If you cannot visualize three space, can you work on algebraic topology? Classical mechanics? E&M? Engineers could not understand the nature of mathematical talent. You said 证明的能力其实来自于对数理逻辑系统的理解, that is only partially true. It is originality and new ideas that makes a great mathematician. Engineers could never understand that. Have you ever created and proved new theorems? Obviously not. Have you ever done maths research? clearly never. All you do is to use other's results and never create new theorems yourself. That is not research maths. The reality is, maths talents (天赋)determine how far you can go. There is never long enough life span to do any significant amount of modern maths. Maths talents have meaning. Otherwise, why would they give Field's Medal to some people and not others. Given that engineering and maths are totally completely different subjects, believe me, the old Chinese saying "隔行如隔山“ is truth. Within different subdivisions of maths, you still have "隔行如隔山“. In the modern age, there is no such person as a Renaissance man in mathematics. So, how well you can do (do as in create new maths) maths is a function of your familiarity with a given subject, what kinds of new ideas do you have in the subject, how mentally free are you on the subject, how many concrete examples you know about the subject, how well you can shoehorn other disparative maths fields to your service, etc. Knowing how to do proofs in number theory does not ensue knowing how to do proofs well in topology and geometry. Hell! Knowing well in algebraic number theory does not mean you know how to do proofs in analytical number theory. Maths is like that. There are no general statements that you can make about any of them. And you try to talk about maths when you are still learning and have never done maths research. I have one experience and I spent at six years getting a PhD in pure maths. I may not be any good in maths but I know what is maths and I know what research in maths means. Remember, it is "隔行如隔山“. This does not mean I know everything you know. However, I understand the mechanism and the philosophy of maths a lot better than you because I had real experience in mathematical research; that is all. My dissertation can be published as a couple page short paper. There! I am much more qualified at talking about mathematics than any of you amateurs. Let me put it this way. You also said "所以其实数学作为一门证明技术，和其它所有技术同样，天赋走一时，坚持走一世。" That is not true. If you had wrong ideas, no matter how much time you spend on the wrong path, you will never be able to solve the problem at hand. If you are on the right path and got the right idea, the problem is immediately solved. Just look at the career of 张益唐 and you would understand what I mean. He formulated the problems correctly and so he was able to solve an important probelm in analytic number theory. Others did not formulate the problem correctly so they fail. Life long effort has nothing to do with it. Formal system of logic that include the arthmetic sytem are essential incomplete. You also have the Turing undecidability. So whether you can prove a statement in your system within your system is not always decidable. For example, the Riemann Hypothesis. Better be lucky than good. I am not in mathematical logic and not in computer science. Please forgive my misquotes. 非常抱歉， 中文谈数学，我力有不逮也，盖10岁移美64年，吾已大半个ABC矣，中文欠缺，尚祈见谅。
@@hockng5610Sir, I read your comment. That is a big decision to work with Pure Math, but math is too damn important. Although difficult math is, I decide to get a master of it.
The main problem is that in advanced math and statistics, both textbooks and professors skip many crucial steps and tend to overcomplicate the ideas. Otherwise, with proper explanations and examples, many average students can also learn advanced topics.
我是工科，现在工作了还在学数学，因为半导体行业偏微分和概率论还是很有用的。同时理论计算机也需要理解大量形式系统相关知识。数学能不能学好我感觉是看直觉逻辑和形式逻辑的思维方式能不能熟练切换。懂一阶系统和集合论和不懂学的可能是两个数学。证明的能力其实来自于对数理逻辑系统的理解。所谓的天赋指的就是自发形成的形式化的思维方式和证明的maturity。这种思维方式可以后天通过阅读训练获得。所以其实数学作为一门证明技术，和其它所有技术同样，天赋走一时，坚持走一世。
If you cannot visualize three space, can you work on algebraic topology? Classical mechanics? E&M? Engineers could not understand the nature of mathematical talent. You said 证明的能力其实来自于对数理逻辑系统的理解, that is only partially true. It is originality and new ideas that makes a great mathematician. Engineers could never understand that. Have you ever created and proved new theorems? Obviously not. Have you ever done maths research? clearly never. All you do is to use other's results and never create new theorems yourself. That is not research maths. The reality is, maths talents (天赋)determine how far you can go. There is never long enough life span to do any significant amount of modern maths. Maths talents have meaning. Otherwise, why would they give Field's Medal to some people and not others. Given that engineering and maths are totally completely different subjects, believe me, the old Chinese saying "隔行如隔山“ is truth. Within different subdivisions of maths, you still have "隔行如隔山“. In the modern age, there is no such person as a Renaissance man in mathematics. So, how well you can do (do as in create new maths) maths is a function of your familiarity with a given subject, what kinds of new ideas do you have in the subject, how mentally free are you on the subject, how many concrete examples you know about the subject, how well you can shoehorn other disparative maths fields to your service, etc. Knowing how to do proofs in number theory does not ensue knowing how to do proofs well in topology and geometry. Hell! Knowing well in algebraic number theory does not mean you know how to do proofs in analytical number theory. Maths is like that. There are no general statements that you can make about any of them. And you try to talk about maths when you are still learning and have never done maths research. I have one experience and I spent at six years getting a PhD in pure maths. I may not be any good in maths but I know what is maths and I know what research in maths means. Remember, it is "隔行如隔山“. This does not mean I know everything you know. However, I understand the mechanism and the philosophy of maths a lot better than you because I had real experience in mathematical research; that is all. My dissertation can be published as a couple page short paper. There! I am much more qualified at talking about mathematics than any of you amateurs.
Let me put it this way. You also said "所以其实数学作为一门证明技术，和其它所有技术同样，天赋走一时，坚持走一世。" That is not true. If you had wrong ideas, no matter how much time you spend on the wrong path, you will never be able to solve the problem at hand. If you are on the right path and got the right idea, the problem is immediately solved. Just look at the career of 张益唐 and you would understand what I mean. He formulated the problems correctly and so he was able to solve an important probelm in analytic number theory. Others did not formulate the problem correctly so they fail. Life long effort has nothing to do with it. Formal system of logic that include the arthmetic sytem are essential incomplete. You also have the Turing undecidability. So whether you can prove a statement in your system within your system is not always decidable. For example, the Riemann Hypothesis. Better be lucky than good. I am not in mathematical logic and not in computer science. Please forgive my misquotes.
非常抱歉， 中文谈数学，我力有不逮也，盖10岁移美64年，吾已大半个ABC矣，中文欠缺，尚祈见谅。
我理解的所谓的天赋就是指你的RAM，大脑并行计算的能力。
@@hockng5610Sir, I read your comment. That is a big decision to work with Pure Math, but math is too damn important. Although difficult math is, I decide to get a master of it.
感謝小姊姊在YT上傳影片🙏
太谦虚了，也许你在清华数学系是个普通学生，但在普通人中已经不“普通”了
The main problem is that in advanced math and statistics, both textbooks and professors skip many crucial steps and tend to overcomplicate the ideas. Otherwise, with proper explanations and examples, many average students can also learn advanced topics.
马上分享给我喜欢数学的儿子们。谢谢Iris
我是个普通人，我更喜欢普通的方法和普适性的分享，谢谢
只要努力，勤奋，不懂就问，多多练习，数学系不是一件难事。
这话说的，有这态度，学啥不行啊。
专注和强大的追求，不计一辈子穷困潦倒都不后悔的追求决心
数学逻辑+运算习惯+耐心
真的，一般人都還沒天才那麼努力。
我台灣清華數學系1982級畢業的
數學系的確有不少天才大神
但是如果有興趣用功努力的人，也會有不錯的成績
天賦不夠的人的確要克服很多資質落差壓力
才能在數學系存活下來
我打比赛的队友也是我的同班同学，数学分析课堂上前半段睡觉后半段听课，到头来给我讲题。。。。等到大四上岸解开心结后，才告诉我们他当年本来是稳上南大，结果跑肚子发挥失常才来了我们这。
找准定位，端正态度。祝福你。
说的很好
无论搞什么，都需要天赋。如何知道有无天赋，那就看你喜欢否。记住，那些参加数学竞赛获奖者未必是个天才。实践告诉我，真正在数学有成就的学者其学习期间的成绩未必很优秀。
另外，数学分析真的不是一般人能吃透的，就连讲课的讲师也不一定都能吃透里面的内涵。
出生就决定智商了，这种肯定是要天赋的，就跟我就记忆能力特别强，但是数学就一般般
学数学不需要太多智商，但是研究数学的的确确是需要智商的。我下铺的同学，他是天生会微积分的，书都没怎么看，微积分核心思想已经领悟了。他是四川省理科第七名。 面对一道题目，智商差5，在花费的时间上相差真是太多了。所以，很多考入清华北大的学生，最终只收获到了一份自卑。一般而言，保送生强于高考的考生强于北京户籍的考生。
哪有這麼多天才，天才背後同樣是熱愛某件事，例如醫學、科學、物理研究，下的苦工不比普通人小，只不過他們是熱愛這件事，所以人們看不見他們的辛苦和苦惱，加上在他們善長的領域確實比普通人快，同樣他們都有不善長的事，沒有人是萬能的。每個人都有其獨特之處，因此世界才變得美妙！
书没看代表什么？我天生就会开大炮呢？谁都不用教，不用学，天生就会
一般而言是如此，但同樣是稅吏，有人成了門徒，有人成了物理長老，有人成了夏洛克，腫麼者也是不公平@@藍小小-g2l 😅
对，就是有人天生就会微积分。天才就是他们天生懂很多，或者一接触到某个领域他们就觉得已经有一种熟悉的感觉
好老师
天才是天生的是可遇不可求
Iris学姐好！能不能请您为我们这些喜欢数学但非数学专业的人规划一个学习路径呢？谢谢啦！
以后出一期视频讲讲~
@@Iris-mj4nr 谢谢学姐！
@@Iris-mj4nr 好期待，尤其贝叶斯统计这块
其实只要把数学教科书写得好上那么一点点，学数学的难度马上就下降一大半。数学难，就难在数学教科书都是垃圾。
個人是讀物理，但在學數學歷程中最困難的是抽象代數，什麼微分、幾何、拓撲看幾個chapter還沒有看一頁抽象代數那麼辛苦...😭
同感。此外还有图论
需要 下一个
两个字:佩服！
四个字:佩服美女！
六个字:佩服数学美女！
我从小学就怕算术，中学数学更别提，就是绕不过湾儿来，学校挨老师说回家挨老爹打，长大了算不过账来。唉，天生对数字不敏感。
相比之下，国外的大学数学系，知识面很广泛，但是难度不深；中国一个相反，面窄，难度太大了，没有这个必要，真的。
不是说要学习日本的匠人精神吗？
數學一個好答案值千金。
顶起。给我家少爷看看，报了数学上了师大实验班，9月10号报到。
加油！
數學系的別稱：
「有字天書」系！
感謝分享~
請問要念統計碩士博士，會需要加強哪些數學底子呢？
概率论和数理统计都是比较重要哒~
@@Iris-mj4nr 非數學本科念統計碩，看統計paper數學多一點的，發現有很多數學符號，覺得本身數學底子不夠深，有點痛苦>
你不去当老师可惜了， 你可以培养很多的人才
数学是一门很有逻辑性，条理性和辩证法的学科，它也是最古老的学科之一。数学的最高形式是哲学。
中国的数学教育一直存在着误区，数学老师总是要把学生们培养成数学家，数学竞赛什么的，久而久之，学生都觉得会数学的都是高材生，神童， 胡说八道！
其实人人都能学好数学，主要是中国的数学教育不是真正学习数学，而是灌输处处竞争的思维，这是非常错误的。
数学老师不是数学家！ 数学家也不会教数学。 数学老师最主要的不是解难题，而是如何把数学概念教清楚，题型讲透， 提高学生的信心和兴趣，这是最主要的，而不是一味地竞赛。
看你说的头头是道， 既然你已经知道发生什么事，有没有想过做点什么改变这一现象？
Iris妹妹，我是華清池造化弄人系的。😂
我崇拜 俄罗斯大神 格里戈里.佩雷尔曼
你居然称自己为普通人，我去！