Many times, we have been unconsciously applying mathematical knowledge, but it is because we don't understand mathematical knowledge that we unconsciously move towards the opposite situation. For example, we sometimes think something is valuable, so we take it for granted that the more the better. This is a kind of linear reasoning. And many times, linear thinking is often a one-sided way of thinking. For example, government taxes, many people think that government taxes are directly proportional to the tax rate (that is, linear), so they think that raising the tax rate will increase government taxes. In fact, too high a tax rate will reduce people's enthusiasm for work, make the total value smaller, and then the tax revenue will also decrease. Therefore, the relationship between tax rate and tax revenue is nonlinear.
Nonlinear thinking shows that the right direction depends on your current position. As shown in the above figure, if the current tax rate is to the left of the highest tax rate, then the tax rate can be increased. If the current tax rate is on the right side of the highest tax rate, then lowering the tax rate is a good choice.
For linear thinking and nonlinear thinking, we won't care too much about the difference between them before we know these mathematical knowledge. But like the following example, we know how shocking mathematics is.
A stockbroker sent you an industry message on his own initiative, revealing the news that a stock is about to skyrocket. A week later, you find that the stock has really gone up. The following week, the broker predicted that a stock would fall, but it did not come true. For 10 week in a row, all the forecasts made by brokers about the rise and fall of different stocks have come true. So, will you trust this broker and pay him a lot of money to manage your funds?
I think many people can't wait to hand over their property to this agent. However, if you tell a story from the perspective of an agent, the situation is very different. In the first week, brokers issued 10240 pieces of industry information, half of which predicted that stocks would rise and the other half predicted that stocks would fall. In this way, in the second week, 5 120 people who received the wrong prediction information were ignored, and 5 120 people who received the correct prediction information, including you, will continue to be divided into two groups. One group received the message that the stock would go up, while the other group did the opposite. So repeatedly. By the week of 10, there will be 10 lucky people who receive the correct forecast from the securities firm for 10 times in a row. No matter what the stock market is like, this result will not change.
Even if the stockbroker is a layman who knows nothing, eventually 10 people will think that the stockbroker is a genius after receiving 10 correctly predicted stock information, thus defrauding a lot of money. This scam works because it tells you not false information, but clever use of mathematical principles to make you form a wrong conclusion. The case of 10 consecutive guesses is a small probability event, but it does happen if the sample is large enough.
Small probability events are not uncommon. There is little possibility of being struck by lightning or winning the lottery, but such things keep happening. This is because the world has a large population and many people buy lottery tickets. If the vision is wide enough, most coincidences are not surprising.
It is precisely because many people do not understand the mathematical knowledge of probability that they believe that buying lottery tickets is predictable and they are at a loss in one scam after another. It can be seen that mathematical knowledge should indeed have a decisive position in life. This book gave me a lot of inspiration, let me re-understand mathematics, and also let me have a strong interest in mathematics.