正文翻譯

What is mathematics?

什么是數(shù)學(xué)？

評(píng)論翻譯

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For me pure mathematics is philosophy & all mathematical expressions - a language/syntax for logical deduction.
There is a hypothesis we start with and try to prove it through logical deduction or already accepted mathematical expressions. You need to be creative and/or knowledgeable in order to be come up with ground breaking hypothesis but you need to be highly analytical in order to validate/prove the hypothesis.

對(duì)我來(lái)說，純數(shù)學(xué)就是哲學(xué)。所有數(shù)學(xué)表達(dá)式是一種語(yǔ)言或語(yǔ)法上的邏輯演繹。
這兒有一個(gè)假設(shè)，然后我們?nèi)ネㄟ^邏輯演繹去證明真?zhèn)?，或去?yàn)證是否符合已知的數(shù)學(xué)表達(dá)式。當(dāng)然，你需要極具創(chuàng)造力或精通多個(gè)領(lǐng)域，才能提出突破性的假設(shè)，但你還需要繼續(xù)分析，才能證明假設(shè)真?zhèn)巍?/p>

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Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.
Since the beginning of recorded history, mathematic discovery has been at the forefront of every civilized society, and in use in even the most primitive of cultures. There needs of math arose based on the wants of society. The more complex a society, the more complex the mathematical needs. Primitive tribes needed little more than the ability to count, but also relied on math to calculate the position of the sun and the physics of hunting.
*****History of mathematics(FOR KNOWLEDGE).

數(shù)學(xué)是研究形狀、數(shù)量和排列的邏輯科學(xué)。數(shù)學(xué)就在我們身邊，在我們做的每一件事中。它是我們?nèi)粘Ｉ钪兴惺挛锏慕M成部分，包括移動(dòng)設(shè)備、建筑(古代和現(xiàn)代)、藝術(shù)、金錢、工程，甚至體育。
自有記錄的歷史開始以來(lái)，數(shù)學(xué)就一直處于每一個(gè)文明社會(huì)的前沿，甚至在最原始的文化中也得到應(yīng)用。數(shù)學(xué)的需求是建立在社會(huì)需求的基礎(chǔ)上的。社會(huì)越復(fù)雜，數(shù)學(xué)需求就越復(fù)雜。原始部落所需要的僅僅是數(shù)數(shù)的能力，但有的部落也能靠數(shù)學(xué)來(lái)計(jì)算出太陽(yáng)的位置和狩獵的一些物理原理。
一些數(shù)學(xué)歷史(知識(shí))。

****Several civilizations — in China, India, Egypt, Central America and Mesopotamia — contributed to mathematics as we know it today. The Sumerians were the first people to develop a counting system. Mathematicians developed arithmetic, which includes basic operations, multiplication, fractions and square roots. The Sumerians’ system passed through the Akkadian Empire to the Babylonians around 300 B.C. Six hundred years later, in America, the Mayans developed elaborate calendar systems and were skilled astronomers. About this time, the concept of zero was developed.
*****As civilizations developed, mathematicians began to work with geometry, which computes areas and volumes to make angular measurements and has many practical applications. Geometry is used in everything from home construction to fashion and interior design.

1，一些早期文明——中國(guó)、印度、埃及、中美洲和美索不達(dá)米亞——對(duì)我們今天所知的數(shù)學(xué)做出了巨大的貢獻(xiàn)。蘇美爾人是最早發(fā)展計(jì)數(shù)系統(tǒng)的人。接著數(shù)學(xué)家依于此發(fā)展出了算術(shù)，包括基本運(yùn)算、乘法、分?jǐn)?shù)和平方根。大約在公元前300年，蘇美爾人的歷法通過阿卡德帝國(guó)傳到了巴比倫人的手中。大約600年后，在美洲，瑪雅人發(fā)展出了復(fù)雜的歷法系統(tǒng)，并成為了熟練的天文學(xué)家。于此同時(shí)，零的概念也被發(fā)明了出來(lái)。
2，隨著文明的發(fā)展，數(shù)學(xué)家開始研究幾何，應(yīng)用幾何來(lái)計(jì)算面積體積和測(cè)量角度，并有許多實(shí)際應(yīng)用。幾何學(xué)被被用于家庭建筑、時(shí)尚、室內(nèi)設(shè)計(jì)等等諸多領(lǐng)域。

****Geometry went hand in hand with algebra, invented in the ninth century by a Persian mathematician, Mohammed ibn-Musa al-Khowarizmi. He also developed quick methods for multiplying and diving numbers, which are known as algorithms — a corruption of his name.
****Algebra offered civilizations a way to divide inheritances and allocate resources. The study of algebra meant mathematicians were solving linear equations and systems, as well as quadratics, and delving into positive and negative solutions. Mathematicians in ancient times also began to look at number theory. With origins in the construction of shape, number theory looks at figurate numbers, the characterization of numbers, and theorems.

3，幾何與代數(shù)是緊密聯(lián)系在一起的，代數(shù)是9世紀(jì)由波斯數(shù)學(xué)家穆罕默德.花拉子米發(fā)明的。他還發(fā)明了乘法和除法的快速方法，即所謂的算法——這個(gè)單詞也是由他的名字發(fā)展來(lái)的。
4，代數(shù)為文明提供了一種劃分繼承和分配資源的方式。對(duì)代數(shù)的研究意味著數(shù)學(xué)家要解線性方程組。需要解二次方程，鉆研正解和負(fù)解。于是古代的數(shù)學(xué)家也開始研究數(shù)論。數(shù)論著眼于數(shù)字的具象化、數(shù)字本身的表征和定理。

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Some revealing quotes , opinions and definitions about mathematics.
"Mathematics is the most beautiful and most powerful creation of the human spirit."--Stefan Banach.
"Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth."
--Galileo Galilei .

關(guān)于數(shù)學(xué)的一些有啟發(fā)性的名言。
“數(shù)學(xué)是人類精神中最美麗、最強(qiáng)大的創(chuàng)造物。”——斯蒂芬·巴拿赫。
“哲學(xué)是寫在我們面前的那本偉大的書里的——我指的是宇宙——但如果我們不先學(xué)習(xí)它的語(yǔ)言和掌握它的符號(hào)，我們就無(wú)法理解它。這本書是用數(shù)學(xué)語(yǔ)言寫的，里面的符號(hào)是三角形、圓形和其他幾何圖形，沒有這些符號(hào)的幫助，一個(gè)字也理解不了;沒有它，人就在黑暗的迷宮中徒勞地游蕩?！?br>——伽利略

“Neglect of mathematics work injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy.”
--Roger Bacon.
"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."
--David Hilbert .

“數(shù)學(xué)是科學(xué)的大門鑰匙，忽視數(shù)學(xué)必將傷害所有的知識(shí)，因?yàn)楹鲆晹?shù)學(xué)的人是無(wú)法了解任何其他科學(xué)乃至世界上任何其他事物的。更為嚴(yán)重的是，忽視數(shù)學(xué)的人不能理解他自己這一疏忽，最終將導(dǎo)致無(wú)法尋求任何補(bǔ)救的措施。”
——羅杰·培根
“數(shù)學(xué)不分種族，不分國(guó)界。對(duì)于數(shù)學(xué)來(lái)說，整個(gè)文明世界就是一個(gè)國(guó)家?！?br>——戴維·希爾伯特

“Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley.”
--Yōko Ogawa
“[When asked why are numbers beautiful?]

“解決一個(gè)你知道答案的問題，就像在向?qū)У闹敢?，沿著別人鋪好的小路爬山。在數(shù)學(xué)中，真理就在某處，在一個(gè)沒有人知道的地方，在所有人走過的路之外。而且并不總是在山頂。它可能在最光滑的懸崖上的裂縫里，也可能在山谷深處的某個(gè)地方?！?br>——小川洋子
(當(dāng)被問到為什么數(shù)字是美麗的?)

It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.”
--Paul Erd?s .
"Beauty is the first test: there is no permanent place in the world for ugly mathematics."--G.H.Hardy .

這就像問為什么貝多芬的第九交響曲很美。如果你不明白為什么，別人是不會(huì)告訴你的。我知道數(shù)字是美麗的。如果它們不美，那什么都不美?！?br>——保羅·愛多士
“數(shù)學(xué)的優(yōu)美至關(guān)重要, 丑陋參差的數(shù)學(xué), 在世界毫無(wú)立足之地.”——G.H.哈代。

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
--Felix Klein .
“But in my opinion, all things in nature occur mathematically.”
--René Descartes .

“每個(gè)人都知道曲線是什么，直到他學(xué)習(xí)了足夠多的數(shù)學(xué)知識(shí)，就會(huì)對(duì)無(wú)數(shù)可能的例外情況感到困惑?！?br>——菲利克斯·克萊因。
“但在我看來(lái)，自然界的所有事情都是用數(shù)學(xué)方法來(lái)實(shí)現(xiàn)的?！?br>——勒奈·笛卡爾。

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Maths has that certain panache to present the ideas of physics. Broadly speaking, pure mathematics is a science that studies entirely abstract concepts.
What do mathematicians do ?
Mathematicians seek out pattern and use them to formulate new conjectures.
Mathematics arises from many different kinds of problems.

數(shù)學(xué)是表現(xiàn)物理學(xué)思想的華麗外表。廣義上說，純數(shù)學(xué)是一門研究完全抽象概念的科學(xué)。
數(shù)學(xué)家是做什么的?
數(shù)學(xué)家們尋找邏輯規(guī)律，并用它們來(lái)形成新的猜想。
數(shù)學(xué)產(chǎn)生于許多不同類型的問題。

And it's all about finding patterns. And by "pattern" I mean a connection, a structure, some regularity a fluidity, some rules that govern what we see. Second of all, I think it is about representing these patterns with a language. We make up language if we don't have it, and in mathematics, this is essential. It's also about making assumptions and playing around with these assumptions and just seeing what happens.
When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight about reality of nature with the help of abstraction and logic. Maths helps us to understand the logic of life. The most Beautiful and Powerful creation ever created by Humans.

這一切都是為了尋找邏輯規(guī)律。我所說的“邏輯規(guī)律”是指一種聯(lián)系，一種結(jié)構(gòu)，一種規(guī)律性，一種流動(dòng)性，一種支配我們所看到的事物的規(guī)則。其次，我認(rèn)為數(shù)學(xué)也是表示這種邏輯規(guī)律的語(yǔ)言。如果我們之前沒有這種語(yǔ)言，那我們就需要把他創(chuàng)造出來(lái)，在數(shù)學(xué)中，這是非常重要的。數(shù)學(xué)還包括：去做一些假設(shè)，對(duì)這些假設(shè)的要素進(jìn)行修改，然后看看會(huì)發(fā)生什么。
當(dāng)數(shù)學(xué)表達(dá)式是真實(shí)現(xiàn)象的良好模型時(shí)，數(shù)學(xué)推理就可以在抽象和邏輯的幫助下對(duì)自然現(xiàn)實(shí)產(chǎn)生新的洞察。數(shù)學(xué)可以幫助我們理解生活的邏輯。而數(shù)學(xué)正是人類創(chuàng)造出來(lái)的最美麗、最強(qiáng)大的發(fā)明。

And it about the perspective the way you look at something in some way
For Example x + x = 2 · x.
This is a very nice pattern, and it's true, because 5 + 5 = 2 · 5. We've seen this over and over, and we represent it like this. But think about it this is an equation. It says that something is equal to something else, and that's two different perspectives. One perspective is, it's a sum. It's something you add together. On the other hand, it's a multiplication, and those are two different perspectives.
Every mathematical equation where you use that equality sign is actually a metaphor. It's an analogy between two things. You're just viewing something and taking two different points of view, and you're expressing that in a language. And I believe that you understand something if you have the ability to view it from different perspectives.

它是關(guān)于你看待事物的角度：
例如x + x = 2x。
這是一個(gè)很好的表達(dá)式，它是正確的，因?yàn)? + 5 = 2·5。我們應(yīng)該已經(jīng)見過很多次了，我們也知道如何表示它。但是想想看，這本質(zhì)是一個(gè)方程，意思是左邊等于右邊，但有兩個(gè)不同的角度。一種觀點(diǎn)是，它是一個(gè)和，它表達(dá)的是加的過程。另一方面，這表達(dá)的是乘法，于是就有了兩個(gè)不同的角度。
每個(gè)使用等號(hào)的數(shù)學(xué)方程實(shí)際上都是一種隱喻。這是兩個(gè)事物之間的相互解釋。你用了兩種不同的觀點(diǎn)來(lái)看待事物，卻只用了一種語(yǔ)言來(lái)表達(dá)。我相信，如果你能從不同的角度看待問題，你就能理解它。

Mathematicians describes Mathematics as
Gauss referred to mathematics as "The Queen of the Sciences".
Poisson "Life is good only for two things - discovering mathematics and teaching mathematics".

數(shù)學(xué)家們把數(shù)學(xué)描述為：
高斯把數(shù)學(xué)稱為“科學(xué)的皇后”。
泊松說:“人生只有兩件事是美好的——發(fā)現(xiàn)數(shù)學(xué)和教授數(shù)學(xué)。”

The mathematicians does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful --Henri Poincare
Andre Weil describes "We know that God exists because mathematics is consistent and we know that devil exists because we cannot prove the consistency".
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
Maths is the only reason that world is so well arranged well versed. It is the way to understand nature, formulate things, seeing a pattern. And the most scintillating and admired thing that how it fraternizes with everything.
And Yes It is the cheapest science. All one need is a pencil and paper.

數(shù)學(xué)家學(xué)習(xí)純數(shù)學(xué)不是因?yàn)樗杏?他研究它只是因?yàn)樗矚g它，他喜歡它只是因?yàn)閿?shù)學(xué)是美麗的
安德雷·韋依描述道:“上帝是存在的，因?yàn)閿?shù)學(xué)顯然是自洽的，但是魔鬼也是存在的，因?yàn)槲覀儫o(wú)法證明這種自洽?！?br>如果人們不相信數(shù)學(xué)是簡(jiǎn)單的，那只是因?yàn)樗麄儧]有意識(shí)到生活是多么的復(fù)雜。
數(shù)學(xué)是世界如此井然有序的唯一原因。它是理解自然、形成物體、看到規(guī)律的方法。而最耀眼和最令人欽佩的是它如何與萬(wàn)物合二為一。
是的，這是最便宜的科學(xué)。一個(gè)人所需要的只是一支鉛筆和一張紙。

MATHEMATICS may not teach us how to breathe oxygen and how to exhale Carbon-dioxide Or to love a friend and forgive an enemy. It may not even help us find our way to our one true love.
But it gives us every reason to HOPE that every problem has a solution.
And there is no Nobel Prize for outstanding contributions of Mathematicians.
He is the real BATMAN out there!!

數(shù)學(xué)可能不會(huì)教我們?nèi)绾魏粑鯕?，如何呼出二氧化碳，或者如何去愛一個(gè)朋友，去原諒一個(gè)敵人。它甚至不能幫助我們找到通往真愛的路。
但它給了我們每個(gè)問題都有解決方法的理由。
數(shù)學(xué)家擁有杰出的貢獻(xiàn)，但可惜卻沒有諾貝爾數(shù)學(xué)獎(jiǎng)來(lái)表彰他們。
數(shù)學(xué)家才是真正的蝙蝠俠!!

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As my friend Anurag wrote what could be described as almost poetic descxtion of mathematics, I would like to add some more thoughts and musings to his.
As he said mathematics literally just means counting. I would like to expand on that. Every branch or subset or mathematics compliments and supplements each other. There are so many ways of solving a problem using different approaches. Essentially every problem when broken down turns into a counting problem.

我的朋友Anurag所寫的內(nèi)容，幾乎可以被形容為是對(duì)數(shù)學(xué)的詩(shī)意描述，但是我想在他的文章中加入更多的思考。
正如他所說，數(shù)學(xué)的字面意思就是數(shù)數(shù)。我想就此展開討論。每一個(gè)數(shù)學(xué)的分支、子學(xué)科都是相互補(bǔ)充的。每一個(gè)問題都有很多不同的方法來(lái)解決。本質(zhì)上，每一個(gè)問題，都可以分解成一個(gè)數(shù)數(shù)問題。

We start from the base and build up as we go. First comes number theory -natural, whole,fractions, integers, fractional integers, rational numbers, irrational numbers, real numbers, imaginary numbers, complex numbers. Similary we start with algebra (Who put the alphabet in mathematics? - a medi Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī ) which builds up from elementary algebra, abstract algebra... boolean algebra.... and so on.
But when we look closely at algebra we find that is nothing but fancy counting.

我們從基礎(chǔ)開始，隨著我們的發(fā)展而不斷壯大。首先是數(shù)論——自然數(shù)、整體、分?jǐn)?shù)、有理數(shù)、無(wú)理數(shù)、實(shí)數(shù)、虛數(shù)、復(fù)數(shù)。同樣的，我們也可以從代數(shù)開始(是誰(shuí)在數(shù)學(xué)中加入了字母表?)——一個(gè)中世紀(jì)的波斯數(shù)學(xué)家穆罕默德花拉子米,建立從初等代數(shù),抽象代數(shù)…布爾代數(shù)…等等。
但當(dāng)我們仔細(xì)研究代數(shù)時(shí)，我們發(fā)現(xiàn)這其中只不過是更加花哨的數(shù)數(shù)。

Same goes for Calculus, Geomtry, Trigonometry, Complex numbers and Combinatories. But if you have kept a keen eye you will realise from early on that this new mathematics is essentially redundant. The problem would seem familiar. Trigonometric problems would look algebraic, integral problems would look geometrical and so on and so forth. You can solve any problem from one branch in mathematics using the other you just have to translate it from one language to another.

微積分、幾何學(xué)、三角學(xué)、復(fù)數(shù)和組合學(xué)也是如此。但如果你一直保持敏銳的眼光，你就會(huì)從一開始就意識(shí)到許多新的數(shù)學(xué)子學(xué)科本質(zhì)上是多余的。許多問題似乎很熟悉。三角問題看起來(lái)像代數(shù)問題，積分問題看起來(lái)像幾何問題等等。你可以用數(shù)學(xué)的一個(gè)分支來(lái)解決任何問題你只需要把它從那個(gè)子支的語(yǔ)言轉(zhuǎn)換成自己這個(gè)子支的語(yǔ)言。

Oh! I forgot to add, the greatest thing about mathematics is that you can invent your own mathematics out of the blue.
It is the language of the universe, we just haven't learn all of its words.

哦!我忘了說了，數(shù)學(xué)最偉大的地方，是你可以沿著路徑繼續(xù)發(fā)展創(chuàng)造屬于你冠名的真理。
數(shù)學(xué)是宇宙的語(yǔ)言，我們只是還沒有學(xué)會(huì)它所有的單詞！