140623
能把“家丑”如此坦然地对 the general public 公开，M. Kline 确实拥有比一般数学家更广阔的眼界。在数学史方面作了一个很好的叙述，尤其是略带八卦的笔调常常使人会心一笑。最为出彩的还是一针见血地指出了数学作为一种（与绘画、音乐等并无本质差异的）人类创作活动的存在，以及几千年来一直被理想化的“数学王国”的崩塌。这与我所期待的结论一致。
0 有用 霜晚 20161227
挺有意思的艺术史
0 有用 乐无言 20160124
有些哲学观点引用的不错，某些地方梳理的还挺有意思的。可惜既没有入门的乐趣，对熟悉的人也不大有醍醐灌顶的作用，有点鸡肋啊。只能纯当看数学史了
0 有用 蔡继民 20100220
很不错的书，建议对数学有兴趣的同学读一下，一定会对数学的发展史有很多收获
0 有用 Love Saoirse 20080602
Wonderful casual math history book.
0 有用 霜晚 20161227
挺有意思的艺术史
0 有用 乐无言 20160124
有些哲学观点引用的不错，某些地方梳理的还挺有意思的。可惜既没有入门的乐趣，对熟悉的人也不大有醍醐灌顶的作用，有点鸡肋啊。只能纯当看数学史了
0 有用 SlowMover 20140623
能把“家丑”如此坦然地对 the general public 公开，M. Kline 确实拥有比一般数学家更广阔的眼界。在数学史方面作了一个很好的叙述，尤其是略带八卦的笔调常常使人会心一笑。最为出彩的还是一针见血地指出了数学作为一种（与绘画、音乐等并无本质差异的）人类创作活动的存在，以及几千年来一直被理想化的“数学王国”的崩塌。这与我所期待的结论一致。
0 有用 蔡继民 20100220
很不错的书，建议对数学有兴趣的同学读一下，一定会对数学的发展史有很多收获
0 有用 Love Saoirse 20080602
Wonderful casual math history book.