数学的作用

我们爱学习
9 Feb 202411:33

Summary

TLDRThe video script delves into the history and evolution of mathematics education, discussing the limitations of current curriculums that primarily focus on concepts developed centuries ago. It explores the recent paradigm shift towards understanding mathematics as the 'science of patterns,' encompassing diverse branches like topology, game theory, and differential geometry. The script highlights the challenges faced in past reform attempts and suggests incorporating conceptual understanding alongside computational skills in mathematics pedagogy.

Takeaways

  • 🔑 The math curriculum in middle and high schools primarily covers the mathematical developments up until about 300 years ago, omitting the major advancements made in the last three centuries.
  • ⌛ The history of mathematics can be traced back to ancient civilizations like Babylonians and Egyptians, who used mathematics for practical applications like measurement and accounting.
  • 🔺 The ancient Greeks viewed mathematics as a subject of study in its own right and introduced the concept of proving theorems, as exemplified in Euclid's 'Elements'.
  • ➕ Algebra was developed by Arab mathematicians, and the word 'algebra' is derived from the Arabic term 'al-jabr'.
  • 🌐 In the last 300 years, mathematics has undergone a profound transformation, with the emergence of numerous new branches and a shift in focus towards studying patterns and abstract concepts.
  • 🧩 Mathematics is now considered the 'science of patterns', with various traditional fields categorized based on the patterns they study, such as arithmetic (patterns of computation), geometry (patterns of shapes), and calculus (patterns of motion).
  • 🔣 Mathematical symbols are abstract representations of concepts, and true mathematics only emerges when these symbols are interpreted and understood by individuals capable of 'performing' them.
  • 🎓 At the university level, mathematics delves into highly abstract and counterintuitive concepts, such as paradoxes and axiom systems, which challenge our intuitive understanding of mathematics.
  • 🚫 A failed attempt at reforming mathematics education in the 1960s, known as 'New Math', tried to introduce advanced concepts prematurely without sufficient emphasis on computational skills, leading to its eventual failure.
  • 🌐 To improve mathematics education, schools could offer elective courses on mathematical concepts and history, allowing students to gain a deeper understanding beyond computational skills.

Q & A

  • What is the main topic discussed in the script?

    -The script discusses the evolution of mathematics education, focusing on the historical development of mathematics and how it should be incorporated into modern curriculum at the high school and university levels.

  • What are the major periods in the early development of mathematics mentioned in the script?

    -The script mentions the following major periods: the origin of numbers and arithmetic (around 10,000 years ago), the incorporation of geometry by Babylonians and Egyptians (a few hundred years later), the Ancient Greek period (500 BC to 300 BC), and the algebraic contributions of the Arabs (8th-9th century AD).

  • What is the significance of the Ancient Greek period in the development of mathematics?

    -The Ancient Greeks approached mathematics as a subject of study, moving beyond its practical applications in measurement and accounting. They introduced the concept of proving theorems through formal logic and reasoning, exemplified by Euclid's 'Elements'.

  • What major shifts have occurred in mathematics over the past 200-300 years?

    -Over the past 200-300 years, mathematics has seen a significant increase in the number of sub-branches (from around 4 to over 60), and a paradigm shift towards viewing mathematics as the 'science of patterns'. This includes studying patterns in computation, shapes, motion, reasoning, topology, and self-similarity.

  • What is the significance of mathematical symbols according to the script?

    -The script compares mathematical symbols to musical notes, stating that they represent abstract concepts and only come to life when 'performed' or understood by someone capable of interpreting them. Mathematical symbols allow us to perceive patterns in the universe that are otherwise invisible.

  • What is the relationship between mathematics and physics mentioned in the script?

    -The script states that physics is the image of the universe seen through the lens of mathematics. Mathematical abstractions make invisible forces and fields visible, as in the case of understanding the forces that keep an airplane aloft.

  • What challenges or paradoxes are discussed in relation to advanced mathematics?

    -The script mentions paradoxes such as Russell's paradox, the barber paradox, and the Banach-Tarski paradox, which led to a 'third mathematical crisis' and the need for additional axioms (like the ZF and ZFC axiom systems) to maintain the consistency of mathematical foundations.

  • Why did the 'New Math' educational reform in the 1960s fail, according to the script?

    -The 'New Math' reform failed because it focused too much on introducing new concepts to students while neglecting the importance of developing computational skills. The script states that mastering calculations is crucial for truly understanding mathematical concepts.

  • What does the script suggest as a way to improve mathematics education?

    -The script suggests that while computational skills should still be emphasized, schools could offer elective courses on mathematical concepts and the history of mathematics for students who have already mastered calculations, as a way to deepen their understanding.

  • How does the script categorize people who use mathematics in an industrial society?

    -The script divides people into two categories: those who can find solutions to given mathematical problems (the majority), and those who can translate real-world problems into mathematical models and use mathematical tools for analysis (a valuable minority).

Outlines

00:00

📚 The History and Evolution of Mathematics Education

This paragraph discusses the historical perspective of mathematics education in high schools and universities, as presented by Stanford researcher Keith Devlin. It traces the origins of mathematics from ancient times, highlighting the contributions of various civilizations, such as the Babylonians, Egyptians, and Greeks. The paragraph emphasizes that the mathematics taught in high schools today largely covers content from before the last 300 years, despite significant advancements in the field during that time period.

05:01

🔢 The Abstraction and Symbolism of Mathematics

This paragraph explores the abstract nature of mathematics and its symbolic representation. It compares mathematical symbols to musical notation, stating that both require interpretation to convey their true essence. The paragraph discusses how mathematics, through its abstract symbols, allows us to comprehend invisible patterns in the universe. It also reframes physics as the perception of the universe through the lens of mathematics. Additionally, it touches upon the significant transition to more abstract and conceptual mathematics in university-level studies.

10:01

🎓 The Challenges and Reforms in Mathematics Education

This paragraph examines the failed attempt at reforming mathematics education in the 1960s, known as the "New Math" movement. It discusses the pitfalls of focusing solely on new concepts while neglecting computational skills, which are crucial for developing a deeper understanding of mathematical concepts. The paragraph suggests incorporating courses on mathematical concepts and history for students who have already mastered computational techniques. It also emphasizes the importance of training individuals who can translate real-world problems into mathematical models and solve them using mathematical tools.

Mindmap

Keywords

💡Mathematics

Mathematics refers to the study of quantities, structures, space, and change. In the video, mathematics is discussed as a subject that has evolved significantly over time, from its origins in arithmetic and geometry to modern branches like topology and abstract algebra. The video traces the historical development of mathematics and its changing paradigms, highlighting its role in understanding patterns and modeling the natural world.

💡Paradigm

A paradigm is a fundamental model or framework that shapes how we understand and approach a subject. The video discusses how mathematics has undergone several paradigm shifts throughout history, from the Greek focus on geometric proofs to the modern view of mathematics as the study of patterns. These paradigm changes have significantly influenced the direction and scope of mathematical research.

💡Irrational numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. The video cites the discovery of irrational numbers, such as the square root of 2, by ancient Greek mathematicians as a significant milestone in the development of mathematics. This discovery challenged the prevailing notion that all lengths could be represented by rational numbers and paved the way for a more abstract understanding of numbers.

💡Algebra

Algebra is a branch of mathematics that deals with the manipulation of symbols and the study of abstract structures. The video highlights the Arabic origin of the word "algebra" and its development as a computational method arising from trade practices. Algebra introduced new concepts and techniques that expanded the scope of mathematical inquiry.

💡Calculus

Calculus is a branch of mathematics that deals with the study of rates of change and accumulation. The video mentions that calculus, along with probability theory, was one of the major developments in mathematics during the 17th century. These subjects laid the foundation for much of the mathematical progress that followed and are still essential components of secondary and tertiary mathematics education.

💡Paradoxes

Paradoxes are statements or scenarios that lead to contradictory or seemingly absurd conclusions when analyzed logically. The video discusses various mathematical paradoxes, such as Russell's paradox and the Banach-Tarski paradox, which challenged the foundations of set theory and led to the development of axiom systems like ZF and ZFC. These paradoxes highlighted the need for careful formalization and axiomatization in mathematics.

💡Axioms

Axioms are fundamental principles or statements that are accepted as true without proof in a given system or theory. The video explains how axiom systems like ZF (Zermelo-Fraenkel) and ZFC (Zermelo-Fraenkel with the Axiom of Choice) were developed to address paradoxes and provide a consistent foundation for set theory and, by extension, much of modern mathematics.

💡Patterns

The video presents the modern view of mathematics as the "science of patterns." This perspective shifts the focus from specific numerical calculations to the study of abstract structures, relationships, and patterns that underlie various mathematical concepts. By understanding patterns, mathematicians can model and analyze phenomena across various disciplines.

💡Mathematical modeling

Mathematical modeling refers to the process of translating real-world problems or phenomena into mathematical language and using mathematical tools to analyze and solve them. The video highlights the importance of mathematical modeling in fields like manufacturing, where individuals can translate practical problems into mathematical form, analyze them using mathematical techniques, and provide solutions.

💡Curriculum reform

The video discusses attempts at reforming mathematics curriculum, such as the "New Math" movement in the 1960s, which aimed to introduce more modern mathematical concepts into secondary education. While well-intentioned, these reforms often failed due to an overemphasis on abstract concepts at the expense of computational skills, which are essential for developing a deeper understanding of mathematical ideas.

Highlights

Mathematics education in schools today focuses primarily on the developments of the past 300 years, neglecting the significant progress made in recent centuries.

The origins of mathematics can be traced back to counting and arithmetic practices dating back 10,000 years, closely tied to the use of currency.

During the Ancient Greek period (500 BC - 300 BC), mathematics shifted from a purely practical endeavor to a field of study, with a focus on geometry and the discovery of irrational numbers.

The Arabic world drove the next major leap in mathematics during the 8th-9th centuries, introducing algebra as a calculation method derived from trade practices.

In the past 200 years, mathematics has undergone a profound transformation, with numerous new subfields emerging, such as Boolean algebra, non-Euclidean geometries, set theory, and many more.

The modern view of mathematics is as the 'science of patterns,' encompassing patterns in computation, shapes, motion, reasoning, topology, and self-similarity.

Mathematical symbols represent abstract concepts, akin to musical notes, and only when 'interpreted' by those with the ability do they reveal the true nature of mathematics.

Physics can be understood as the universe viewed through the lens of mathematics, making the invisible forces and fields visible through abstract mathematical symbols.

University-level mathematics introduces concepts entirely detached from physical reality, such as the comparison of infinities, the division of infinitesimals, and paradoxes like Russell's paradox.

The set theory crisis led to the development of axiomatic systems like ZF and ZFC, which introduced additional axioms to resolve paradoxes, potentially leading to different mathematical 'worlds.'

The Banach-Tarski paradox demonstrates that, under the Axiom of Choice, a solid sphere can be dissected and reassembled into two spheres of the same volume.

The 1960s 'New Math' reform in the US aimed to introduce recent mathematical developments into school curricula but failed due to an overemphasis on concepts over computational skills.

Mastering computational techniques is crucial for truly understanding mathematical concepts, which the 'New Math' reform overlooked.

In an industrial society, two types of mathematicians are needed: those who can find solutions to given problems, and the more valuable ones who can translate real-world problems into mathematical models and solve them.

To improve mathematics education, schools could offer elective courses on mathematical concepts and history, complementing the existing focus on computational skills.

Transcripts

00:00

斯坦福大学的高级研究员——英国数学家凯斯·德弗林(Keith Devlin)最近写了一篇关于数学教育的文章

00:07

这篇文章从数学发展史的角度谈论了中学和大学的数学教育

00:11

很有启发

00:13

今天这期《科技参考》,

00:14

我就来解读一下

00:16

看看上了这么多年的数学课

00:18

我们接触到了数学的哪些部分

00:20

今后的数学课应该怎么改进

00:22

中学数学课不涉及最近300年数学发展

00:23

延续至今的数学,只有一条发展主线

00:26

数学史学家认为

00:28

数和算术是数学的起源

00:30

这个时间点是从一万年前开始的

00:33

那时出现的数学和货币的使用紧密相连

00:37

接下来几百年

00:39

古巴比伦人和古埃及人把几何扩充到数学中

00:43

那时候的数学是纯实用的

00:45

就像一本菜谱,告诉你每一步怎么做

00:48

然后准会得到答案

00:51

公元前500年到公元前300年这段时间是古希腊数学时期

00:56

古希腊人非常注重几何

00:58

所有问题都尽量转化到几何图形上来解决

01:02

最典型的就是把“数”看作长度的测量值

01:06

也因为古希腊人有这样另类的思考方法

01:09

所以当他们发现存在一些无法对应真实长度的值时

01:14

无理数就被发现了

01:16

比如说,两个直角边为1的三角形

01:18

斜边等于根号2

01:20

根号2这个值就无法对应真实长度

01:23

只能无限近似

01:25

虽然还有阿基米德羊皮卷这种在20世纪才发现的重大数学考古遗迹

01:31

证明他老人家已经非常接近发现微积分了

01:35

但由于没有被继承下去

01:37

所以从整体的文明进展上看

01:39

古希腊的数学就止于无理数的发现

01:43

古希腊对数学的影响不只是具体知识

01:46

更是把数学变成了一个研究领域

01:49

在此之前

01:50

数学完全局限在测量、计数、会计这些应用领域

01:55

比如公元前500年

01:56

米利都的泰勒斯就把数学领域的活动范式定义成:

02:01

一切具体数字表达的论断都能通过形式化的论证

02:06

以符合逻辑的方式加以证明

02:09

而这个范式就是后来所谓的“证明定理的过程”,

02:13

一个定理的证明背后就是无限多具体数字的成立

02:18

在古希腊

02:19

这个范式的巅峰就是欧几里得的《几何原本》。

02:23

这里全部采用了证明定理的方式阐述数学思想

02:27

并不拘泥在具体数字的计算上

02:30

公元8、9世纪

02:31

数学的再次阶跃式发展是由阿拉伯人推动的

02:35

代数(algebra)这个词就来源自阿拉伯语“al-jabr”,

02:39

这个词在阿拉伯语中是“某些东西”的意思

02:42

代数是在阿拉伯人和世界各地人的贸易中不断衍生出来的计算方法

02:48

除了上面这些之外

02:50

世界上其他地区虽然也出现过零星的数学研究

02:53

但它们都没有延续下去

02:56

并没有对今天的数学产生重大的影响

03:00

这些数学早期发展中出现的知识点

03:02

再加上2个来自十七世纪的数学进展

03:06

也就是微积分和概率论的初级知识

03:08

大体上就是全世界的中学数学课的内容基础

03:13

也就是说,最近300年的数学进展

03:16

根本没有走进中学数学课堂

03:19

但今天

03:20

数学领域内所有研究对象都是最近两、三百年内发展出来的

03:25

这就会让绝大部分人对数学世界有错误的印象

03:29

认为它不是一个繁荣发展的领域

03:32

那么,最近两百年

03:33

数学世界发生了怎样的转变呢?

03:36

首先是,数学子分支一下多了很多

03:40

300年之前的数学

03:41

大约可以分为4个不同的领域——算数、几何、微积分、代数

03:47

而一百年前

03:48

布尔代数、各种非欧几何、集合论等集中出现

03:52

数学可以划分成12个领域

03:55

如今

03:55

数学的子分支已经有六七十个了

03:58

微分几何、博弈论、拓扑学、测度论、李群、李代数等等是大家熟悉的

04:05

上面这些都是表面所见的变化

04:08

更深刻的变化总结起来

04:10

可以用研究范式的变化来描述

04:13

这个变化就像

04:14

古希腊数学提出数学围绕定理研究

04:18

改变了之前古巴比伦和古印度对围绕具体数字的研究那样

04:23

那么强烈

04:25

理解数学这门学科,最先进的概念是

04:28

数学是一门关于模式的科学(science of patterns)

04:32

在模式的角度下重新划分数学

04:34

数学就是另外的样子——

04:37

传统中与数有关的内容

04:39

比如算术、数论

04:40

都属于计算的模式

04:42

传统中的几何都属于形状的模式

04:46

传统中的微积分属于处理运动的模式

04:49

传统中逻辑属于推理的模式

04:52

传统中拓扑学属于研究封闭性与位置的模式

04:56

传统中的分形属于自相似的模式

04:59

数学符号的新理解

05:00

数学使用一套特殊的符号系统

05:03

每个符号都是抽象的概念

05:05

数学符号和音乐中的音符有高度类似的地位

05:09

一页蝌蚪一样的音符虽然代表着一份音乐作品

05:13

但只有当这些蝌蚪被演奏出来的时候才是音乐本身

05:18

数学符号也是这样

05:20

当它们只在纸上的时候并不是数学本身

05:24

只有当有能力“演绎”它们的人在脑中把数学符号理解后

05:29

数学才露出真身

05:31

由于每个符号都是抽象的

05:33

所以当数学露出真身时

05:35

也是抽象的

05:37

这种抽象的存在却能帮我们理解宇宙中压根看不见的模式

05:43

1623年伽利略的一句话是关于数学符号最好的描述:“只有那些懂得自然是用什么语言书写的人

05:51

才能读懂自然这本巨著

05:54

而这种语言就是数学

05:55

05:57

知道了这些

05:57

我们也可以重新理解什么是物理学

06:01

物理学就是宇宙经过数学这个镜片后看到的像

06:06

当飞机从头顶飞过

06:08

我们看不到飞机下方有任何支撑

06:11

只有通过数学

06:12

我们才看“看到”使飞机维持在天空的那些力、场

06:17

数学的抽象符号把自然界里的不可见变为可见

06:21

大学数学,巨大的跨越

06:22

如果进入大学,学习的是数学专业

06:24

那内容就大幅超越了中学数学的知识范畴

06:28

出现了很多完全脱离与物理现实对应的重要思想

06:33

简单一些的问题

06:34

比如“一个无穷与另一个无穷到底谁更大”,

06:39

或者“一个无限小除以另外一个无限小到底等于多少”。

06:44

复杂一些的问题就更多了

06:46

比如

06:46

如果“一个集合S由所有不是自身元素的集合构成”这样一个定义成立的话

06:53

那么集合S包含自身吗?

06:56

这就是罗素悖论

06:58

如果S属于S,根据S的定义

07:01

S就不能当作集合了

07:03

于是S就不属于S,矛盾了

07:06

反过来,如果S不属于S

07:08

那S就满足集合的定义

07:10

于是根据它的定义

07:12

S是集合,于是又矛盾了

07:15

总之,正反说都不对

07:17

这个悖论通俗又等效地表达就是

07:20

如果“世界上没有一句话是绝对的”,

07:23

那么这句话是绝对的吗?

07:26

一个理发师发誓给不自己理发的人理发

07:29

那这个理发师该不该给自己理发呢?

07:33

这些悖论导致了第三次数学危机

07:35

最后是通过修修补补

07:37

让数学大厦没有倒塌

07:39

修修补补的方式就是

07:41

在定义集合的同时添加一些额外限制

07:45

防止刚刚那些悖论出现

07:47

这些额外限制天然正确吗?

07:49

其实只能暂且默认它们正确

07:52

然后才能让其后的数学大厦保持不倒塌

07:57

而到底应该额外设定哪些限制才能让数学大厦不倒塌呢?

08:01

方法并不唯一

08:03

比如最著名的是ZF公理化(Z是策梅洛的缩写

08:07

F是弗兰克尔的缩写)

08:10

需要额外添加9条公理——外延公理、分类公理、配对公理、并集公理、正则公理等等

08:16

还有其他方法,比如ZFC公理

08:19

就在上面9条公理基础上又增加了选择公理

08:23

这样做也可以保证原有已经被证明的数学定理继续有效

08:28

还有冯·诺伊曼提出的NBG公理体系

08:32

选择不同的公理体系

08:33

就会导出不同的世界

08:35

比如

08:36

“分球悖论”(banach tarski paradox)说的是:

08:38

在选择公理成立的情况下

08:40

可以将一个三维实心球分成有限部分

08:44

然后仅仅通过旋转和平移到其他地方重新组合

08:48

就可以组成两个半径和原来相同的完整的球

08:53

通俗地说就是,只要选择公理成立

08:56

它就可以一个球变两个球

08:58

两个球变四个球,四个球变八个球

09:01

你说,这实在太违背常识了

09:03

于是不承认选择公理

09:06

不承认选择公理

09:07

还能导出更奇特的结论

09:09

比如一个空间具有两种不同的维度

09:13

而选择公理说的又是什么呢?

09:15

通俗地说就是:哪怕有无穷多个集合

09:19

每个集合都有无穷多个元素

09:22

也总是存在一个选择元素的规则能够从每一个集合里不多不少只选出来一个元素

09:29

至于规则是什么样的,不知道

09:32

反正存在就是了

09:34

以上内容的证明和思考与中学数学截然不同

09:38

不知道你是不是已经听懵了

09:40

但这也才刚刚接触到了最近200年数学发展的边缘

09:44

改革失败

09:45

1960年代

09:46

美国曾经搞过一次数学的教学改革

09:49

叫做新数学(New Math)

09:51

希望把那些最近200年来的数学内容引入教材

09:55

其实也就是把今天大学里学的数学内容往中学阶段挪一挪

10:00

初衷虽好

10:01

但实际执行的时候出现了偏差

10:03

偏差主要是由这样的念头导致的:“让学生忘掉各种复杂的计算技巧吧

10:09

只关注新概念就好”。

10:11

但对数学来说,有一点是比较残酷的:

10:15

你只有对某类数学概念对应的复杂计算掌握到一定水平

10:20

才能在之后理解这种数学概念真正的性质

10:24

在不精通计算之前

10:26

自以为的理解都是包含大量错误的

10:29

所以

10:29

那次数学教学改革没过几年就失败了

10:33

此后大学数学的内容一直都没有进入过中学课本

10:37

数学课的改革方法

10:39

学数学到底有什么用呢?

10:41

工业社会的进步需要大量掌握数学知识的人

10:44

这些人会自然地分成两类:

10:47

第一类人能找到给定数学问题的解

10:50

他们占比大多数

10:51

第二类人可以把

10:52

比如说制造业里遇到的问题翻译成数学语言

10:56

然后使用数学工具对问题做精确地分析

11:00

这个工作也被称为建立数学模型

11:02

然后再提出问题,解决问题

11:05

这类人占比很少,却是非常珍贵的

11:08

第二类人更要求对数学概念有深入的理解

11:12

今天的数学教材

11:14

虽然还是侧重训练学生的计算技巧

11:17

但学校可以适当开设数学概念和数学史的课程

11:21

供他们选修

11:22

这对那些已经熟练掌握计算技巧的学生来说

11:26

是更上一层楼的好方法

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