《数学沉思录》让我深刻地感受到数学的魅力和奥秘。作者以简洁清晰的语言，探讨了数学的本质和意义，让我对数学有了全新的认识。通过阅读，我不仅对数学有了更深刻的理解，也激发了我对数学的兴趣和探索欲望。这本书让我受益匪浅。
《数学沉思录》读后感(篇一)
相当一般。无论是深度、广度还是作者想给读者的哲学提示，都远不如克莱因的《数学：确定性的丧失》一书。如果真想对数学的本质有所了解，这本书虽说不上是误人子弟，但是绝对太过流于表面了。
豆瓣怎么也这么不地道了，已经写了很多了，还说简短，不给发表！！！
对一本好书，有的说；对一本不好的书，让我们说些什么呢？？？
1/11页《数学沉思录》读后感(篇二)
这本书有人认为是关于数学史的著作，我通体读完之后觉得应该将它归于数学哲学著作比较稳妥，就数学哲学而言，它的难度比张景中的《数学与哲学》的稍难一点，比夏皮罗的《数学哲学》要简单，但是说他是数学哲学的著作吧，又不地道，其实就是择要写了数学史以及数学哲学史上举足轻重的几位人物的思想，如阿基米德、柏拉图、伽利略、牛顿、笛卡尔……当然，后面几章又不是按照人物编排，而是按照主题编排，比如非欧几何的诞生、20世纪数学哲学的三大流派等，值得略读，若要细读还是推荐另选其他系统性较强的著作，这本书普及意义明显，学术意义不高。
2/11页《数学沉思录》读后感(篇三)
对于学过微积分随机过程各种代数几何的学生而言，一直没读过像这种讲数学的big picture的书。当我看到笛卡尔坐标系把代数和几何完美地结合在一起的时候，心里真的很是激动。就感觉从小到大学的数学那些点点都被连接成了线，很好的结合在了一起。
这本我觉得挺难的，尤其是最后说纽结和非欧几何那里，但是依然不影响整体阅读的愉悦性。曾经买了这本书的英文原版，但是被里面各式专有名词吓cry了，尤其是第一章的亚里士多德柏拉图那里，完全看不懂，现在打算把原版再对照着读一下。
由于这本书，也让我打算看看欧式几何的elements。
3/11页《数学沉思录》读后感(篇四)
这本书很好的介绍了数学史上的一些内容，包括数学的一些历史以及一些数学背后的哲学思想。
千万不要不要被这本书的名字所吓倒，这本书里其实没有什么晦涩难懂的数学证明，连公式都没有。但是还是有很值得一读的内容的，书中的思想在我这样看过莫里斯克莱因的相关作品的人看来，也是有相当水准的。
这本书读起来很轻松，它不是一本数学史，也不是一本关于数学哲学的书，而是像在讲故事一样讲述作者对数学的思考。以前还没有看到过类似风格的作品。
这本书是写给业余数学爱好者、想提高自身素养的非理工科学生的，但对于度过更专业的书籍的朋友也是会有所收获的。
4/11页《数学沉思录》读后感(篇五)
这本书中，作者想澄清这两个问题：数学的本质，以及数学和物质世界、人类思维之间关系的本质。它不是一本包罗万象的数学史，而是按照数学中的一些关键概念的演化过程来组织结构，这些概念可以有效帮助我们认识数学的本质，同时作者引经据典，传奇的历史人物和神秘的古老传说让本书妙趣横生。
数学无处不在，无所不能，它“无理由的有效性”让人类能成功的解释这个世界的各个领域。上帝是个数学家吗，数学是发明的还是发现的？作者从哲学、历史、科学的各个角度娓娓道来，本着探讨的态度，最终能够让每个人得出自己的结论。
其实，数学的形式是人类发明的，而数学的本质是人类发现的。
5/11页《数学沉思录》读后感(篇六)
柏拉图原本不是个数学家而是个科技哲学家，柏拉图其实没什么可圈可点的数学成果，但他推崇数学并能对数学做出有见地的评论，柏拉图学院的石碑上书“不懂几何者勿入”。柏拉图的哲学观念在数学家中影响深远，彪炳史册的G.H.哈代、保罗·埃尔德什、罗杰·彭罗斯等数学家或物理学家都是柏拉图主义者（尽管当代数学大师阿蒂亚却不赞同柏拉图主义）。甚至连伟大的美国开国总统乔治·华盛顿都赞同柏拉图的某些观点，比如华盛顿总统认同数学是培养理性思维的最有效的方式。
与E.T.贝尔和M·克莱因一样，Mario Livio也盛誉阿基米德无与伦比的天才，对阿基米德的推崇似乎是西方科学史的共识，即阿基米德的数学成果具备现代数学难能而可贵兼严格性的标准。而伽利略、牛顿、麦克斯韦、爱因斯坦、狄拉克等都是秉承阿基米德“用数学语言的理路解释自然”的定量科学精神。
原作应该是本很不错的数学史/科技哲学著作，但翻译似乎有些问题，比如第234页居然把量子场论（QFT）翻译为“量子领域理论”，在Google翻译上机器翻译一下都不至于犯此低级错误。
6/11页《数学沉思录》读后感(篇七)
春节间各种饭局与赶路的间隙读完的书。
一个很有意思的问题：数学是客观存在的还是人类发明。但论述远不如标题吸引人。花了近半篇幅在古希腊与中世纪，而近现代数学部分篇幅太少。最后一章总结更是有种“哥裤子都脱了你就给我说这个？”的感觉。
没读过《古今数学思想》（大一起就放在想读列表里了），因此也无法对这类数学哲学类的书做比较。但起码这本书的感觉是：
那些已经懂的东西，非要包装上哲学的外壳显得太矫情——况且许多那个时代的哲学思想在今天看来已经太幼稚了。
而那些不懂的东西，由于没有深入介绍，也能难让读者体会到其内在的美观与这个世界的神秘关联。——能明白的就是黎曼几何与广义相对论这样从小听到大的烂梗了
一句话说就是“懂的人不用看也能知道讲什么，不懂的人看了后还是不懂”的节奏。
个人比较推荐第七章，对于近代数理逻辑的脉络梳理的很清晰。建议CS从业人员读读。还有那些想读GEB但没有毅力的人（比如我）做个铺垫。
几个8g很好看。比如笛卡尔去世是因为无法睡懒觉，哥德尔申请美国居留权（简直萌翻了！）。
参考文献很详细。
7/11页《数学沉思录》读后感(篇八)
这本书简直解答了我的许多困惑。数学无理由的有效性究竟从何而来，数学既是发明又是发现，比如数学里面一些发明，竟然可以在其他领域里，被应用，比如研究随机现象的布朗运动竟然可以无缝衔接金融里面的随机现象，一个是人类研究物理世界时的发明和创造，一个是人类经济活动，完全受人类的精神意志活动的影响，为何会如此吻合？ 数学是建立在抽象的概念以及其逻辑推理。
说到数学无理由的有效性，这一概念正是康德应对休谟攻击提出的，数学是理性逻辑最高才智的产物，而那些诸如芝诺悖论，休谟提出的问题，都指向一个问题，理性的，逻辑的是否可靠，（以理性和逻辑为起点却产生了与现实不符的悖论）这个问题追溯下去，有新康德主义，有现象学。（我的天呐）
再回到数学上，我对数学开窍比较晚，还是在高中的时候，数感的神经元才被激活。数学是一种语言，只不过这种语言不同于其他语言，另外各种数学概念都是人类应对特定的问题发明出来的，数学里面对于概念的清晰有非常深的执着（对概念的清晰定义是一个非常有意义做法，以后再说）
人类逻辑是物理世界强加给我们的，并且因此而与物理世界保持一致。数学源自于逻辑。这就是数学与物理世界一致的原因，这没有什么神秘的。即使这样，我们也不应失去对自然事物的好奇和怀疑，哪怕是在我们可以更好地理解这些事物之后也应如此。
8/11页《数学沉思录》读后感(篇九)
岁末年初，除了吃喝就是看了几部电影拍了些照片，还有就是读了这本书翻译自《Is God a Mathematician？》的《数学沉思录》。本书原作者力图以浅显的文字回顾数学发展的历史，同时梳理数学史上不同阶段不同人物对“数学是发明还是发现？”这个问题的回答。内容偏哲学和科普，涉及到从古希腊的古典算术、欧几里德几何到近现代的概率统计、微积分、非欧几何以及拓扑学等等数学分支学科，不过所涉内容没有太多深奥的数学知识，以逻辑思辨和历史为主，同时不乏奇闻轶事。对有一定阅读能力、对数学有兴趣的人，阅读本书不会有难度。另外，按照本书提供的注解知识通过互联网进行扩充的话，相信会有收获的。对于本书总体评价是，原作者写的不错（没读过英文原文，靠推测的），翻译显得比较粗糙（后面槽点中会说几点）。
本书中，比较有趣的且讲的比较多的几个片段有，阿基米德的故事，笛卡尔的解析几何，牛顿和莱布尼茨的微积分，非欧几何，纽结理论，以及黄金分割比率。从本书的内容来看，数学史上的每一次暴发性的发展，都跟思想史的大发展息息相关，他们的共同特点都是冲破各种阻力，打破现实桎梏，解放禁锢的思想。例如笛卡尔的解析几何，将几千年来的欧几和算术的不相关给关联起来，而非欧几何甚至冲击当时的整个哲学体系（书中提到的片段是，那个著名的哲学家康德就深信欧几里德几何是一切现实的基础）。在读到这些章节的时候，让我想到，其实不仅仅数学，其他学科（如物理学）和其他领域（如互联网）也同样是这样的。
就本书主题“数学是发现还是发明？”，本书并未给出一个明确的答案，这点可能会让那些等待“参考答案”的中国“学生”们有一点点失望。不过我总结了下我的观点：数学所揭示的真理是人类的发现，但描述这些真理的文字体系（符号体系等）则是人类的发明。如果有朝一日，人类真的跟宇宙中的其他文明联系上了，我相信，如算术、几何、数论等所描述的“真理”，其他文明也有，但很大的可能是，他们用另一种“数学”去描述这些“真理”。就像，同样表示中国人所谓的“吃饭”，美国人用的是“eat”。
最后，虽然比较推荐这本书，但本书还是有不少槽点的。首先是这个书名翻译是我最不满意的，一个好好的科普书非得装深沉，直接翻译为“上帝是数学家么？”不更好么？至于副书名“古今数学思想的发展和演变”，好像也有点过了。其次，本书翻译的略显生硬，作为科普的话，还是需要生动点的语言。还有，发现有些专有名词翻译有误，这个是硬伤，比如把“quantum field theory”（量子场论）翻译为“量子领域理论”，还好书中很多专有名词和名字注有英文原文。
9/11页《数学沉思录》读后感(篇十)
作为一本思想史，它为柏拉图的“理型世界”注入了炫彩斑斓的活力。而数学的神秘，就在于它“不讲道理”的有效性和扩展性，以及作为一种哲学，在上帝和人类之间的摇摆不定。
符号，是数学思想的精华所在。二十世纪哲学的语言学转向，也再一次表明，当人类试图在基于不同认知背景的前提下实现“主体间客观”的交流并达成共识时，就不可避免地会遇到语言工具（这里指日常语言）本身所带来的系统误差。
之前在《孤独六讲》的书评里也提到，我们每一个人都是孤独的，只因我们的思想、意识被束缚于自己的身体之中，不得脱。为了实现不同思想的交流，必然要用到语言工具，而这日常语言的语法体系是不严格的。同样的话语，每个人的理解都会有所差异。
要消除误会，实现近乎完满的相互理解，语言，并不可少。作为日常语言的某种补充，数学语言必然是符号化、严格化的，尽管数学发展之初，关于很多符号的理解，依然停留在直觉的基础上。
可即使是朴素的数学观念，依然在最初帮助我们建立起了一整套“看似完整的”、“严格的”数学符号体系——欧几里得几何。至少，它符合我们的直觉，它似乎“不证自明”。
在柏拉图和他的继任者眼中，理型世界充满了对现实世界的暗喻。尽管我们从未在生活中见到过严格的直线、圆，但依然会说，太阳是圆的，尺子是直的。这就是我们生活的现实世界——一个伴随着观察尺度的不同，伴随着观察工具的不同，伴随着观察角度的不同，而得到不同景象的世界。
甚至于我们的眼睛，也是一种带有“偏见”的工具：明明构成我们身体的分子、原子如此小，而分子与分子、原子与原子之间的空隙有那么大，可我们的眼睛，却能神奇地“忽视”那占据空间99.9%的微粒间空隙，而只看到那0.1%的实体粒子集合体！
与此相反的是，高频电磁波总能轻而易举地穿透我们的身体，仿佛那个身体所占据的空间是“空”的，这些粒子可看不到这个世界上有我们人类的存在~
这样的“现实悖论”数不胜数，它充分说明了，我们对现实世界的观察，以及基于观察的理解，往往很难做到真正的客观、准确。相反，我们所使用的大部分概念，都是现实世界的某种程度的“近似”。是的，在不同观察尺度下，需要使用不同的“近似”工具。
在认识世界的道路上，数学，作为人类思想的抽象产物，具有趋于严格化、符号化的特征。理型世界投下了阴影，我们循着影子去寻找洞口的光明。那是上帝的领域吗？至少，在面对最初的那几组公理时，人类是这样认为的，因为它们在直觉里似乎不可被证伪，而只有“上帝创造了数学”才能解释这种神秘性。
不幸的是，非欧几何打破了这一长久的学科信仰。如果数学只是一种可以自行修改的游戏规则，如果理型世界也只是“人类眼中”的、带有“立场偏见”的index，那我们这一路走来，还需要寻求些什么？
这是有幸被智慧闪电劈中的我们必须面对的焦虑。智慧，总是驱使着我们去寻求经验范围之外的“客观世界”，去探求这个世界的根本，去反思“我是谁”“我为何在此”的意义。
智慧总是在寻求答案，永不停止。至于终点在哪里？我只想说，科学的尽头就是信仰。
思考数学的过程，实则是为了定位我们存在于世的那个坐标。
10/11页《数学沉思录》读后感(篇十一)
Mathematics is the language with which God wrote the Universe.
—-Galileo Galilei
There is something very mathematical about our universe. The more carefully we look at our universe, the more mathematics we seem to find. From financial modeling, weather forecasting, archetactures to electronic design, mathematical equations appear to be everywhere. However, thoughout history, no one was able to offer a clear explanation on why mathematics seems to be omnipotent in explaning everything. The nature of mathematics remains as a mystery. Even so, a great number of philosophers, mathematicians and physicists preserve in their journeys in exploring all facets of mathematics. Mario Livio is one of these people intrigued by the mysterious beauty of mathematics. In this short, accessible book titled Is God a Mathematician, Mario offers a profound exploration of the nature of mathematics. He centers his book around two questions: 1. Why is mathematics so effective in explaining the world around us that it even yields new knowledge？ and 2. Is mathematics ultimately discovered or invented？
Unreasonable Effectiveness
Livio traces the development of mathematics from ancient Greeks, Archimedes, Pythagoras, Aristotle, Plato through the middle ages and the enlightenment to the present, and provides mini-bibliographies on Galileo, DeCartes, Newton, Gauss, Riemann, Boole and Russell etc.. He goes through various anecdotes about how these great figures came up with their great ideas that have reshaped the world. What truly fascinates Livio, however, is the role that pure math played in all these discoveries. He was bewildered by the amazing fact that concepts once explored by mathematicians with absolutely no application in mind have turned out decades later to be unexpected solutions to real-world problems. A long time ago when Pythagoras studied the construction of right triangles, he did not know that several years later, his theorem would play such an important role in the day to day life of engineers, architects and construction workers; When Euclid came up with his famous Euclidean Algorithm, he did not related that to the basis of musical intervals; When Newton deduced the law of gravity, he could hardly have known that the predictions made by these mathematical laws would perfectly overlap the empirical data he was originally trying to match; When Gauss wondered the sum of the nth odd numbers, he might not associate that with the motion of free fall bodies, which later becomes the core of classical mechanics. Livio also mentions the “group theory”, developed by Galois in 1832 to determine the solvability of algebraic equations, has become the language used by physicists, linguists, and even anthropologists to describe all the symmetries of the world (such as the structure of solids, the organization of elementary particles, etc.). Another often cited example is the Maxwell Equations, the most beautiful set of formulas ever that models the fundamentals of electromagnetism demonstrating the behavior of electrons to the nature of light. It is amazing that what guided Maxwell to derive such a powerful set of formulas was mathematical analogy, not empirical evidence. Just as all these stories, there are numerous instances in which mathematical principles, previously considered as purely logical curiosities, turns out to be uncannily productive in yielding new discoveries that are even unexpected on physical grounds.
Livio’s review suggests that mathematics cannot be separated from the physical world; they are essentially interrelated in some sense. It is impossible for us, for example, to learn physics without referring to mathematics. What is exactly the relationship between mathematics and physics then？ Some people describe the situation by saying that mathematician plays a game in which he himself invents the rules, whereas physicist plays a game in which the rules are provided by Nature. However, as what Mario Livio tries to indicate in his book, mathematics and physics are highly interrelated, and it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. Mathematics also helps to correct physicists’ mistakes. There are many stories in which our ancestors made mistakes: Earth was once the center of the universe; heavy bodies were once supposed to fall faster than light bodies; chemical elements were once the basic units of all matters…It was the sense of mathematic principles that helped them correct their mistakes, and guided them towards the eternal truth.
Mathematics is not only applicable to areas of physics, but also to chemistry, biology and other disciplines such as geography, economics, psychology, not to mention our daily lives. The increasing use of computing machines testifies the idea that everything can be broken down to series of 0s and 1s. Mathematics appears as the fundamental units of nearly everything. This simple, undeniable fact gives rise to several philosophical questions that remain unresolved for years: What is mathematics？ Why mathematical concepts have applicability far beyond the context in which they were originally developed？ Why scholars in all disciplines find that they are unable to even state their theories without referring to abstract mathematical theories？
Challenged with these questions, Livio replies, “I don’t know”. It is not surprising that he is as baffled as other outstanding physicists who have attempted to answer these questions, most notably Eugene Wigner, Steven Weinberg and Mark Steiner. Wigner, in his The Unreasonable Effectiveness of Mathematics in the Natural Sciences, concluded his paper with the same question with which he began: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." The unreasonable effectiveness of mathematics seems to be inexplicable at the moment; however, I would like to offer one possible way of analyzing it. In my view, the unreasonable effectiveness of mathematics is in fact the unreasonable effectiveness of language. Mathematics constitutes a particular kind of description of the world, just like any language does. That mathematics does a better job than other natural languages in explaining physical reality is perhaps best explained by the predictive success of sciences based on mathematics. In other words, it is the predictive success of sciences based on mathematics that gives the most important validation of mathematics. As for why mathematics appears to be a perfect language is still a mystery. To figure it out we should perhaps first consider the nature of mathematics itself.
Discovered or Invented？
It is already stunning that mathematics provides the solid structure that holds together any theory of the universe. It is even more interesting as we realize that the nature of mathematics itself is not entirely clear. Where does mathematics come from, after all？ There is much to suggest that mathematics comes from our observations of the physical world and that there is an innate way our brains making sense of those observations through mathematics. However, we also have the impression that mathematicians tend to work in very abstract areas with no intention of seeing applicability in the physical world. They regard mathematics as purely intellectual enjoyment until they see its direct application to our physical world in an unexpected way.
The unreasonable effectiveness of mathematics thus creates another set of puzzles regarding the nature of mathematics itself: Does mathematics have an existence that is entirely independent of human mind？ In other words, are we merely discovering mathematical verities, just as chemists discover previously unknown elements？ Or, is mathematics a human invention, a pure product of human reasoning？ Was the equation “2+2=4” an invention by ancient mathematicians to best organize quantitative relationships, or is it an objective truth that can never be violated experientially？ Was Fibonacci Sequence created on purpose to explain the growing pattern of sunflowers, or that they just happen to overlap？
In discussing the nature of mathematics, many scholars have taken the Platonic view, the idea that mathematics has its own existence regardless of whether human knows it or not, and we just have to discover it. Once a particular mathematical concept has been grasped, say, 2+2=4, then we are up against undeniable facts. For example, if two apples are placed together another two apples, there would be four there, even though no humans were there to observe it, or that we do not think there are four. Similarly, the growing pattern of sunflowers is not a subjective interpretation made by the human mind, but an inalterable, pre-determined fact once the sunflowers were born. In the Platonic view, mathematics indeed exists in some abstract fairyland that is part of the physical reality. This view, however, remains arguable. If mathematics is part of the physical reality, then why does it appear to be immutable while the physical world is constantly changing？ How does the human brain, with its limitations, get access to such an immutable, abstract and mysterious fairyland？
Opponents to the Platonic view thus argue that mathematics does not arise from the physical world but from the human brain. Human invented mathematics by idealizing and abstracting elements of the physical world. Such a view is able to explain why mathematics appears heavenly, but it fails to explain why math happens to fit the physical world so well. If mathematics has absolutely no existence outside our mind, how can we explain the fact that invention of so many mathematical truths remarkably anticipated real-world problems not even occurred until many years later？
Livio’s answer to this puzzle is both—Mathematics is both discovered and invented! Some people argue that Livio simply failed to answer the question as he offered no exact conclusion about the nature of mathematics. Though his conclusion is indeed a bit of a cop out, I do think, however, it is the best attempt and the most convincing answer that Livio could offer. What Livio suggests is that humans invented certain basic concepts, and then they discover the implications of those axioms. He demonstrated this idea using the evolution of golden ratio, an example which I find quite convincing. Euclid, in his monumental work The Element, defined the “golden ratio” as the length ratio of two segments that takes its algebraic form as (1+√5)/2. Livio considers Euclid’s definition of the concept “golden ratio” as an invention because Euclid’s inventive act singled out this ratio and attracted the attention of later generations to it. It came as no surprise when people later discovered that dodecahedron has the golden ratio written all over it, that the five-pointed star has its segment divided by the golden ratio, that leaf arrangements and structure of crystals embody patterns that resemble the golden ratio. These discoveries are only true because of the initial acceptance of the foundational axioms that Euclid invented. That is why, for Livio, mathematics is a combination of invention and discoveries.
Mystery As Ultimate Beauty
Livio does a remarkably good job in showing that historical flights of mathematical imagination, no matter how trivial and random they once were, are often found to be incredibly useful in describing the mysteries we are curious about several centuries later. As for why mathematics acts as such a prophet, Livio perhaps was not “successful” enough to provide us with a definite, convincing answer. But to quote from Einstein, “The most beautiful experience we can have is the mysterious-—the fundamental emotion which stands at the cradle of true art and true science”. We should be grateful for the fact that there are so much we do not know, as mysteries give rise to our willingness to learn as well as the pleasure we get from it.
I want to end with a quote from Bertrand Russell—-the influential philosopher, mathematician, logician, social critic, the person whom I most admire. To me, this quote speaks to the reason for studying almost everything:
"Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.”
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