什么是数学？（1）
正文翻译
What is mathematics?
什么是数学？
What is mathematics?
什么是数学？
评论翻译
"Mathematics is the art of giving the same name to different things." -Henri Poincaré.
I would define mathematics as the study of structure divorced from context.
“数学是给不同事物取相同名字的艺术。”——亨利·庞加莱
我将数学定义为脱离具体环境的结构研究。
I would define mathematics as the study of structure divorced from context.
“数学是给不同事物取相同名字的艺术。”——亨利·庞加莱
我将数学定义为脱离具体环境的结构研究。
In mathematics, we study various structures: numbers, groups, geometric obxts, etc. We study their patterns and figure out how they work and interconnect. I would make the argument that anything existing in the universe and anything that can be cooked up by the human mind that has some sort of structure to it can be studied mathematically.
Of course, what one might argue is that disciplines like physics, chemistry, and biology do the same thing: they search for the physical patterns and structures that exist out in the world. What is the key difference between these pursuits and mathematics?
在数学中，我们要去研究各种结构：数字、组合、几何对象等等。我们研究它们的模式，弄清楚它们是如何工作和相互联系的。我的观点是，任何存在于宇宙中的东西，任何可以被人类大脑制造出来的东西，只要具有某种结构特性，就可以用数学方法来研究。
当然，有人可能会说，像物理、化学和生物学这样的学科也是在做同样的事情：它们在发现存在于世界上的某种物理模式或结构。这些与数学之间的关键区别是什么呢?
Of course, what one might argue is that disciplines like physics, chemistry, and biology do the same thing: they search for the physical patterns and structures that exist out in the world. What is the key difference between these pursuits and mathematics?
在数学中，我们要去研究各种结构：数字、组合、几何对象等等。我们研究它们的模式，弄清楚它们是如何工作和相互联系的。我的观点是，任何存在于宇宙中的东西，任何可以被人类大脑制造出来的东西，只要具有某种结构特性，就可以用数学方法来研究。
当然，有人可能会说，像物理、化学和生物学这样的学科也是在做同样的事情：它们在发现存在于世界上的某种物理模式或结构。这些与数学之间的关键区别是什么呢?
The key is that in all of the above-mentioned physical sciences, any problem that you consider has a certain context, a certain specific interpretation. An oscillating pendulum is not the same thing as a vibrating string which is not the same thing as a spring with a mass attached to it. However, from a mathematical point of view, all of these systems are essentially the same thing.
Once you strip away all of the physical details and particular context of a problem, what remains is its mathematical content. The beauty of this is that by considering things so abstractly, you begin to see connections that you would not otherwise recognize. The motion of a pendulum, a string, and a spring are all described using sinusoidal functions. That same periodic behavior might be observed just as well for sound waves, and light waves, and ocean waves.
关键在于，在上述所有的这些类物理学科中，你所考虑的任何问题都存在着具体的背景和特定的解释。一个摆动的钟摆和一个振动的弦是不一样的，同样一个震动的弦和一个正在伸缩的弹簧也不一样。然而，在数学的角度来看，所有这些系统的本质上都是一样的。
一旦你剥去一个问题的所有物理细节和特定背景，剩下的就是它的数学内容。这样做的美妙之处在于，通过对事物进行抽象的思考，你开始看到一些许多事物之间原本意识不到的联系。摆、弦和弹簧的运动都可以用正弦函数来描述。同样的周期循环也可以在声波，光波和海浪中观察到。
Once you strip away all of the physical details and particular context of a problem, what remains is its mathematical content. The beauty of this is that by considering things so abstractly, you begin to see connections that you would not otherwise recognize. The motion of a pendulum, a string, and a spring are all described using sinusoidal functions. That same periodic behavior might be observed just as well for sound waves, and light waves, and ocean waves.
关键在于，在上述所有的这些类物理学科中，你所考虑的任何问题都存在着具体的背景和特定的解释。一个摆动的钟摆和一个振动的弦是不一样的，同样一个震动的弦和一个正在伸缩的弹簧也不一样。然而，在数学的角度来看，所有这些系统的本质上都是一样的。
一旦你剥去一个问题的所有物理细节和特定背景，剩下的就是它的数学内容。这样做的美妙之处在于，通过对事物进行抽象的思考，你开始看到一些许多事物之间原本意识不到的联系。摆、弦和弹簧的运动都可以用正弦函数来描述。同样的周期循环也可以在声波，光波和海浪中观察到。
When you have a perspective this broad, you can begin by looking at a problem one way, flip it around, realize that you can contextualize it in a completely different way, and then start using the tools of a seemingly unconnected theory to solve your problem.
In my mind, that is what makes mathematics beautiful, and what makes it different to just about every other endeavor that mankind has ever attempted.
当你在用具象的视角，你可以从某个角度看待事物，但反过来看，在抽象下你可以将事物在多个维度与其它东西产生联系，然后你就可以开始用一个看似不相干的理论工具来解决你的问题。
在我看来，这就是使数学美丽的原因，也是使它不同于人类曾经使用过的任何其它工具的原因。
In my mind, that is what makes mathematics beautiful, and what makes it different to just about every other endeavor that mankind has ever attempted.
当你在用具象的视角，你可以从某个角度看待事物，但反过来看，在抽象下你可以将事物在多个维度与其它东西产生联系，然后你就可以开始用一个看似不相干的理论工具来解决你的问题。
在我看来，这就是使数学美丽的原因，也是使它不同于人类曾经使用过的任何其它工具的原因。
------------------
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.
对我来说，纯数学就是哲学。所有数学表达式是一种语言或语法上的逻辑演绎。
这儿有一个假设，然后我们去通过逻辑演绎去证明真伪，或去验证是否符合已知的数学表达式。当然，你需要极具创造力或精通多个领域，才能提出突破性的假设，但你还需要继续分析，才能证明假设真伪。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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.
对我来说，纯数学就是哲学。所有数学表达式是一种语言或语法上的逻辑演绎。
这儿有一个假设，然后我们去通过逻辑演绎去证明真伪，或去验证是否符合已知的数学表达式。当然，你需要极具创造力或精通多个领域，才能提出突破性的假设，但你还需要继续分析，才能证明假设真伪。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
I don’t know if you game a lot but I find the universe surrounding us much similar to the worlds created in a video game. Like every world in a video game built using code our universe has certain rules and definitions which must remain valid at all points of time.
Philosophy is a first step to brain-storm and envision such rules and definitions. These need to be validated through logical deduction(mathematics) and empirical evidence (experimental proof).
Just like there are cheats codes in a video game that give the player certain advantages (like infinite health or strength etc), we in the real world develop technology leveraging these validated mathematical hypothesis.
我不知道你是不是经常玩游戏，但我发现我们周围的宇宙和电子游戏中创造的世界是非常相似的。就像电子游戏中使用代码构建的世界一样，我们的世界也有着一定的规则和逻辑，这些规则和逻辑会在任何时间地点都有效。
通过哲学概念上的头脑风暴去理解和设想规则和逻辑是第一步，接着需要通过逻辑演绎(数学)和经验证据(实验证明)来验证这些设想是否正确。
就像电子游戏中的作弊代码会给玩家一些优势(比如无限的健康或力量等)，我们在现实世界中也是在利用这些已验的数学假设来开发新技术。
Philosophy is a first step to brain-storm and envision such rules and definitions. These need to be validated through logical deduction(mathematics) and empirical evidence (experimental proof).
Just like there are cheats codes in a video game that give the player certain advantages (like infinite health or strength etc), we in the real world develop technology leveraging these validated mathematical hypothesis.
我不知道你是不是经常玩游戏，但我发现我们周围的宇宙和电子游戏中创造的世界是非常相似的。就像电子游戏中使用代码构建的世界一样，我们的世界也有着一定的规则和逻辑，这些规则和逻辑会在任何时间地点都有效。
通过哲学概念上的头脑风暴去理解和设想规则和逻辑是第一步，接着需要通过逻辑演绎(数学)和经验证据(实验证明)来验证这些设想是否正确。
就像电子游戏中的作弊代码会给玩家一些优势(比如无限的健康或力量等)，我们在现实世界中也是在利用这些已验的数学假设来开发新技术。
Today’s computational & artificial intelligence related technologies allows us to fast forward or automate the steps of logical deduction & empirical evidence through simulations, big data, machine learning etc.
But people from all over the ancient world were able to make very accurate hypothesis with limited technology. And the only way I could think they were able to achieve this is through focused philosophical discourse with self or with their peers. Through deep meditation and dedicating ones life to understand why the world behaves the way it does, I believe mankind of that time were able to achieve what they did. And mathematics was an essential tool for them in their enlightening journey.
今天的计算和人工智能相关技术允许我们通过模拟、大数据、机器学习等方式，快速或自动化的进行逻辑推理和实验证明的步骤。
但是来自古代世界各地的人们，只能利用有限的技术做出假设，并验证正确。我认为他们能做到这一点的唯一方法，是通过与自我或同龄人进行集中的哲学讨论。通过深刻的思考，甚至穷尽一生去理解这个世界为什么会这样运行。而数学就是那个时代的人类理解这个世界的过程中必不可少的工具。
But people from all over the ancient world were able to make very accurate hypothesis with limited technology. And the only way I could think they were able to achieve this is through focused philosophical discourse with self or with their peers. Through deep meditation and dedicating ones life to understand why the world behaves the way it does, I believe mankind of that time were able to achieve what they did. And mathematics was an essential tool for them in their enlightening journey.
今天的计算和人工智能相关技术允许我们通过模拟、大数据、机器学习等方式，快速或自动化的进行逻辑推理和实验证明的步骤。
但是来自古代世界各地的人们，只能利用有限的技术做出假设，并验证正确。我认为他们能做到这一点的唯一方法，是通过与自我或同龄人进行集中的哲学讨论。通过深刻的思考，甚至穷尽一生去理解这个世界为什么会这样运行。而数学就是那个时代的人类理解这个世界的过程中必不可少的工具。
---------------
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).
数学是研究形状、数量和排列的逻辑科学。数学就在我们身边，在我们做的每一件事中。它是我们日常生活中所有事物的组成部分，包括移动设备、建筑(古代和现代)、艺术、金钱、工程，甚至体育。
自有记录的历史开始以来，数学就一直处于每一个文明社会的前沿，甚至在最原始的文化中也得到应用。数学的需求是建立在社会需求的基础上的。社会越复杂，数学需求就越复杂。原始部落所需要的仅仅是数数的能力，但有的部落也能靠数学来计算出太阳的位置和狩猎的一些物理原理。
一些数学历史(知识)。
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).
数学是研究形状、数量和排列的逻辑科学。数学就在我们身边，在我们做的每一件事中。它是我们日常生活中所有事物的组成部分，包括移动设备、建筑(古代和现代)、艺术、金钱、工程，甚至体育。
自有记录的历史开始以来，数学就一直处于每一个文明社会的前沿，甚至在最原始的文化中也得到应用。数学的需求是建立在社会需求的基础上的。社会越复杂，数学需求就越复杂。原始部落所需要的仅仅是数数的能力，但有的部落也能靠数学来计算出太阳的位置和狩猎的一些物理原理。
一些数学历史(知识)。
****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——中国、印度、埃及、中美洲和美索不达米亚——对我们今天所知的数学做出了巨大的贡献。苏美尔人是最早发展计数系统的人。接着数学家依于此发展出了算术，包括基本运算、乘法、分数和平方根。大约在公元前300年，苏美尔人的历法通过阿卡德帝国传到了巴比伦人的手中。大约600年后，在美洲，玛雅人发展出了复杂的历法系统，并成为了熟练的天文学家。于此同时，零的概念也被发明了出来。
2，随着文明的发展，数学家开始研究几何，应用几何来计算面积体积和测量角度，并有许多实际应用。几何学被被用于家庭建筑、时尚、室内设计等等诸多领域。
*****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——中国、印度、埃及、中美洲和美索不达米亚——对我们今天所知的数学做出了巨大的贡献。苏美尔人是最早发展计数系统的人。接着数学家依于此发展出了算术，包括基本运算、乘法、分数和平方根。大约在公元前300年，苏美尔人的历法通过阿卡德帝国传到了巴比伦人的手中。大约600年后，在美洲，玛雅人发展出了复杂的历法系统，并成为了熟练的天文学家。于此同时，零的概念也被发明了出来。
2，随着文明的发展，数学家开始研究几何，应用几何来计算面积体积和测量角度，并有许多实际应用。几何学被被用于家庭建筑、时尚、室内设计等等诸多领域。
****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，几何与代数是紧密联系在一起的，代数是9世纪由波斯数学家穆罕默德.花拉子米发明的。他还发明了乘法和除法的快速方法，即所谓的算法——这个单词也是由他的名字发展来的。
4，代数为文明提供了一种划分继承和分配资源的方式。对代数的研究意味着数学家要解线性方程组。需要解二次方程，钻研正解和负解。于是古代的数学家也开始研究数论。数论着眼于数字的具象化、数字本身的表征和定理。
****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，几何与代数是紧密联系在一起的，代数是9世纪由波斯数学家穆罕默德.花拉子米发明的。他还发明了乘法和除法的快速方法，即所谓的算法——这个单词也是由他的名字发展来的。
4，代数为文明提供了一种划分继承和分配资源的方式。对代数的研究意味着数学家要解线性方程组。需要解二次方程，钻研正解和负解。于是古代的数学家也开始研究数论。数论着眼于数字的具象化、数字本身的表征和定理。
-----------------
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 .
关于数学的一些有启发性的名言。
“数学是人类精神中最美丽、最强大的创造物。”——斯蒂芬·巴拿赫。
“哲学是写在我们面前的那本伟大的书里的——我指的是宇宙——但如果我们不先学习它的语言和掌握它的符号，我们就无法理解它。这本书是用数学语言写的，里面的符号是三角形、圆形和其他几何图形，没有这些符号的帮助，一个字也理解不了;没有它，人就在黑暗的迷宫中徒劳地游荡。”
——伽利略
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 .
关于数学的一些有启发性的名言。
“数学是人类精神中最美丽、最强大的创造物。”——斯蒂芬·巴拿赫。
“哲学是写在我们面前的那本伟大的书里的——我指的是宇宙——但如果我们不先学习它的语言和掌握它的符号，我们就无法理解它。这本书是用数学语言写的，里面的符号是三角形、圆形和其他几何图形，没有这些符号的帮助，一个字也理解不了;没有它，人就在黑暗的迷宫中徒劳地游荡。”
——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 .
“数学是科学的大门钥匙，忽视数学必将伤害所有的知识，因为忽视数学的人是无法了解任何其他科学乃至世界上任何其他事物的。更为严重的是，忽视数学的人不能理解他自己这一疏忽，最终将导致无法寻求任何补救的措施。”
——罗杰·培根
“数学不分种族，不分国界。对于数学来说，整个文明世界就是一个国家。”
——戴维·希尔伯特
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
--Roger Bacon.
"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country."
--David Hilbert .
“数学是科学的大门钥匙，忽视数学必将伤害所有的知识，因为忽视数学的人是无法了解任何其他科学乃至世界上任何其他事物的。更为严重的是，忽视数学的人不能理解他自己这一疏忽，最终将导致无法寻求任何补救的措施。”
——罗杰·培根
“数学不分种族，不分国界。对于数学来说，整个文明世界就是一个国家。”
——戴维·希尔伯特
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
“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?]
“解决一个你知道答案的问题，就像在向导的指引下，沿着别人铺好的小路爬山。在数学中，真理就在某处，在一个没有人知道的地方，在所有人走过的路之外。而且并不总是在山顶。它可能在最光滑的悬崖上的裂缝里，也可能在山谷深处的某个地方。”
——小川洋子
(当被问到为什么数字是美丽的?)
--Yōko Ogawa
“[When asked why are numbers beautiful?]
“解决一个你知道答案的问题，就像在向导的指引下，沿着别人铺好的小路爬山。在数学中，真理就在某处，在一个没有人知道的地方，在所有人走过的路之外。而且并不总是在山顶。它可能在最光滑的悬崖上的裂缝里，也可能在山谷深处的某个地方。”
——小川洋子
(当被问到为什么数字是美丽的?)
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 Erdős .
"Beauty is the first test: there is no permanent place in the world for ugly mathematics."--G.H.Hardy .
这就像问为什么贝多芬的第九交响曲很美。如果你不明白为什么，别人是不会告诉你的。我知道数字是美丽的。如果它们不美，那什么都不美。”
——保罗·爱多士
“数学的优美至关重要, 丑陋参差的数学, 在世界毫无立足之地.”——G.H.哈代。
--Paul Erdős .
"Beauty is the first test: there is no permanent place in the world for ugly mathematics."--G.H.Hardy .
这就像问为什么贝多芬的第九交响曲很美。如果你不明白为什么，别人是不会告诉你的。我知道数字是美丽的。如果它们不美，那什么都不美。”
——保罗·爱多士
“数学的优美至关重要, 丑陋参差的数学, 在世界毫无立足之地.”——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 .
“每个人都知道曲线是什么，直到他学习了足够多的数学知识，就会对无数可能的例外情况感到困惑。”
——菲利克斯·克莱因。
“但在我看来，自然界的所有事情都是用数学方法来实现的。”
——勒奈·笛卡尔。
--Felix Klein .
“But in my opinion, all things in nature occur mathematically.”
--René Descartes .
“每个人都知道曲线是什么，直到他学习了足够多的数学知识，就会对无数可能的例外情况感到困惑。”
——菲利克斯·克莱因。
“但在我看来，自然界的所有事情都是用数学方法来实现的。”
——勒奈·笛卡尔。
---------------
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.
数学是表现物理学思想的华丽外表。广义上说，纯数学是一门研究完全抽象概念的科学。
数学家是做什么的?
数学家们寻找逻辑规律，并用它们来形成新的猜想。
数学产生于许多不同类型的问题。
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.
数学是表现物理学思想的华丽外表。广义上说，纯数学是一门研究完全抽象概念的科学。
数学家是做什么的?
数学家们寻找逻辑规律，并用它们来形成新的猜想。
数学产生于许多不同类型的问题。
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.
这一切都是为了寻找逻辑规律。我所说的“逻辑规律”是指一种联系，一种结构，一种规律性，一种流动性，一种支配我们所看到的事物的规则。其次，我认为数学也是表示这种逻辑规律的语言。如果我们之前没有这种语言，那我们就需要把他创造出来，在数学中，这是非常重要的。数学还包括：去做一些假设，对这些假设的要素进行修改，然后看看会发生什么。
当数学表达式是真实现象的良好模型时，数学推理就可以在抽象和逻辑的帮助下对自然现实产生新的洞察。数学可以帮助我们理解生活的逻辑。而数学正是人类创造出来的最美丽、最强大的发明。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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.
这一切都是为了寻找逻辑规律。我所说的“逻辑规律”是指一种联系，一种结构，一种规律性，一种流动性，一种支配我们所看到的事物的规则。其次，我认为数学也是表示这种逻辑规律的语言。如果我们之前没有这种语言，那我们就需要把他创造出来，在数学中，这是非常重要的。数学还包括：去做一些假设，对这些假设的要素进行修改，然后看看会发生什么。
当数学表达式是真实现象的良好模型时，数学推理就可以在抽象和逻辑的帮助下对自然现实产生新的洞察。数学可以帮助我们理解生活的逻辑。而数学正是人类创造出来的最美丽、最强大的发明。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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.
它是关于你看待事物的角度：
例如x + x = 2x。
这是一个很好的表达式，它是正确的，因为5 + 5 = 2·5。我们应该已经见过很多次了，我们也知道如何表示它。但是想想看，这本质是一个方程，意思是左边等于右边，但有两个不同的角度。一种观点是，它是一个和，它表达的是加的过程。另一方面，这表达的是乘法，于是就有了两个不同的角度。
每个使用等号的数学方程实际上都是一种隐喻。这是两个事物之间的相互解释。你用了两种不同的观点来看待事物，却只用了一种语言来表达。我相信，如果你能从不同的角度看待问题，你就能理解它。
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.
它是关于你看待事物的角度：
例如x + x = 2x。
这是一个很好的表达式，它是正确的，因为5 + 5 = 2·5。我们应该已经见过很多次了，我们也知道如何表示它。但是想想看，这本质是一个方程，意思是左边等于右边，但有两个不同的角度。一种观点是，它是一个和，它表达的是加的过程。另一方面，这表达的是乘法，于是就有了两个不同的角度。
每个使用等号的数学方程实际上都是一种隐喻。这是两个事物之间的相互解释。你用了两种不同的观点来看待事物，却只用了一种语言来表达。我相信，如果你能从不同的角度看待问题，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".
数学家们把数学描述为：
高斯把数学称为“科学的皇后”。
泊松说:“人生只有两件事是美好的——发现数学和教授数学。”
Gauss referred to mathematics as "The Queen of the Sciences".
Poisson "Life is good only for two things - discovering mathematics and teaching mathematics".
数学家们把数学描述为：
高斯把数学称为“科学的皇后”。
泊松说:“人生只有两件事是美好的——发现数学和教授数学。”
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.
数学家学习纯数学不是因为它有用;他研究它只是因为他喜欢它，他喜欢它只是因为数学是美丽的
安德雷·韦依描述道:“上帝是存在的，因为数学显然是自洽的，但是魔鬼也是存在的，因为我们无法证明这种自洽。”
如果人们不相信数学是简单的，那只是因为他们没有意识到生活是多么的复杂。
数学是世界如此井然有序的唯一原因。它是理解自然、形成物体、看到规律的方法。而最耀眼和最令人钦佩的是它如何与万物合二为一。
是的，这是最便宜的科学。一个人所需要的只是一支铅笔和一张纸。
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.
数学家学习纯数学不是因为它有用;他研究它只是因为他喜欢它，他喜欢它只是因为数学是美丽的
安德雷·韦依描述道:“上帝是存在的，因为数学显然是自洽的，但是魔鬼也是存在的，因为我们无法证明这种自洽。”
如果人们不相信数学是简单的，那只是因为他们没有意识到生活是多么的复杂。
数学是世界如此井然有序的唯一原因。它是理解自然、形成物体、看到规律的方法。而最耀眼和最令人钦佩的是它如何与万物合二为一。
是的，这是最便宜的科学。一个人所需要的只是一支铅笔和一张纸。
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!!
数学可能不会教我们如何呼吸氧气，如何呼出二氧化碳，或者如何去爱一个朋友，去原谅一个敌人。它甚至不能帮助我们找到通往真爱的路。
但它给了我们每个问题都有解决方法的理由。
数学家拥有杰出的贡献，但可惜却没有诺贝尔数学奖来表彰他们。
数学家才是真正的蝙蝠侠!!
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!!
数学可能不会教我们如何呼吸氧气，如何呼出二氧化碳，或者如何去爱一个朋友，去原谅一个敌人。它甚至不能帮助我们找到通往真爱的路。
但它给了我们每个问题都有解决方法的理由。
数学家拥有杰出的贡献，但可惜却没有诺贝尔数学奖来表彰他们。
数学家才是真正的蝙蝠侠!!
-------------------
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所写的内容，几乎可以被形容为是对数学的诗意描述，但是我想在他的文章中加入更多的思考。
正如他所说，数学的字面意思就是数数。我想就此展开讨论。每一个数学的分支、子学科都是相互补充的。每一个问题都有很多不同的方法来解决。本质上，每一个问题，都可以分解成一个数数问题。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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所写的内容，几乎可以被形容为是对数学的诗意描述，但是我想在他的文章中加入更多的思考。
正如他所说，数学的字面意思就是数数。我想就此展开讨论。每一个数学的分支、子学科都是相互补充的。每一个问题都有很多不同的方法来解决。本质上，每一个问题，都可以分解成一个数数问题。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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.
我们从基础开始，随着我们的发展而不断壮大。首先是数论——自然数、整体、分数、有理数、无理数、实数、虚数、复数。同样的，我们也可以从代数开始(是谁在数学中加入了字母表?)——一个中世纪的波斯数学家穆罕默德花拉子米,建立从初等代数,抽象代数…布尔代数…等等。
但当我们仔细研究代数时，我们发现这其中只不过是更加花哨的数数。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
But when we look closely at algebra we find that is nothing but fancy counting.
我们从基础开始，随着我们的发展而不断壮大。首先是数论——自然数、整体、分数、有理数、无理数、实数、虚数、复数。同样的，我们也可以从代数开始(是谁在数学中加入了字母表?)——一个中世纪的波斯数学家穆罕默德花拉子米,建立从初等代数,抽象代数…布尔代数…等等。
但当我们仔细研究代数时，我们发现这其中只不过是更加花哨的数数。
原创翻译：龙腾网 https://www.ltaaa.cn 转载请注明出处
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.
微积分、几何学、三角学、复数和组合学也是如此。但如果你一直保持敏锐的眼光，你就会从一开始就意识到许多新的数学子学科本质上是多余的。许多问题似乎很熟悉。三角问题看起来像代数问题，积分问题看起来像几何问题等等。你可以用数学的一个分支来解决任何问题你只需要把它从那个子支的语言转换成自己这个子支的语言。
微积分、几何学、三角学、复数和组合学也是如此。但如果你一直保持敏锐的眼光，你就会从一开始就意识到许多新的数学子学科本质上是多余的。许多问题似乎很熟悉。三角问题看起来像代数问题，积分问题看起来像几何问题等等。你可以用数学的一个分支来解决任何问题你只需要把它从那个子支的语言转换成自己这个子支的语言。
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!我忘了说了，数学最伟大的地方，是你可以沿着路径继续发展创造属于你冠名的真理。
数学是宇宙的语言，我们只是还没有学会它所有的单词！
It is the language of the universe, we just haven't learn all of its words.
哦!我忘了说了，数学最伟大的地方，是你可以沿着路径继续发展创造属于你冠名的真理。
数学是宇宙的语言，我们只是还没有学会它所有的单词！
很赞 1
收藏