頂級數學家可以恐怖到什麼程度？

禁酒不禁菸者

每一個有成就的數學家，絕不只是在自己的領域有所成就和鑽研。

直到大學的時候才明白，數學和哲學是分不開的。

數學比較直接，最好是有哲學來輔助。

其實，時間萬物，都離不開哲學，更離不開數學。

數學能解決一切問題，毋庸置疑。

數學的成就，取決於哲學。哲學的觀點，需要數學驗證。

因為，數學是最實事求是的科學。而哲學，又是最重要的唯心理論。

對於兩者先有雞還是先有蛋，考究起來已經沒有意義。但是頂級的數學家，完全是唯物和唯心的結合體。那種可怕程度，情商和智商頂尖的結合體，簡直是無敵的存在。

胖油油的老冰棍

很多年前，我還在讀研究生。

有一次我們討論班，講一篇丘成桐先生的文章。一個師兄經過將近一個小時的熟練演算，用了七八塊黑板，把兩項變成了三項。

小老闆看了之後，頷首。我們一堆人暗自喝彩：師兄好強。

小老闆說：XX算的很好啊，功底很紮實，不過當年丘先生掃了一眼，覺得這個東西就是最後這個結果。。。

集體石化。

賊叉

你跟“最強大腦”冠軍選手之間的差距。再翻上5倍，差不多就是“最強大腦”們跟入門級數學家之間的差距。而入門級數學家和頂尖數學家之間的差距可以用下面這個公式計算：

lim∞（1+1/n）^½iπn

如果你們看懂，就該知道你們不在一個維度上，不要試圖去理解對方。

帖木兒

1、早在愛因斯坦提出關於廣義相對論之前，黎曼幾何就已經創造出了它的理論模型

2、隨手列兩個比較著名的數學家

牛頓，十七世紀最接近上帝的存在

歐拉，數學之神

高斯，數學王子

（具體多牛逼自己查，都是神一樣的存在）

3、數學指引了人類現代科技的發展

ps：

沒有深入的學習數學，就別爭論什麼科學、數學、哲學、神學的了。

別成天瞎掰掰，省的丟人現眼。

某些連高中數學及格水準都達不到，沒受過絲毫高等數學教育的人，何來的資格評論數學？連數學世界的門都沒有邁進去，能懂什麼是數學？

一言以蔽之，數學是本質！

最後補一篇比較出名的英文專著《What is mathematics？》 Richard Courant Herbert Robbins

（大家要的翻譯在圖片裡）漢語版叫《什麼是數學》復旦大學出版社出版

第一章 引論

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?

動中之靜靜中之動

恕我直言：數學，是人類文明大樹上最不堪用的一個分枝……甚至連一片葉子也算不上。當然，你將之理解成金字塔的塔尖，葉子上的葉尖這倒是合理的。但金字塔的塔尖，沒有寬與厚的塔基，它何以聳立？？

若以此論，頂尖數學家的恐怖只體現在個的人瘋魔狀態，腦漿裡只有0~9任意組合的存在。若論頂尖數學家能力的恐怖，則幾近於無。

或許有人用數學概率的博弈論來說明頂尖數學家的能力。但我說，這恰恰說明的是數學無用論，因為這是竊取。真正的能力是創造。而數學，毫無疑問，在現實生活中只為創造提供輔助，並非是創造的主要，必要因素。

此論並非為詆譭我們人類的數學研究者。只是希望我們能夠清楚地認知人類的文明什麼為根？什麼為枝？什麼為葉？的主次關係。並進一步地探究根本，讓人類文明健康茁壯地成長……

致！

長弓善道

普林斯頓的 J. B. Conway。

我們研究生的時候學過他寫的《複分析》。

還活著。

他的辦公室手稿堆成山，柴火垛一樣。

人家問，這麼亂，需要的時候怎麼找啊？

他說，不找。

想起什麼需要的時候都是現推導，比找快。

上世紀初，印度有個年輕人，十幾歲。

擅長做夢。

夢到的都是數學公式。

醒來就嘩嘩嘩地寫。

寫了兩厚本。

隨便一個公式就是頂級猜想的答案。

寫信給英國一數學院士。

倆人書信合作一年。

被邀請到英國。

能證明的公式都是正確的。

哥們後來也成了院士。

回印度省親，染上肺結核。30出頭就死了。

好多公式現在還沒證明。估計都是正確的。

其他的耳熟能詳的一個是伽羅華，一個是阿貝爾。

阿貝爾22歲一篇六頁紙論文取代世界級數學家幾百頁的證明。

頂級貢獻無數。

27歲病逝。

伽羅華自學成才。

16歲才開始接觸數學。

18歲創造了群論。

臨終前夜，把數學的畢生研究拼命手寫下來。

後人整理，發現他開闢了嶄新的一門代數學科。

怎麼死的？

為了女人爭風吃醋，與職業槍手比槍法。

死的時候20週歲半。

拂茵

數學對我們來說可是一個重要的科目，從上幼兒園我們就開始學算數，生活中處處也都離不開數學，毫無疑問，數學家就是對數學有一定深入研究的人，他們研究數論算法，數學建模，理論物理，幾何算法，代數的變換等等。我們所知道的數學家有愛因斯坦，華羅庚，陳景潤，陶哲軒等等。那麼，頂級數學家可以恐怖到什麼程度呢？

比如愛因斯坦，他就是一個傳奇一樣的存在，我們都知道，他是一位偉大的物理學家，但他同時又是一位數學家，與伽利略，牛頓一樣，他們都是在物理學與數學上有很大的造詣。可以說，他的特殊之處就在於將數學與物理的結合，他常常用數學去解答物理問題，數學就像是他科學研究的一個工具，這也是他在物理學作出極大成就的原因之一。

著名數學家歐拉在雙目失明後仍然堅持對數學的研究，後又經大火燒點他的手稿，他仍沒倒下，憑藉自己超強的心算能力和記憶力用口授的方式進行數學研究，在數論，代數，無窮級數，函數等方面做出了重大的貢獻。

數學家高斯在小時候就展現出他天才的一面了，他十分地聰明。上小學時，班裡同學們太吵，老師為了讓他們安靜下來，就出了一道難題，從一加到一百，沒想到小高斯卻當了真，別的小朋友都算不出來時，高斯用短短几分鐘的時間就算出來了答案5050。可見他是多麼的厲害，從小數學就那麼好，怪不得能成為大數學家。也就是從他之後，發現了數學累加的定理，也就是從1+2+3+……+n的方法。

之前報道過一則新聞，澳大利亞的19名天才數學家組成了一個叫做“龐特沙龍”的賭博小組，運用他們的數學知識和高智商在世界各國進行賭博，他們贏的幾率很高，幾乎是“十賭九贏”，結果在短短不到三年時間裡，就搖身一變為了大富翁。

時間史

數學，尤其是高等數學是人類認識客觀世界與實現技術進步的基礎學科，其重要地位不容置疑，但是如果說數學是一切科學之母，是一切科學的本質，本人認為是值得商榷的，至少在人文社科領域，是值得商榷的。

近代以來，西方世界人文社科領域逐漸形成數學化的趨勢，很多人文社科領域的著作和學術文章，到處充斥著看似高深莫測的數學模型，彷彿沒有用到數學模型的理論與概念，就登不了大雅之堂，就不能說明作者的水準。

自然科學研究者必須要具備深厚的數學基礎，因為自然科學的本質就是數學的，數學是一切自然科學的邏輯基礎。

而人文社科領域則不然，人文社科是研究人類社會的發展變遷的歷史及規律的，本質上是研究人與人、人與社會、人與客觀存在的關係的科學，因此，人文社科的本質是哲學的，哲學才是一切人文社科的邏輯基礎。

比如愛因斯坦，他就是一個傳奇一樣的存在，我們都知道，他是一位偉大的物理學家，但他同時又是一位數學家，與伽利略，牛頓一樣，他們都是在物理學與數學上有很大的造詣。可以說，他的特殊之處就在於將數學與物理的結合，他常常用數學去解答物理問題，數學就像是他科學研究的一個工具，這也是他在物理學作出極大成就的原因之一。

數學是一個重要學科，保險公司的精算師，華爾街的數量交易設計師，原子彈，飛機，幾乎所有高大上的東西都離不開數學！要說未來能怎麼恐怖？能把人腦思維決策過程，用數學方式完成那就恐怖了。普天之下，莫非人腦。人腦是設計一切利用一切工具的起始。

估計思維模仿光靠數學也不行，還要化學，生物這些學科一起。現在孩子讀書很辛苦，一個不懂事的孩子要成為一個高材生，需要讀十多年書，這生命大好時光就浪費在重複人類已有的知識上了，類似炒剩飯，不是什麼新鮮知識點

大多數人對數學的理解只是用來計算幾個數字和各種公式之類的。但真正的數學其實是一門可以把邏輯、哲學、推理、驗證、空間、方位、趨勢、預測、概率、統計等所有已知的知識，用數學的語言和方式進行嚴謹的驗證、表達和應用的藝術，不分中西。

電力工程技術

數學家是物理學家最好的朋友和幫手，頂級數學家的記憶力和計算能力都是遠勝普通人的，因此在普通人看來數學家們就是“神一樣的存在”

我們從幼兒園開始就在學習數學了，隨著年齡的增大所學習的數學知識也越來越多，然而需要注意的是：在初高中以及大學難倒我們的那些數學知識只是幾百年前數學知識，而數學的最前沿是一般人一輩子都接觸不到也用不到的。

雖然數學中的大部分對平常人來說用處十分有限，而且很多人也都不喜歡數學，但數學確實是人類所有科學的基礎，尤其是對於物理學而言，一個沒有數學基礎支撐的物理學理論是十分不穩固的，因此一些物理學大神們往往也是數學大神，比如牛頓就獨立創立了微積分。

麥克斯韋和法拉第最大的區別就是一個數學好一個數學渣，所以名震天下的是“麥克斯韋方程組”而不是“法拉第方程組”

我國的華羅庚當年被人傳說“看一眼就知道蚊帳有多少小孔”“插一筷子就能知道碗裡有幾粒米飯”這種說法雖然有誇張的成分在裡面，但華羅庚本人用估算法確實也能八九不離十。

宇宙探索未解之迷

不要動不動說恐怖，數學是一個重要學科，保險公司的精算師，華爾街的數量交易設計師，原子彈，飛機，幾乎所有高大上的東西都離不開數學！要說未來能怎麼恐怖？能把人腦思維決策過程，用數學方式完成那就恐怖了。普天之下，莫非人腦。人腦是設計一切利用一切工具的起始。估計思維模仿光靠數學也不行，還要化學，生物這些學科一起。現在孩子讀書很辛苦，一個不懂事的孩子要成為一個高材生，需要讀十多年書，這生命大好時光就浪費在重複人類已有的知識上了，類似炒剩飯，不是什麼新鮮知識點，學習是類似一個只有硬件的計算機要安裝操作系統，灌輸軟件的過程，如果數學能設計出大腦思維，把這個人類已知的知識做成穿戴芯片，把人類已有的知識灌輸到孩子大腦裡。輔助一個孩子大腦變高材生，省去十多年學習過程，我認為就算恐怖了。

關鍵字: 數學 哲學 科學