Written by | forest
1 Non-mathematics majors are not suitable for studying mathematics
Are non-mathematicians suitable for studying mathematics? If we ask people from the class, we will find that the vast majority of the answers are that non-mathematics students are not suitable for studying mathematics. The reason is the lack of training in mathematical foundations. For example, if you start reading graduate school classes directly, you will not even understand the textbooks. What do we think of this realization? There is some truth to their view, but it is too one-sided. Non-mathematics majors to learn mathematics, his motivation for his interest will be relatively strong, this is one aspect. On the other hand, when you haven't even started thinking about anything, giving you too much basic training is a good thing and a bad thing at the same time.
The good part is that you have mastered some mathematical language and understand some basic logic. The bad side is that it will limit your thinking in a fixed way. For example, we don't know what mathematics wants to do, we don't know what mathematics wants to do, we don't know what mathematics is all about, and then we are told that this problem should be solved this way, and this solution is correct. If you ask why you're so solved, a lot of times you won't get any answers you want. Non-math students may be more open-minded when they look at many problems.
Figure 1
For example, if you talk to someone majoring in mathematics, his first reaction may be whether your logic is right or not, and whether there is a problem with your details. This was his first reaction. If you talk to people in other professions, they may first focus on what your idea is, what problem you want to solve, and whether your idea can solve these problems, without sticking to some very detailed things. If you're right, strictness is just a means. Rigorous proofs can guarantee that we are not wrong, but strict proofs do not guarantee that we have the idea to solve the problem.
In fact, students from the class, due to the influence of today's mathematical education, formal thinking will be very serious. Let's take an example of n-dimensional Euclidean space. Historically, people first discovered real numbers, that is, one-dimensional European-style spaces, and then generalized to high-dimensional n-dimensional spaces according to the needs of reality. But from a mathematical point of view, n-dimensional space is a more basic thing, and one-dimensional space is just a special case where n equals 1. So when we talk about it, we can start with the European space of n-dimensionality, and then make n equal to 1 to get the special result of one-dimensional space. And to put it this way, the problem is even more profound.
If this is such thinking, why does mathematics arise, what kind of problems mathematics wants to solve, what kind of connection between abstract mathematics and concrete practical problems, these very important questions will not cause thinking, for example, the physical space of reality is three-dimensional, with the theory of relativity plus time to study four-dimensional space-time, why should people study general n-dimensional space, where does its power come from, and why is its application correct?
From this question, on the one hand, I want to tell everyone that the mathematics major itself has its advantages in learning mathematics, and it is necessary to know where its advantages are, but it is also necessary to know its own shortcomings and find ways to make up for these deficiencies. On the other hand, if there are non-mathematical majors who are committed to the study of mathematics, then there are also his advantages, but there are also his shortcomings, but also to give full play to their own advantages, while finding ways to make up for their shortcomings through certain training.
2 Origins of probability theory
What problem does probability theory solve? Why do we humans engage in this discipline?
Figure 2
The emergence of probability theory in history stems from the allocation of gambling money, and from this time there have been probability-related problems, but the real formation and development of probability theory is not simply due to these problems, and the greater source of motivation is production practice. Let's look at a light bulb manufacturer. Customers who want to buy light bulbs must be concerned about how long the manufacturer's light bulbs can burn and whether the quality of the product is guaranteed. If it's a specific light bulb, how long it's going to burn, we can let it burn all the time and see when it burns out. The problem is that after testing, the bulb is scrapped. In real life, we can't solve problems with such a line of thinking. We must find a way to solve it by other means and give convincing conclusions.
If for the same material, the same production process, the same batch of light bulbs produced, we think that one of its qualities is probably stable. How is it called probably stable? That is to say, this batch of bulbs are all taken over and burned, some of which may burn for a particularly short time, and some may burn for a particularly long time, but for the vast majority of light bulbs, the time it burns will be stable near a value.
Based on this idea, if 20,000 light bulbs are produced at a time, although 20,000 cannot be burned, it is no problem to smoke 100 out and burn them. After burning, 10 of these 100 may not even burn enough for 200 hours, and there may be 15 directly burned for 8,000 hours, but the vast majority of the remaining ones are about three or four thousand hours. At this time, it is said that this batch of light bulbs is about a relatively stable burning time of 4,000 hours. For example, more than 80% qualified, then for such a production process, it is to reach a passing line, of course, need to improve, but can continue to use. When you go buying light bulbs, you may also accidentally buy bad ones, but you won't say that I bought 100 in a row and all of them are bad. This is a class of problems that we need to solve with probability theory.
The first very basic step in the formation of the discipline of probability theory is the emergence of the concept of probability. There are many problems in reality, and although every examination will produce a definite result, we cannot deal with the problem in this way. We found that when considering multiple times or more, it has a stable thing, which is the concept of probability. Or the concept of probability, which abstracts this stability. But abstraction is not yet mathematics, it is still just an idea, after abstraction, there must be a mathematical language to express it, there are mathematical methods to deal with it.
Other problems were also identified during the study. For example, if you study different problems, the patterns it satisfies are not the same. Grab a small ball from a bag, grab it out and put it back, and you will find that different probabilities have different patterns. In fact, a generalization is a type of probability problem, and this concept is used to solve this type of problem. After these basic concepts and theories are formed, they abstract some numbers, some quantitative relations, and then need to do some calculations of the corresponding probabilities.
We have just said that the allocation of stakes has led to the accidental occurrence of probability theory, and the real formation and development of probability theory should be attributed to the practice of production. Everyone thinks, let's first clarify the historical origin and basic understanding of probability theory, how to calculate the probability of a specific general type, why is it calculated, and so on, little by little, is this kind of learning idea more clear and efficient?
3 What is to be studied in advanced mathematics
The core of higher mathematics is calculus, so what problem does calculus come from? The answer is movement. Before the 15th and 16th centuries, mathematics as a whole was in the period of constant mathematics, that is, the period of elementary mathematics, and gradually transitioned from the budding of capitalism to variable mathematics. In order to rebel against religious theology, the pioneers of modern natural science, represented by Galileo and Descartes, began to study a large number of movement problems. You say, for example, they study celestial motion, and theology says that celestial bodies move along a circle, because God is pushing. And why is it a circumference? Because God is perfect, and the most perfect curve is the circumference, the celestial bodies move along the circumference. Through certain studies, people have found that this trajectory is not a circumference, but an ellipse. Then through Newton and others to make a summary, tell people that the movement of celestial bodies is not driven by God, but the influence of gravity, which is the effect of the laws of nature.
Figure 3
Later, people found that the study of movement also has its productive significance, and the study of movement is the realistic requirement that the development of the whole society puts forward to people.
Studying mechanical motion from the perspective of physics is a situation we are familiar with, and chemistry will also study motion from a chemical point of view, such as considering the splitting and binding of chemical bonds at the molecular and atomic levels. Different disciplines have different research objects and study from different perspectives, but they all involve a common type of problem, that is, the problem of quantitative relations. Mathematics is precisely to solve this problem, so it puts forward a realistic need for mathematics. That is to say, one needs to study motion from and only from the point of view of quantitative relations, which is exactly the problem that mathematics is trying to solve. When social development puts forward the need for research movements to people, it gives mathematics a corresponding task and a historical mission.
In order to accomplish such a task, the discipline that undertakes such a historical mission as the study of movement from quantitative relations is calculus, and calculus comes from this place, and that is what it is going to do.
When mathematics entered the era of variables, its research objects and research problems were different, and its research methods also changed. Analytic number theory has emerged in number theory, and analytic geometry and differential geometry have also emerged in geometry.
4 Descartes' contribution
Before the advent of calculus, there was a mathematician who had to be mentioned, Descartes. What do you think of Descartes' mathematical achievements? In other words, how to evaluate the significance of coordinate systems?
Without the Cartesian coordinate system, what would mathematics look like today? Numbers and shapes are not combined, nor do they produce variables, and we may always be in the period of elementary mathematics.
Figure 4 Cartesians
The most immediate significance of the establishment of the Cartesian coordinate system is the birth of the discipline of analytic geometry, which contains a change of concept, that is, the introduction of a dynamic point of view. People no longer think of a curve as simply a curve, but also as the trajectory of a point, which is said from a morphological point of view. From the perspective of numbers, it will be found that numbers can change, and there can be a range of changes, that is, the concept of variables arises. In the era of variable mathematics, if there are no variables, even variables cannot be portrayed, and there is no stage for natural development, which is the real mathematical meaning of the Cartesian coordinate system. All variable mathematics can be seen as originating from this place in Descartes.
His work did not prove any profound theorems, but he engaged in this thing that opened the entire era of variable mathematics. Thus, Descartes was a world-class mathematician, a great mathematician of the same level as Newton and Leibniz.
5 The birth of the function
After variables, there are functions, and the concept of functions was first defined by Euler, but mathematicians have been studying functions before that. Functions are actually the attachment of variables to variables, which are specific quantities in other disciplines, such as speed, current, etc., but in mathematics, variables only retain the characteristics that the number can change within a certain range. Functions are the object of study of calculus, and the main thing here is the continuous function. So what kind of methods do we need to do research?
The concept of limits is not intuitive, because limits come backwards. For example, if a person goes from place A to place B, the extreme view is that I first go to place B, and then look back at where I have gone in 1/2 of the time and where 1/4 of the time has gone. The limit is looked back when this exercise process is completed, and it is not the same as people's intuitive way of thinking. Newton had the idea of limits, but he wasn't very sure himself, and he still wanted to use it in his later years, but after all, calculus didn't follow Newton's thinking. At that time, Newton and Leibniz competed for the priority of the invention of calculus, which led to the division of mathematics in europe into two parts, one part of the Group of British mathematicians represented by Newton, and the other part of the mathematicians of continental Europe represented by Leibniz. The British faction did not go down, although there were also McLaughlin, Taylor and others, but what these people did obviously could not be compared with Euler, Lagrange and others.
Figure 5 Leibniz (left) and Newton (right)
If you want to study movement, y changes with x, the natural way is to examine how y changes when x changes a little. This is the infinitesimal method of our calculus study, and along this line the entire development of calculus before the 19th century was formed. After the birth of calculus, it can be used as a tool to study differential equations, and it can also use differentiation to study geometry (differential geometry). In the 18th century, the solution of ordinary differential equations has been studied relatively clearly, and physics at this time is still mainly in the era of mechanical research, so the partial differential equations are mainly some simple wave equations represented by the string vibration equation. In the 19th century, physics entered the era of research in electromagnetism and thermal science, so the development of partial differential equations in the 19th century was represented by the heat conduction equation and laplace equation, and solutions such as the Green function method were also born at this time.
From these things we can see where calculus comes from, what problems to solve, how to translate some of these ideas into mathematical ways in the process of studying calculus, and what kind of concepts it needs. We need to clarify these problems, and these problems need to be combined with the history of the evolution of mathematics itself, that is, the problem that we are talking about the consistency of history and logic.
Why do humans study mathematics? Through the explanation of the history of mathematics just now, we reduce the logical process to the process of cognition, and we can find that mathematics in history is not what it is now. In fact, the future of mathematics is not what it is today, mathematics must have a historical development. Many people regard mathematics as an established look, but we know that mathematics has its own tasks to complete, because mathematics has to solve the real quantitative relationship, spatial form, etc., and reveal the law of quantitative relations and spatial form to it, so when it develops to a certain stage, it may have some qualitative changes, which is the development law of mathematics itself. The development of mathematics will eventually move in the direction of better serving humanity.
This article is reprinted with permission from the WeChat public account "Mathematical Graticule".
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