Turn over, King of Mathematics —— Notes on Devil Mathematics

This week, I continued to read the thinking training module with the theme of "mathematical thinking". Intensive reading is "Devil's Mathematics" written by Jordan Allen Berg, a professor of mathematics at the University of Wisconsin.

When it comes to math, many people may frown, as if they had returned to the afternoon when they put down their pencils. When they pick up, they can't understand the math teacher's deduction, which really makes people anxious and depressed. What we learn in school seems to be just a bunch of boring rules, laws and axioms. We studied trigonometric function in middle school and calculus in college. However, how many cotangent functions or indefinite integrals can most adults use in their daily lives? Then why should we learn these seemingly unquestionable mathematics handed down by our predecessors?

In this book "The Devil's Mathematics", the author abandons complicated technical terms and uses anecdotes, basic equations and simple charts in the real world to tell the charm of mathematics and how to acquire the skills of solving problems in life with mathematical principles. Jordan? Allen Berg believes that mathematics is one of the most important basic sciences for human beings, and it is also the most useful thinking tool in life. Mathematics can help us better understand the structure and essence of the world and should be put in everyone's toolbox with ideas. Especially in the current era of big data, we need the power of mathematical thinking to better solve problems and avoid fallacies and mistakes.

At the beginning of the book, the author puts forward a view that mathematical knowledge can be divided into four quadrants, and we only need to pay attention to one of them.

The first quadrant is simple and simple mathematical knowledge. These mathematical knowledge looks complicated, but in terms of understanding difficulty, it is actually very simple.

The second quadrant is complex but simple mathematical knowledge. These mathematics need some problem-solving skills and more care, but these are still simple mathematical knowledge. We spent a lot of time learning problem-solving skills at school, which actually didn't help us understand the beauty of mathematics. On the contrary, it may make us lose interest in mathematics.

The third quadrant is complex and profound mathematical knowledge. This is an area of interest to people who specialize in mathematical research. If you want to enter this field, you need a certain mathematical talent, and you must be very devoted, make hard efforts and pursue your life. We ordinary people may only glance at the door and don't know what the mysterious world is like inside. This knowledge is for us ordinary people to worship.

What is most worth learning is the mathematics knowledge of the fourth quadrant, that is, simple and profound mathematics knowledge. Simple, because this is all introductory knowledge; Profound, because this knowledge is against our intuition, or requires us to reason more carefully. For example, the understanding of randomness, causality and regression all belong to this category. Here, the author gives a story of "disappearing bullet holes": If it is necessary to armor the fighter, should it be added to the fuselage with dense bullet holes or to the engine with fewer bullet holes? During World War II, Abraham Wald, a member of the U.S. Statistical Research Group, thought that the place where armor should be installed should not be the fuselage with more bullet holes, but the engine with fewer bullet holes. Why is this happening? Let me start with a theoretical hypothesis. Theoretically, the probability of being shot in all parts of the plane should be the same. So, why are there more bullet holes in the fuselage of the returning plane than in the engine? In other words, where are the bullet holes that should have been in the engine? Wald believes that this is because all the planes whose engines were hit crashed. The plane that came back, although there were many bullet holes in the fuselage, was able to withstand the blow, so it could return safely. For example, if we go to a field hospital to count the wounded, you will find that more soldiers were shot in the leg than in the brain. Soldiers shot in the brain rarely survive, and soldiers shot in the leg are more likely to survive. This is the so-called "survivor prejudice", that is to say, we only see the survivors, but we don't see those who have failed and died.

Therefore, the focus of this book is to introduce how to use the fourth quadrant mathematical method to analyze and solve problems in daily life. The author uses entertaining cases and methods to help us re-understand five concepts related to mathematics, namely linearity, reasoning, regression, existence and expectation.

The best way to predict the future is to start with certainty. Economists often have to make predictions. There is a joke that economists like to do forecasting the most, but they are not good at forecasting the least. It is relatively easy to predict the short-term or long-term, but the most difficult thing is to predict the medium term.

Because the simplest method is linear extrapolation, there will be greater certainty in short-term and long-term forecasting. The method of linear extrapolation means that what happened today will happen tomorrow. In the real world, there are indeed many linear changes or similar linear changes. For example, the aging of population, the growth of information and the irreversible development of industrialization and urbanization in China. In linear trends, we can also distinguish between hard trends and soft trends. Hard trends are trends that you can measure or perceive; Soft trends are speculations that you seem to see and predict. For example, after the end of World War II, a large number of American soldiers returned to China, and there was a baby boom, so population data is a hard trend that we can see and predict; However, people thought that business orders would be temporarily reduced and the economy would decline after the war, but the expected economic recession did not happen, which is a soft trend that is more difficult to predict.

Relatively speaking, the short-term and long-term technical difficulties are relatively small, while the medium-term prediction is more complicated. Not surprisingly, there will be more fluctuations in the medium term, and the inflection points of these fluctuations are difficult to predict. For example, even if it is known that there is a bubble in the stock market, it is difficult to predict when the bubble will burst. Even if you know that the stock price is undervalued, it is difficult to predict when it will rebound.

Therefore, we must be cautious when predicting the medium-term trend. When forecasting the medium-term trend, there is more noise and the law is more complicated. We will encounter fluctuations and cycles. So although the linear trend is the simplest and most intuitive, we should remind ourselves that not all phenomena are linear trends. Blind application of linear trends sometimes leads to very absurd conclusions.

Give another example. When discussing Trump's tax cuts recently, the media often mentioned the Laffer curve. Laffer curve says that with the increase of tax rate, the tax will increase at first, but if the tax rate is too high, it will affect people's enthusiasm for work, and the tax rate will decrease, but the tax will decrease. Is the Laffer curve right? Mathematically, the Laffer curve may be right. Laffer curve points out that the relationship between tax rate and tax revenue is not linear. It seems reasonable to explain the relationship between tax rate and willingness to work from common sense. But why do most economists scoff at the Laffer curve?

Because the Laffer curve lacks a solid theoretical basis. First of all, the tax rate is not necessarily the most important factor in determining government tax revenue. A more useful way to increase taxes may be to improve tax efficiency. Moreover, after tax reduction, people's enthusiasm for work may not necessarily increase. After all, the factors that affect people's work enthusiasm are very complicated. There are two factors that determine our work enthusiasm, one is the basic factor and the other is the motivation factor. Money income is only the basic factor, and the driving factors include challenge, sense of identity, sense of responsibility and personal growth.

Most economists are not saying that the shape of the Laffer curve is wrong, but that we can't simply use things when looking at tax reform. At present, the tax rate of high income in the United States is much lower than most of the 20th century. In other words, few economists believe that the United States is now in the downward region of the Laffer curve.

If we simply evaluate the effect of Trump's tax cuts, the impact of Trump's tax cuts on the US economy may not be as great as some friends think. First of all, Trump's tax cuts did not happen when the US economy was relatively depressed. Economics tells us that only when the economy is depressed, the stimulating effect of tax reduction on economic growth is more obvious; Second, Trump's tax reduction policy obviously has the color of "robbing the poor and helping the rich". This will aggravate the gap between the rich and the poor in the United States and further divide the already torn American society; Third, if we cut taxes without reducing government spending, it is likely to lead to increasing debt pressure in the United States.

The United States, on the other hand, allows multinational companies to return their overseas profits through tax cuts. With the pressure of capital outflow, the return of RMB to the depreciation channel, the pressure of passive tax reduction and the possible passive contraction of asset price bubbles, how long can China remain "immune"? I won't say too much this time, but I'll talk about it later in the reading module about "the game of big countries" (allow me to charge first and then share, covering my face hhh).

One day, you suddenly received an email from a Baltimore stockbroker recommending a stock that promised to rise in a week, but you ignored it. For the next ten weeks, he recommended a new stock every week, and you were surprised to find that all the stocks he predicted actually went up. So, in the eleventh week, will you choose to buy his stock? This is the story of a famous Baltimore stockbroker. However, you may find it amazing, even a miracle. Baltimore stockbrokers have guessed the rise and fall of stocks for ten times in a row, but this is a scam with hidden probability behind it. Knowing the method, the stock market idiot can easily realize it, because there is more than one recipient. Only 10240 emails need to be sent in the first week. Half of the recipients' emails predict the stock rise, and the other half make the opposite prediction. Next week, the following recipients will not receive the mail, and the remaining 5 120 people will continue to receive the mail with different forecast scores in two batches, and so on. By the tenth week, only 10 people will receive the email with accurate prediction for ten consecutive weeks. what do you think? Therefore, when we do mathematical reasoning, we should take this story as a warning: we must be careful in the analysis of big data. There may be more than one root of quadratic equation, and the same observation may produce many theories. It is not the truth that leads us astray, but the omission of some assumptions in reasoning.

The chapter "Inference" also mentions two interesting concepts: "null hypothesis" and "significance test".

The null hypothesis is that the hypothesis has no effect, or the hypothesis has no effect at all, or the hypothesis has no correlation. When we do research, we should start with the null hypothesis, and then do experiments or collect data to see if we can overturn the null hypothesis. How to overturn the null hypothesis? This requires significance test, which is actually a fuzzy reduction to absurdity.

The idea of reducing to absurdity is that in order to prove that a proposition is incorrect, we first assume that the proposition is true, and then see if we can draw any conclusions. If this conclusion is obviously wrong, then the hypothesis is a false proposition. In other words, we assume that h is true. According to H, a fact F does not hold, but F does, so H does not hold. However, in most studies, it is impossible for us to draw such a definite conclusion, so the significance test appears.

We assume that H is true, and the possibility of getting an O result based on H should be very small. Unfortunately, we have seen the event O happen, so it is very unlikely that H will be established. For example, let's assume that Mr. S works hard. If he works hard, the chances of finding him beating the glory of the king at work will be very small. However, we found that this man did have an important meeting, and he was still hitting the glory of the king. What does this mean? Explain our original assumption, that is to say, the assumption that he works actively and conscientiously is probably wrong.

So the significance test can be divided into four steps:

1, start the experiment; 2. Assume that the null hypothesis holds; 3. Observe the probability of the occurrence of event O in the experimental results, which we call P value. P value reflects the possibility of zero hypothesis; 4. If the p value is very small, we think that the experimental results are unlikely to meet the zero hypothesis. You can judge by this reduction to absurdity that the conjecture you originally wanted to test is statistically significant. If the p value is very large, we have to admit that the zero hypothesis has not been overthrown.

Of course, the significance test also has potential pitfalls to pay attention to:

How small are the values of 1 and p that are significant? There is no clear boundary between significant and insignificant.

2, we can't assume that one factor will have an impact. If we want to draw influential conclusions too much, we may manipulate the experiment.

3. Don't misunderstand the meaning. Many scientific terms are misleading, and the word salience is a typical example. Distinguish between "salience" and "effectiveness" (the key point of paper writing is get√).

Research shows that the probability of tall parents giving birth to tall children is not 100%. In fact, the height of parents and children is affected by the regression effect. Everything that is affected and random on the vertical axis of time follows this law. As long as the data is large enough, human height or IQ tends to be average, which is the familiar "law of large numbers". For example, chestnuts, the annual birth rate of same-sex babies in big hospitals will be closer to 50% than that in small hospitals. what do you think?

The principle of "the minority is subordinate to the majority" is simple and seemingly fair, but only when two viewpoints are involved can the best effect be achieved. As long as there are more than two views, most people's preferences will be contradictory. So it can be said that public opinion does not exist at all. More precisely, public opinion will exist only if most people agree with it. If we act according to logic, we often need to go against the opinions of the majority. For politicians, it is their duty to make rational use of inconsistent public opinion just to satisfy most people.

The purchase value of lottery tickets is different from the winning value. The purchase value is the amount you spend on buying a lottery ticket, while the winning value is the real value of the lottery ticket after the introduction of probability theory. We can express it by expectation. The expected value of lottery tickets is not worth buying only if it is lower than the purchase value. If it is higher than the purchase value, the lottery ticket is worth buying when your purchase amount reaches a certain amount.

Mathematical thinking is actually an instinct of ours, and it is actually the same origin as language. Our ancestors used to live in trees and often needed to jump from one branch to another. They need a good sense of three-dimensional space. When they reach the open grassland, they need to judge the distance, which requires a two-dimensional sense of space. As the living environment became more and more complicated, our ancestors began to have the consciousness of judging causality. However, why doesn't natural mathematical thinking solidify into our daily thinking? Why do most of us still think math is too difficult? The key here is abstraction.

Abstraction is the most powerful tool in the mathematics toolbox. Mathematicians try to abstract whenever they get a chance. In the end, they will completely forget the real world and focus on abstract definitions and concepts. Therefore, the author will say that there are two moments when children begin to give up learning mathematics, one is when they are exposed to fractions, and the other is when they are learning algebra. This is a two-step abstract process. Abstraction can be divided into four levels: seeing is believing, thinking is believing, seeing is not believing, thinking is not believing. Finally, "thinking empty" is the level of mathematical thinking. Mathematical objects are completely abstract, and they have no simple or direct connection with the real world. Mathematics is an abstract level above abstraction. For example, we first come into contact with the commutative law and associative law in addition and subtraction, extending to multiplication, geometry, function, set and matrix. If we study mathematics, we will also consider when groups can satisfy the commutative law. The essence of mathematics is the same. This is a science about patterns. Some patterns are relatively simple, while others are relatively complex. Complex patterns are only patterns of patterns, even patterns of patterns, so we begin to be confused. We can think of mathematics as a magnificent building built with Lego bricks. Although it looks complicated, if you look closely, you will find that it is assembled from simple modules. The essence of mathematics is that simple things are complex, while complex things are actually simple. This goes back to the theme of this book. Why should we learn simple and profound mathematics knowledge?

After reading the Laffer curve, we can understand the relationship between tax rate and government. Only by knowing "linear centralism" can we understand that "proportional conversion" is so absurd; "Law of Large Numbers" is a merciless hand; The "pie chart bigger than the plate" reflects the "true but inaccurate" digital dislocation ... These common mathematical knowledge tell us that we must pay attention to the occasions where mathematics appears. Without collateral, mathematics will become your tool, political vote, market data and profit report. This and that, they are often wrapped in complicated and overlapping numbers. What can break them is the insight cultivated by mathematical thinking. This is what the author wants.

Above.