This article is a personal essay , Share my personal experience in the past few years , Hope to give readers inspiration .
I'm a math fan , I've read a lot of math books , But there is a question that often haunts my mind ： What is mathematical thinking ? Is it possible to prove a mathematical problem only with strict mathematical methods ?
of course , The answer is No . This can be seen from the role and influence of mathematics in many disciplines . Here are some examples of yourself learning , Understanding in life and work , It is reflective , Let's share it here .
1, Why does it fail ?
The expert I learned from my childhood was the essence of learning cattle. , The best of a hundred . Until graduation , I also use this idea as a compass . Now? , There is no denying it , But there are many reflections . By right “ systematicness ” Reflection on the word , Make yourself right “ The best of a hundred ” With a new understanding . It's hard to define systematicness （
It's like how to define a chair
）, Examples are easier to understand . For example, calculus textbooks , It contains a differential , Integral and series , There is a connection between them , There is a cross again . You can't just study differential calculus , I think I've done calculus myself . A more vivid example is to think of the system as a complex machine ：
The machine is made up of many parts , Most of the parts are simple and ordinary , A few components are very complex and sophisticated , It looks very complicated when it's assembled into a machine . Complexity is a typical characteristic of the system .
The essence part usually refers to the complex and delicate parts . however , If the whole machine wants to work properly , All parts must be closely fitted , Normal operation . If a simple and trivial part breaks down , It will affect the operation of the whole system , There are even serious consequences . A lot of major accidents were investigated afterwards. , The cause of the cup is often some minor problems , For example, a rivet is loose , But the consequences are too serious .
In practice , I realized that learning a skill requires systematic learning , Linked to each other , You have to learn both small and big moves . If we only learn a bunch of so-called essence , Most of them are isolated , I'm afraid it's counterproductive to use it . Because the essence can also be mutually constrained and offset. , For example, the world star team may not be better than Brazil . Only flowers , There are no green leaves , It's not spring , There's no beauty .
A friend of mine complained to me , Their company is inefficient because there are too many smart people . In this regard , I don't understand . later , He explained , All too smart , All want to be safflower , Select a good job to do . Some jobs are hard to please , Little oil and water , But essential , As a result, no one was willing to do it . As a result, the progress of many projects is slow , inefficiency . I realized later , Think of the company as a system , We all want the cream. , As a result, no one wants to do the best. , This is an example of a society . of course , It must be emphasized here , My view is not negation. , It's not that the dross is good , Instead, it points out that we must look at the problem from the perspective of the whole and the system .
It's good to gather the advantages of a hundred families , But if there is no good connection , I'm afraid it's just a castle in the air . actually , One's talent , Both ability and time are limited , I feel that it's better than a hundred , It's better to learn a few familiar things systematically . Many academicians are familiar with this major
2 reach 3 Branches , Other branches are not well understood .
How to learn systematically ?
This is a big problem , There is no unified answer . My personal experience is , If you're in a business , Then we should lay a solid foundation , Learning from classic textbooks , Then go to the manual , And continue to practice . If it's not the main business , The time and energy available is limited . that , You can see the popular science books and operation manual written by experts in this industry . An operation manual of the industry is a system . My personal experience shows that , For the problems that perplex laymen , It's a disgrace , In the eyes of experts, it's nothing , It's easy to handle . therefore , Admiration is like the water of the Yellow River , spout eloquent speeches . Later I came across the operation manual of the industry , What should I do with most of the problems that bothered me , It's clearly written in the manual . therefore , I constantly pay attention to collect the operation manual of various industries , It's like collecting math articles .
In the process of collection , I found that among the vast majority of professionals , Not many people read the industry manual . Many accidents are caused by operators' illegal operation , And what is the correct operation , It's clearly written in the manual . of course , There are also problems with some of the operating manuals .
2, Why does learning other people's methods make them different ?
A lot of people have this experience , Learn from others （ Suppose the method is really good ） after , Find out why people use it well , I used it badly , In the end, it became a four different ? So I exclaimed at my lack of talent and so on . It's true that there are ways to use it , But such cases are rare .
On four dissimilarities , I have also reflected , Here is my immature opinion . In addition to the previous point of systematology , The reason for not only learning essence . Another important reason is that , Many people will make local adjustments or modifications when they learn a certain method , so-called “
Internalization ”（ Similar to the national conditions ）. Usually this kind of modification is very small , It's natural, too , So many people underestimate the impact .
actually , Just think about non Euclidean geometry , It only modifies the parallel axiom , The axiom of don't change . But the difference between non Euclidean geometry and Euclidean geometry is very big . For example, in hyperbolic geometry , The area of a triangle cannot be arbitrarily large , Similarity is congruence , The conclusion that there is no rectangle is surprising .
Let's make an analogy , Consider a systematic approach as a theory . The basis or hypothesis of this method is regarded as principle or axiom . When you take a similar “ In line with national conditions ” When modifying a principle , The new theoretical system is no longer the original system .
More insidious is the modification of inferences , Not the principle . Most of the time , Someone thinks he admits to those principles , But in the subsequent practice, some inference was modified （ Or theorem ）P
, The reason is to conform to their own actual situation . Suppose he adopted “ wrong P” As a theorem . Anyone who has studied mathematics knows that , At this point, he has denied at least one axiom , That is, the proposition is deduced P
The axioms of . however , He always believed that he did not deny those axioms . therefore , He didn't even know , He is no longer like four （ stultify oneself ）.
of course , You can't say that a system approach is an axiomatic system , But it can be compared . The problems mentioned above are more common in the social field .
3, Commonly used by cattle people “ Three system method ”
“ Three system method ” It's my own name , It is to consider the same problem in three independent ways , It can increase the probability of winning . The three system method has an intuitive explanation , Consider a three element parallel system , Only three components are broken , This parallel system will not work .
I observed a phenomenon ： A set of methods can be called a trick , A lot of old hands use this move very skillfully , But it's far from sublime . There are some good people , Did not use a move to achieve success , It's a combination of moves . Specifically , Tauren made three subsystems into a total system . His explanation is , Few real masters play hard , Drive yourself to the brink , A real master should be like a butcher .
It may be more intuitive to use quantitative thinking , Suppose the efficiency of the three subsystems is p1,p2,p3, be
The total effective interest rate of the three subsystems is
for example p1=0.7,p2 =0.6,p3=0.7, Then the total system efficiency is ：
From a realistic point of view , This efficiency is scary .
I have also reflected on why there are three ? My personal opinion is ： In reality , The condition of independence is too harsh , And it's hard to verify . The efficiency of the synthesis of the two subsystems is not enough , Three are just right . In theory, four are better , But independence is hard to guarantee . In practical application, independence is often weakened to low correlation .
Many businessmen don't understand P=1−(1−p1)(1−p2)(1−p3)
This formula , But when doing business, they often use the idea of three systems , such as , When he makes a deal , Will find as many reasons as possible , Then categorize them . very seldom , For one reason . The more reasons , The greater the chance of profit , This is already the thought of three systems of law .
4, Practical redundancy idea
Redundancy is a very common and practical concept in engineering , Similar margin and backup . for instance , You design a bridge , What is the estimated load capacity of the vehicle in normal operation 300 ton , But when it's built, it's the bridge
500 Tons of capacity designed , This is the idea of redundancy in engineering （ Margin of safety in similar investment ）.
I have reflected on why redundancy is so widespread in engineering . Now two reasons have been found ：
1, Accuracy considerations
. You'll find out if you've been exposed to practical problems , The real world is much more complicated than books , Then consider the operational factors , Accuracy is hard to do , Or it can't be done basically . Most projects rely mainly on experiments , Not theory . Theorists are often embarrassed in practice .
2, Stability considerations
. In practice , Stability is often more important than accuracy . Many projects need to consider normal loads , Factors such as errors and extreme cases . It can't be said that if the probability of extreme events is small, it will not be considered , Because the consequences are too serious . Expanding the safety space can increase stability , This idea is natural , It can also be operated .
In theory, the more redundancy, the better , But the costs naturally go up . There are many problems with bridges in China , But some old bridges are quite similar “ The bridge is strong ”, One of the important reasons is that these old bridge building materials , High redundancy .
of course , If every component in a system has a large redundancy , The system may be too large to work . therefore , From the system point of view, where is the redundancy greater , Where is the redundancy less , It's knowledge and art .
The above four questions are raised here , I hope to arouse your reflection . Can't understand mathematical thinking in a narrow sense , We should learn and use flexibly , Transplanting flowers and trees is also an innovation . It doesn't have to be profound to think in mathematics , Ordinary mathematics can also shine the brilliance of wisdom .