顶级数学家可以恐怖到什么程度？

禁酒不禁烟者

每一个有成就的数学家，绝不只是在自己的领域有所成就和钻研。

直到大学的时候才明白，数学和哲学是分不开的。

数学比较直接，最好是有哲学来辅助。

其实，时间万物，都离不开哲学，更离不开数学。

数学能解决一切问题，毋庸置疑。

数学的成就，取决于哲学。哲学的观点，需要数学验证。

因为，数学是最实事求是的科学。而哲学，又是最重要的唯心理论。

对于两者先有鸡还是先有蛋，考究起来已经没有意义。但是顶级的数学家，完全是唯物和唯心的结合体。那种可怕程度，情商和智商顶尖的结合体，简直是无敌的存在。

胖油油的老冰棍

很多年前，我还在读研究生。

有一次我们讨论班，讲一篇丘成桐先生的文章。一个师兄经过将近一个小时的熟练演算，用了七八块黑板，把两项变成了三项。

小老板看了之后，颔首。我们一堆人暗自喝彩：师兄好强。

小老板说：XX算的很好啊，功底很扎实，不过当年丘先生扫了一眼，觉得这个东西就是最后这个结果。。。

集体石化。

贼叉

你跟“最强大脑”冠军选手之间的差距。再翻上5倍，差不多就是“最强大脑”们跟入门级数学家之间的差距。而入门级数学家和顶尖数学家之间的差距可以用下面这个公式计算：

lim∞（1+1/n）^½iπn

如果你们看懂，就该知道你们不在一个维度上，不要试图去理解对方。

帖木兒

1、早在爱因斯坦提出关于广义相对论之前，黎曼几何就已经创造出了它的理论模型

2、随手列两个比较著名的数学家

牛顿，十七世纪最接近上帝的存在

欧拉，数学之神

高斯，数学王子

（具体多牛逼自己查，都是神一样的存在）

3、数学指引了人类现代科技的发展

ps：

没有深入的学习数学，就别争论什么科学、数学、哲学、神学的了。

别成天瞎掰掰，省的丢人现眼。

某些连高中数学及格水准都达不到，没受过丝毫高等数学教育的人，何来的资格评论数学？连数学世界的门都没有迈进去，能懂什么是数学？

一言以蔽之，数学是本质！

最后补一篇比较出名的英文专著《What is mathematics？》 Richard Courant Herbert Robbins

（大家要的翻译在图片里）汉语版叫《什么是数学》复旦大学出版社出版

第一章 引论

WHAT IS MATHEMATICS?

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the stmggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

Without doubt., all mathematical development has its psychological roots in more or less practical requirements. But once started under the pressure of necessary applications, it inevitably gains momentum in it­ self and transcends the confines of immediate utility. This trend from applied to theoretical science appears in ancient history as well as in many contributions to modem mathematics by engineers and physicists.

Recorded mathematics begins in the Orient, where，about 2000 B，C, the Babylonians collected a great wealth of material that we would clas­ sify today under elementary algebra. Yet as a science in the modem sense mathematics only emerges later, on Greek soil, in the fifth and fourth centuries B.C. The ever-increasing contact between the Orient and the Greeks, beginning at the time of the Persian empire and reaching a climax in the period following Alexander’s expeditions, made the Greeks familiar with the achievements of Babylonian mathematics and astronomy. Mathematics was soon subjected to the philosophical dis­ cussion that, flourished in the Greek city states. Thus Greek thinkers became conscious of the great difficulties inherent in the mathematical concepts of continuity, motion, and infinity，and in the problem of mea­ suring arbitrary quantities by given units. In an admirable effort, the challenge was met，and the result, Eudoxus1 theory of the geometrical

continuum, is an achievement that was only paralleled more than two thousand years later by the modem theory of irrational numbers. The deductive-postulational trend in mathematics originated at the time of Eudoxus and was crystallized in Euclid’s Elements.

However, while the theoretical and postulat.ional tendency of Greek mathematics remains one of its important characteristics and has ex­ ercised an enonnous influence, it cannot be emphasized too strongly

that application and connection with physical reality played just as im­ portant a part in the mathematics of antiquity, and that a manner of presentation less rigid than Euclid's was very often preferred.

It may be that the early discovery of the difficulties connected with incommensurable quantities deterred the Greeks from developing the art of numerical reckoning achieved before in the Orient. Instead they forced their way through the thicket of pure axiomatic geometry. Thus one of the strange detours of the history of science began，and perhaps a great opportunity was missed. For almost two thousand years the weight of Greek geometrical tradition retarded the inevitable evolution of the number concept and of algebraic manipulation, which later formed the basis of modem science.

After a period of slow preparation, the revolution in mathematics and science began its vigorous phase in the seventeenth century with ana­ lytic geometry and the differential and integral calculus. While Greek geometry retained an important place, the Greek ideal of axiomatic crys­ tallization and systematic deduction disappeared in the seventeenth and eighteenth centuries. Logically precise reasoning, starting from clear definitions and non-contradictory? “evident” axioms, seemed immaterial

to the new pioneers of mathematical science. In a veritable orgy of in­ tuitive guesswork, of cogent reasoning interwoven with nonsensical mysticism, with a blind confidence in the superhuman power of formal procedure, they conquered a mathematical world of immense riches. Gradually the ecstasy of progress gave way to a spirit of critical self- controL In the nineteenth century the immanent need for consolidation and the desire for more security in the extension of higher learning that was prompted by the French revolution, inevitably led back to a revision of the foundations of the new mathematics, in particular of the differ ential and integral calculus and the underlying concept of limit. Thus the nineteenth century not only became a period of new advances, but was also characterized by a successful return to the classical ideal of precision and rigorous proof. In this respect it even surpassed the model of Greek science. Once more the pendulum swung toward the side of logical purity and abstraction. At present we still seem to be in this period, although it is to be hoped that the resulting unfortunate sepa， ration between pure mathematics and the vital applications, perhaps inevitable in times of critical revision, will be followed by an era of closer unity‘ The regained internal strength and，above all，the enormous simplification attained on the basis of clearer comprehension make it possible today to master the mathematical theory without losing sight of applications. To establish once again an organic union between pure and applied science and a sound balance between abstract generality and colorful individuality may well be the paramount task of mathe­ matics in the immediate future.

This is not the place for a detailed philosophical or psychological analysis of mathematics. Only a few points should be stressed* There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character of mathematics. True, the element of constructive invention, of directing and motivating intuition, is apt to elude a simple philosophical fonnulation; but it remains the core of any mathematical achievement, even in the most abstract fields. If the crys­ tallized deductive form is the goal, intuition and construction are at least the driving forces. A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but oth­ erwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, with­ out motive or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive halftruth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value.

While the contemplative trend of logical analysis does not represent all of mathematics, it has led to a more profound understanding of math­ ematical facts and their interdependence, and to a clearer comprehen­ sion of the essence of mathematical concepts. From it has evolved a modem point of view in mathematics that is typical of a universal sci­ entific attitude.

Whatever our philosophical standpoint may be, for all purposes of scientific observation an object exhausts itself in the totality of possible relations to the perceiving subject or instrument. Of course, mere per­ ception does not constitute knowledge and insight; it must be coordi­ nated and interpreted by reference to some underlying entity, a “thing in itself，” which is not ari object of direct physical observation，but be­ longs to metaphysics. Yet for scientific procedure it is important to dis­ card elements of metaphysical character and to consider observable facts always as the ultimate source of notions and constructions. To renounce the goal of comprehending the uthing in itself,” of knowing the “ultimate truth，’’ of unraveling the innermost essence of the world, may be a psychological hardship for naive enthusiasts, but in fact it was one of the most fruitful turns in modem thinking.

Some of the greatest achievements in physics have come as a reward for courageous adherence to the principle of eliminating metaphysics. When Einstein tried to reduce the notion of “simultaneous events oc­ curring at different places” to observable phenomena, when he un­ masked as a metaphysical prejudice the belief that this concept must have a scientific meaning in itself, he had found the key to his theory of relativity. When Niels Bohr and his pupils analyzed the fact that any physical observation must be accompanied by an effect of the observing instrument on the observed object, it became clear that the sharp si­ multaneous fixation of position and velocity of a particle is not possible in the sense of physics. The far-reaching consequences of this discovery, embodied in the modem theory of quantum mechanics, are now familiar to every physicist. In the nineteenth century the idea prevailed that me- chanical forces and motions of particles in space are things in them­ selves, while electricity, light, and magnetism should be reduced to or “explained” as mechanical phenomena, just as had been done with heat. The “ether” was invented as a hypothetical medium capable of not en­ tirely explained mechanical motions that appear to us as light or elec­ tricity. Slowly it was realized that the ether is of necessity unobservable; that it belongs to metaphysics and not to physics. With sorrow in some quarters, with relief in others, the mechanical explanations of light and electricity, and with them the ether，were finally abandoned.

A similar situation, even more accentuated, exists in mathematics. Throughout the ages mathematicians have considered their objects, such as numbers, points, etc., as substantial things in themselves. Since these entities had always defied attempts at an adequate description, it slowly dawned on the mathematicians of the nineteenth century that the question of the meaning of these objects as substantial things does not make sense within mathematics, if at all. The only relevant asser­ tions concerning them do not refer to substantial reality; they state only the mterrelations between mathematically "undefined objects” and the rules governing operations with thenr What points, lines, numbers fac­ tually^ are cannot and need not be discussed in mathematical science. What matters and what corresponds to “verifiable” fact is structure and relationship, that two points determine a line, that numbers combine according to certain rules to form other numbers，etc A clear insight

into the necessity of a dissubstantiation of elementary mathematical concepts has been one of the most important and fruitful results of the modem postulational development.

Fortunately，creative minds forget dogmatic philosophical beliefs whenever adherence to them would impede constructive achievement. For scholars and layman alike it is not philosophy but active experience in mathematics itself that alone can answer the question： What is mathematics?

动中之静静中之动

恕我直言：数学，是人类文明大树上最不堪用的一个分枝……甚至连一片叶子也算不上。当然，你将之理解成金字塔的塔尖，叶子上的叶尖这倒是合理的。但金字塔的塔尖，没有宽与厚的塔基，它何以耸立？？

若以此论，顶尖数学家的恐怖只体现在个的人疯魔状态，脑浆里只有0~9任意组合的存在。若论顶尖数学家能力的恐怖，则几近于无。

或许有人用数学概率的博弈论来说明顶尖数学家的能力。但我说，这恰恰说明的是数学无用论，因为这是窃取。真正的能力是创造。而数学，毫无疑问，在现实生活中只为创造提供辅助，并非是创造的主要，必要因素。

此论并非为诋毁我们人类的数学研究者。只是希望我们能够清楚地认知人类的文明什么为根？什么为枝？什么为叶？的主次关系。并进一步地探究根本，让人类文明健康茁壮地成长……

致！

长弓善道

普林斯顿的 J. B. Conway。

我们研究生的时候学过他写的《复分析》。

还活着。

他的办公室手稿堆成山，柴火垛一样。

人家问，这么乱，需要的时候怎么找啊？

他说，不找。

想起什么需要的时候都是现推导，比找快。

上世纪初，印度有个年轻人，十几岁。

擅长做梦。

梦到的都是数学公式。

醒来就哗哗哗地写。

写了两厚本。

随便一个公式就是顶级猜想的答案。

写信给英国一数学院士。

俩人书信合作一年。

被邀请到英国。

能证明的公式都是正确的。

哥们后来也成了院士。

回印度省亲，染上肺结核。30出头就死了。

好多公式现在还没证明。估计都是正确的。

其他的耳熟能详的一个是伽罗华，一个是阿贝尔。

阿贝尔22岁一篇六页纸论文取代世界级数学家几百页的证明。

顶级贡献无数。

27岁病逝。

伽罗华自学成才。

16岁才开始接触数学。

18岁创造了群论。

临终前夜，把数学的毕生研究拼命手写下来。

后人整理，发现他开辟了崭新的一门代数学科。

怎么死的？

为了女人争风吃醋，与职业枪手比枪法。

死的时候20周岁半。

拂茵

数学对我们来说可是一个重要的科目，从上幼儿园我们就开始学算数，生活中处处也都离不开数学，毫无疑问，数学家就是对数学有一定深入研究的人，他们研究数论算法，数学建模，理论物理，几何算法，代数的变换等等。我们所知道的数学家有爱因斯坦，华罗庚，陈景润，陶哲轩等等。那么，顶级数学家可以恐怖到什么程度呢？

比如爱因斯坦，他就是一个传奇一样的存在，我们都知道，他是一位伟大的物理学家，但他同时又是一位数学家，与伽利略，牛顿一样，他们都是在物理学与数学上有很大的造诣。可以说，他的特殊之处就在于将数学与物理的结合，他常常用数学去解答物理问题，数学就像是他科学研究的一个工具，这也是他在物理学作出极大成就的原因之一。

著名数学家欧拉在双目失明后仍然坚持对数学的研究，后又经大火烧点他的手稿，他仍没倒下，凭借自己超强的心算能力和记忆力用口授的方式进行数学研究，在数论，代数，无穷级数，函数等方面做出了重大的贡献。

数学家高斯在小时候就展现出他天才的一面了，他十分地聪明。上小学时，班里同学们太吵，老师为了让他们安静下来，就出了一道难题，从一加到一百，没想到小高斯却当了真，别的小朋友都算不出来时，高斯用短短几分钟的时间就算出来了答案5050。可见他是多么的厉害，从小数学就那么好，怪不得能成为大数学家。也就是从他之后，发现了数学累加的定理，也就是从1+2+3+……+n的方法。

之前报道过一则新闻，澳大利亚的19名天才数学家组成了一个叫做“庞特沙龙”的赌博小组，运用他们的数学知识和高智商在世界各国进行赌博，他们赢的几率很高，几乎是“十赌九赢”，结果在短短不到三年时间里，就摇身一变为了大富翁。

时间史

数学，尤其是高等数学是人类认识客观世界与实现技术进步的基础学科，其重要地位不容置疑，但是如果说数学是一切科学之母，是一切科学的本质，本人认为是值得商榷的，至少在人文社科领域，是值得商榷的。

近代以来，西方世界人文社科领域逐渐形成数学化的趋势，很多人文社科领域的著作和学术文章，到处充斥着看似高深莫测的数学模型，仿佛没有用到数学模型的理论与概念，就登不了大雅之堂，就不能说明作者的水准。

自然科学研究者必须要具备深厚的数学基础，因为自然科学的本质就是数学的，数学是一切自然科学的逻辑基础。

而人文社科领域则不然，人文社科是研究人类社会的发展变迁的历史及规律的，本质上是研究人与人、人与社会、人与客观存在的关系的科学，因此，人文社科的本质是哲学的，哲学才是一切人文社科的逻辑基础。

比如爱因斯坦，他就是一个传奇一样的存在，我们都知道，他是一位伟大的物理学家，但他同时又是一位数学家，与伽利略，牛顿一样，他们都是在物理学与数学上有很大的造诣。可以说，他的特殊之处就在于将数学与物理的结合，他常常用数学去解答物理问题，数学就像是他科学研究的一个工具，这也是他在物理学作出极大成就的原因之一。

数学是一个重要学科，保险公司的精算师，华尔街的数量交易设计师，原子弹，飞机，几乎所有高大上的东西都离不开数学！要说未来能怎么恐怖？能把人脑思维决策过程，用数学方式完成那就恐怖了。普天之下，莫非人脑。人脑是设计一切利用一切工具的起始。

估计思维模仿光靠数学也不行，还要化学，生物这些学科一起。现在孩子读书很辛苦，一个不懂事的孩子要成为一个高材生，需要读十多年书，这生命大好时光就浪费在重复人类已有的知识上了，类似炒剩饭，不是什么新鲜知识点

大多数人对数学的理解只是用来计算几个数字和各种公式之类的。但真正的数学其实是一门可以把逻辑、哲学、推理、验证、空间、方位、趋势、预测、概率、统计等所有已知的知识，用数学的语言和方式进行严谨的验证、表达和应用的艺术，不分中西。

电力工程技术

数学家是物理学家最好的朋友和帮手，顶级数学家的记忆力和计算能力都是远胜普通人的，因此在普通人看来数学家们就是“神一样的存在”

我们从幼儿园开始就在学习数学了，随着年龄的增大所学习的数学知识也越来越多，然而需要注意的是：在初高中以及大学难倒我们的那些数学知识只是几百年前数学知识，而数学的最前沿是一般人一辈子都接触不到也用不到的。

虽然数学中的大部分对平常人来说用处十分有限，而且很多人也都不喜欢数学，但数学确实是人类所有科学的基础，尤其是对于物理学而言，一个没有数学基础支撑的物理学理论是十分不稳固的，因此一些物理学大神们往往也是数学大神，比如牛顿就独立创立了微积分。

麦克斯韦和法拉第最大的区别就是一个数学好一个数学渣，所以名震天下的是“麦克斯韦方程组”而不是“法拉第方程组”

我国的华罗庚当年被人传说“看一眼就知道蚊帐有多少小孔”“插一筷子就能知道碗里有几粒米饭”这种说法虽然有夸张的成分在里面，但华罗庚本人用估算法确实也能八九不离十。

宇宙探索未解之迷

不要动不动说恐怖，数学是一个重要学科，保险公司的精算师，华尔街的数量交易设计师，原子弹，飞机，几乎所有高大上的东西都离不开数学！要说未来能怎么恐怖？能把人脑思维决策过程，用数学方式完成那就恐怖了。普天之下，莫非人脑。人脑是设计一切利用一切工具的起始。估计思维模仿光靠数学也不行，还要化学，生物这些学科一起。现在孩子读书很辛苦，一个不懂事的孩子要成为一个高材生，需要读十多年书，这生命大好时光就浪费在重复人类已有的知识上了，类似炒剩饭，不是什么新鲜知识点，学习是类似一个只有硬件的计算机要安装操作系统，灌输软件的过程，如果数学能设计出大脑思维，把这个人类已知的知识做成穿戴芯片，把人类已有的知识灌输到孩子大脑里。辅助一个孩子大脑变高材生，省去十多年学习过程，我认为就算恐怖了。

關鍵字: 数学 哲学 科学