Looking back on my way of learning advanced mathematics, I found that I had gone through the detours that everyone had gone through and did a lot of unnecessary math problems. At that time, I did about one-third of Jimmy Dovich's Mathematical Analysis Problem Set (this set has more than 4,000 problems), but I may not do well in the exam.
Suddenly one day, I decided not to do it, and I got the results of the mathematical analysis exam. At this time, I feel a kind of relief. After doing the time, the basic concepts are clearly understood and the results are not bad.
Later, when I was a graduate student, I took a junior math class in the department of mathematics and found that I had made a complete mistake before, not because the teacher said something wrong, but because I didn't learn what I should have learned. In fact, I don't need to know how to do those questions at all. If I encounter it in the future, I will either look up reference books or use computer tools to solve it.
The key point is that before I was 20 years old, I didn't understand what kind of vision I should use to understand mathematics and look at those concepts and methods in mathematics. Later, when I realized that advanced mathematics is essentially a dynamic description of trends and an abstract expression of various correlations, I thought it was extremely easy to review calculus. Unfortunately, when I first went to college, no one guided me like this.
The world of mathematics, to a great extent, can be regarded as the highly abstract result of our real world. Its concept is the concentration of various objects in our lives, and its laws are abstract expressions of many laws in our lives. So after having some life experience, I look back at math books, and sometimes I sort out some fragmentary ideas and clues. Today, I will share my nine experiences with you:
1. There are finite and infinite.
We spent a lot of time discussing infinity, with special emphasis on the inapplicability of finite life in the infinite world. This feeling sometimes makes me feel small and short-sighted. Many times, people at the bottom can't understand the ideas of the top.
We often say that "poverty limits imagination". In fact, poverty does not limit imagination. The poor people will entertain foolish ideas, but they don't have much experience in some dimensions, which makes people think that Granny Liu has entered the Grand View Garden.
Ten years ago, there was a book about the Rothschild family and the Federal Reserve of China. After reading it, some of my investment friends said that the author must be a poor man. He never had much money and didn't know where to put it in the world.
What the author lacks is not imagination, but the world imagined out of thin air is completely unreliable. Thinking will not increase one's wisdom, but like Socrates, knowing that one's knowledge is infinite, while the unknown world is infinite, will be closer to the truth and it will be easier to improve one's cognition.
2. Static and dynamic
Our world needs to be still. Without stability, we can't stand on our feet. However, many times, we need to make dynamic assumptions when doing things, just like a football match, no one gives you a chance to stand still and shoot calmly except for the penalty kick. From elementary mathematics to advanced mathematics, we should look at mathematics from static numbers and isolated formulas to a dynamic trend.
For example, when we talk about the concept of infinity, I repeatedly stressed that it should be regarded as a dynamic trend, not a big number. I also talked about it specifically, x? This function, in an infinite world, is much larger than 10000x. China has a saying, don't be too poor to love the rich, which is actually very reasonable. We should look at a young man according to his growth trend, not how much money he has now.
When starting a business, many people like to join in the fun and find outlets for fear of missing opportunities. I told them that if that time window is only a few months or half a year, it is not an opportunity at all, just speculation. This is a static opportunity. The real megatrends always last for more than a decade or even decades, and it is not easy to miss them. After decades of compound growth, it is more profitable than any speculation. This is a compliment to those who see the world dynamically.
3. Smart and clever
The relationship between static and dynamic reminds me of this topic.
Some people think that if you are good at math, you can settle accounts, and you can be smart and not suffer. But the result of being too smart is that you focus on immediate interests and can't see long-term interests, so you are not smart.
For example, in terms of investment, many people like to find opportunities to buy low and sell high to earn the difference. Sometimes they succeed, but usually they miss more than they succeed. To make matters worse, these people will never grasp the opportunity of long-term growth. In fact, with the passage of time, any upward and sustained growth trend may rise a lot and quickly.
4. Reality and fiction
Many tools in mathematics are based on imaginary concepts that do not exist in the world, but they are very useful when applied to real society, such as irrational numbers. People should not only have visual thinking, but also abstract thinking, and can understand irrational numbers through abstract thinking, so it will be convenient to learn physics, do signal processing and engage in control systems.
People are particularly good at creating virtual concepts. In fact, our life today is inseparable from all kinds of virtual objects as physical media. For example, the wealth in the world is real, but the money to measure them is actually empty. You can't buy things with real wealth. Everyone just transfers the money in the bank account from one virtual space to another.
Not only is money virtual, but the virtual meaning of many real goods is greater than the actual meaning. It is estimated that Moutai in one's hand changed hands as a gift for more than ten times before it was finished. It's like those imaginary concepts in mathematics, not just alcoholic drinks. Without them, real-world problems will be difficult to solve.
Save money and make money
I once talked about money in the Letter from Silicon Valley, saying that no one can make a fortune by saving money, and making a fortune depends on making money. Saving money today is actually devaluing. Although the CPI published by various countries seems to be low, the CPI does not include the rise in house prices. When we introduced the exponential function, we said that if the purchasing power shrinks by 10% every year, the purchasing power will be reduced by half in a few years.
However, as long as young people work hard and increase their income by 20% a year, they can still do it. In fact, in the past 30 years, the annual salary of Beijing computer graduates has increased by 17% on average, and the growth rate of experienced excellent practitioners is even faster.
As we said last class, multiplying the trend of becoming infinitesimal with the trend of becoming infinitesimal will clear it, keep it unchanged, or enlarge it, depending on whose rank is higher.
Inflation, rising house prices and other factors are the forces that devalue wealth to infinity, not artificially controllable factors; But on the other hand, the growth of income is the power to make wealth grow infinitely, and ultimately it depends on which power is greater for everyone. Therefore, wealth is earned, not saved.
6. intuition and logic
Our intuition is often right, but this is only in the world we are familiar with or can perceive. Many laws of the world are inconsistent with our intuition, such as Zeno's paradox and Becker's infinitesimal paradox, because our intuition is inconsistent with the laws in the infinitesimal world.
Logic can help us analyze things that are invisible, even things that don't exist. In the course, we quoted the example of Galileo. He decided that Aristotle's conclusion that the heavy ball landed first was wrong, because he found the logical error of this statement. There are countless such examples in life.
7. Concepts and expressions
Today, communication plays an important role in our life. It is important to express one thing clearly. Many times, we need to explain it through the metaphor of images that we can understand each other, just like when we talk about the concept of limit, we use "getting closer" to describe it.
But on many occasions, such an image description is not enough, and it needs to be expressed in extremely strict language. The language of mathematics is one, so is the language of law. More generally speaking, any technical term appears for this purpose. To do things professionally, you need to master professional terms.
8. Friends and rational opponents
Many times, small achievements depend on the help of friends, but to achieve amazing achievements, you need a rational opponent. In the history of mathematical development, Zhi Nuo, Becquerel and Russell, whom we will talk about later, all played the role of anti-angle. It is they who find fault with the imperfect mathematical system and make it perfect.
When we talked about the financial crisis before, economists seemed to have no warning. In fact, when everyone agrees with the same idea, there is no loophole in the system at all. Therefore, in our work, even if we don't like the opinions of rational opponents, we should respect their opinions, because those seemingly different opinions are exactly what we need for progress.
9. Honor and wealth
Science has no patent, because the laws discovered by scientists exist objectively, but they have discovered them, so it is difficult for science to bring wealth directly. Technologies are patented because they use science to change the world. They cannot be confused.
The greatest praise for scientists is honor, so today scientists are fighting for who is the first to discover a certain law, not to keep it secret.
The story that the solution of cubic equation was discovered was told earlier. Among them, Philo and tartaglia tried to turn science into an exclusive achievement, which won't work. Today, most mathematicians know the contribution of cardano and Ferrari to solving equations, but little is known about Ferro. The reason is that the former told the world about these methods.
For those who engage in technology, turning science into a product that changes the world may bring wealth, which is their pursuit. Therefore, technology is exclusive. Although we say that science has no national boundaries, technology has always had national boundaries, not only national boundaries, but also enterprise boundaries.
For a person, what he needs to know is what he wants.
I studied calculus more than 30 years ago, and I can't remember many specific contents, but after learning it, people's way of thinking has changed. So from the purpose of general education, I should be considered as having achieved it. Today, I would like to take this opportunity to share with you my inspiration from mathematics, hoping it can be used for your reference.