The history of mathematics is full of brilliant achievements, but it is also a record of disasters. The loss of truth is, of course, the greatest tragedy, because truth is the most precious wealth of mankind, and even losing one is enough to make people feel sad. Another blow to mathematics is to realize that the structure displayed by the achievements of human reasoning is by no means perfect, but there are various defects, which are easy to be attacked by catastrophic paradoxes found at any time. But that's not the only reason that breaks my heart. The deep doubts and differences among mathematicians come from different research directions in the past hundred years. Most mathematicians retreat from the real world and pay attention to the problems in mathematics. They gave up science. As the opposite of applied mathematics, this direction is called pure mathematics. However, these terms, applied and pure, cannot explain these changes very accurately.
What is mathematics? For predecessors, mathematics is first and foremost the most exquisite invention made by people to study nature. The main concepts, extensive methods and almost all important theorems of mathematics are derived in this process. Science has always been the blood to keep the vitality of mathematics. Mathematicians are enthusiastic partners of physicists, astronomers, chemists and engineers in the field of science. In fact, during most of the 17, 18 and 19 centuries, few people noticed the difference between mathematics and theoretical science. Many outstanding mathematicians did more work in astronomy, mechanics, dynamics, electricity, magnetism and elasticity theory than they did in mathematics. Mathematics is the queen of science and their maid.
We have described (chapter 1 to chapter 4) the long-term efforts made since the Greek period to reveal the mathematical mysteries of nature. This kind of research devoted to nature does not bind all applied mathematics to the solution of physical problems. Great mathematicians often go beyond the immediate problems in science, because they have great wisdom, have a deep understanding of the traditional role of mathematics, and can clarify the directions that have been proved to be of great significance in scientific undertakings and clarify the concepts that are helpful to natural research. Poincare spent several years studying astronomy and wrote his masterpiece "Celestial Mechanics". He saw the necessity of exploring new topics in differential equations, which may eventually promote the development of astronomy.
Some mathematical research has led to and improved some disciplines that have proved to be useful. If the same type of differential equation is used in some different applications, mathematicians will study the general type in order to find an improved or general solution, or learn as much as possible about the whole solution family. It is this highly abstract feature of mathematics that makes it possible to express completely different physical phenomena. So water waves, sound waves and radio waves are all represented by a partial differential equation. Actually, this equation is called wave equation. Other mathematical knowledge obtained through further study of wave equation itself originated from the study of sound waves. The rich structure obtained from the problems in the real world can be strengthened by understanding the same mathematical structure and its abstract basis in different situations.
In order to ensure that the mathematical equations of physical problems have solutions, Cauchy took the lead in establishing the existence theorem of differential equations, thus seeking this solution with confidence. Therefore, although this work is completely mathematical, it has far-reaching physical significance. Cantor's work on Guan 286 in infinite set has caused many discussions in pure mathematics, but it was first stimulated by his attempt to solve the extremely useful infinite series of Fourier series.
The development of mathematics needs to explore problems independent of science. We see (see Chapter 8) that mathematicians in the19th century have realized the fuzziness of many concepts and the inadequacy of argument. Of course, this extensive movement of pursuing rigor is neither a discussion of scientific issues nor an attempt by several schools to rebuild the foundation. Although all this work is devoted to mathematics, it is obviously an emergency response to the whole mathematical structure.
In a word, many pure mathematics researches have completed or strengthened old fields and even opened up new fields, which is of great significance to exploration and application. The research in these directions can be regarded as applied mathematics with broad significance.
So a hundred years ago, no mathematics was created entirely for itself, not for practicality? Yes A prominent example is number theory. Although Pythagoras thinks that the study of integers is the study of the composition of real objects (see chapter 1), number theory soon attracted people's interest for its own reasons-this is Fermat's main topic. In order to obtain realism in painting, Renaissance artists created projection geometry, and Descartes engaged in this research. Pascal put forward a more advanced method of Euclidean geometry, which made it a pure aesthetic study in the19th century, although even then it was due to its great connection with non-Euclidean geometry. Many other research topics are simply because mathematicians find them interesting or challenging.
However, pure mathematics completely unrelated to science is not the main consideration. It is just a hobby to get rid of the more vital and interesting problems caused by science. Although Fermat was the founder of number theory, he devoted more energy to analytic geometry, calculus problems and the invention of optics (see Chapter 6). He tried to interest Pascal and Huygens in number theory, but failed. In the17th century, few people will be interested in such a subject.
Euler did spend some time on number theory, but Euler was not only an outstanding mathematician in the18th century, but also an outstanding mathematical physicist. His research ranges from mathematical methods for solving physical problems, such as solving differential equations, to astronomy, fluid motion, ship design, artillery, cartography, musical instrument theory and optics.
Lagrange also spent some time on number theory. But he also spent most of his life on mathematics-analysis, which is very important for application (see chapter 3). The representative work is Analytical Mechanics, which discusses the application of mathematics in mechanics. In fact, in 1777, he complained: "Arithmetic research has brought me great trouble and may be worthless." Gauss has also made remarkable achievements in number theory. His arithmetic research (180 1) is a classic. If you only read this book, it is easy to believe that Gauss is a pure mathematician, but his main energy is applied mathematics (see Chapter 4). Klein called arithmetic research Gauss's work in his19th century history of mathematics.
Although Gauss did return to the study of number theory in his later years, he obviously didn't think this subject was very important. It often bothers him to prove that no integer greater than 2 in Fermat's Last Theorem satisfies xn+yn=zn, but in a letter written to Alpas in March of 2 18 16, Gauss called Fermat's conjecture an isolated theorem, which made no sense. He also said that many conjectures can neither be proved nor falsified, but he was too busy to think about the kind of work he had done in arithmetic research. He hopes that Fermat's conjecture can be proved on the basis of other work he has done, but that would be the most meaningless inference.
Gauss once said: "Mathematics is the queen of science, while number theory is the queen of mathematics. She often condescends to help astronomy and other natural sciences, but in any case, she is always in the most important position. " This shows his preference for pure mathematics. But gauss's lifelong career did not follow this sentence. He may just do it in his spare time. His motto is: "You, nature, my goddess: my contribution to your law is limited." Ironically, through his work in non-Euclidean geometry, his meticulous proof of the consistency between mathematics and nature has had a far-reaching impact on doubting the truth of mathematics. For the mathematics created before 1900, we can draw a general conclusion: there is pure mathematics, but there is no pure mathematician.
Some progress has changed mathematicians' attitude towards their work wonderfully. The first is to realize that mathematics is not a truth system about nature (see Chapter 4). Gauss made this very clear in geometry, and quaternions and matrices forced people to realize this, and Helmholtz understood it more thoroughly-even the general mathematics of numbers is not an available transcendental theory. Although the practicality of mathematics is impeccable, the search for truth no longer proves that the efforts of mathematics are completely correct.
In addition, important developments such as non-Euclidean geometry and quaternion, although inspired by physical thinking, seem to be inconsistent with nature, but their derived inventions are very practical. People realize that man-made inventions are as meaningful as things that seem to obey the inherent laws of nature, which soon becomes the argument of a brand-new mathematical method. So many mathematicians have come to the conclusion that there is no need to study real-world problems. Artificial mathematics comes from the human brain and will certainly prove to be useful. In fact, pure thinking that is not limited by physical phenomena may do better. Imagination without any constraints may create more powerful theories, and they can also find applications in understanding and mastering nature.
Mathematicians escape from the real world for other reasons. The great expansion of mathematics and natural science makes it very difficult to master these two fields, and the scientific problems studied by predecessors are even more difficult. In that case, why not make research easier on the basis of pure mathematics?
Another factor that urges mathematicians to solve pure mathematical problems is that natural science problems can rarely be completely solved. People can get better and better approximate solutions, but they can't get the final solution. A basic problem-such as three bodies, that is, the sun, the earth and the moon, each attract the other two through gravity, and their operating rules have not been solved so far. As bacon said, the exquisiteness of nature far exceeds the intelligence of human beings. On the other hand, pure mathematics allows definite and finite problems to get their complete solutions. It is very interesting to compare definite problems with problems with infinite complexity and depth. Even some unsolved problems like Goldbach's conjecture have attractive simplicity in the discussion.
Another factor that urges mathematicians to study pure mathematical problems is the pressure from universities and other institutions to publish results. Because application problems need rich knowledge of natural science besides mathematics, it makes the problems to be solved more and more difficult, so it is much easier to put forward your own problems and try your best to solve them. Professors not only choose simple math problems that are easy to solve, but also assign them to their own doctors, so that they can finish their papers quickly. At the same time, professors can help them overcome difficulties more easily.
Several examples of the direction followed by modern pure mathematics can make the difference between pure mathematics and applied mathematics clearer. One area is abstraction. Since Hamilton introduced quaternions that he applied to physics in his mind, other mathematicians have realized that there may be many kinds of algebras, regardless of their potential practicality. The research results in this field are all over the field of abstract algebra today.
Another direction of pure mathematics is popularization. Conic curves-ellipse, parabola, hyperbola-are expressed by quadratic equation algebra, and some curves expressed by cubic equation also have practical significance. The generalized research jumps to the curves represented by n-degree equations at once, and studies their properties in detail, although these curves can't appear in natural phenomena at all.
Usually, papers with generality or abstraction have no practical value. In fact, most of these papers are devoted to reformulating the existing formulas described in concrete and clear language with more general, abstract or updated terms, which can neither provide a more powerful method nor provide a deeper insight for those who apply mathematics. Most of these added terms are artificial and have little to do with physical thought, but it is said that new ideas can be put forward, which is of course not a contribution to mathematical application but an obstacle. It is a new language, but it is not new mathematics.
The third direction of pure mathematics research is specialization. Euclid considered and answered whether there are infinite prime numbers. Now the "natural" question is whether there is a prime number in any seven consecutive integers. Pythagoras introduced the concept of affinity number. If the sum of the factors of one number is equal to another number, these two numbers are called affinity numbers. For example, 284 and 220 are affinity numbers. Leonardo Dixon, an outstanding expert in number theory, introduced ternary affinity numbers: "We say that three numbers constitute ternary affinity numbers, if the sum of the true factors of one number is equal to the sum of the other two numbers." He also raised the question of how to find these figures. Another example is about powerful numbers. A powerful number is a positive integer. If it can be divisible by the prime number P, it can also be divisible by p2. Are there any positive integers (except 1 and 4) that can be expressed as the difference between the powerful numbers of two prime numbers in infinite ways?
These special examples are chosen because they are easy to state and understand. They can't fully represent the complexity and depth of such problems. However, specialization has become so extensive and the problem is so narrow that few people can understand it, just like when the theory of relativity came out, only 12 people in the world could understand it.
Specialization is so rampant that most members of the Bourbaki school who are not committed to applied mathematics think it necessary to criticize it.
Many mathematicians occupy a place in the corner of the mathematical kingdom and are unwilling to leave. Not only do they almost completely ignore things that have nothing to do with their professional fields, but they also don't understand the language and terminology used by colleagues in another corner far away from them. Even those who have received the most extensive training are at a loss in some fields of the vast mathematical kingdom. People like Poincare and Hilbert have left the mark of genius in almost every field, even among the greatest winners, there are few extremely great exceptions.
The price of specialization is the exhaustion of creativity, and specialization needs appreciation because it rarely provides anything of value.
Abstraction, generalization and specialization are three activities that pure mathematicians engage in. The fourth category is axiomatization. Undoubtedly, the axiomatic movement at the end of 19 helps to strengthen the mathematical foundation. Although it did not end the solution of the basic problem, some mathematicians began to modify the details of the newly created axiomatic system. Some people can make their statements more concise by restating axioms. Some people combine the three theorems into one through tedious narrative. Others choose new undefined concepts and obtain the same theoretical system by reorganizing those axioms.
We can see that not all axioms are useless, but what we can do is really meaningless. Solving practical problems requires people to go all out because they have to face these problems, but axiomatization allows all kinds of freedom. It is basically the organization of people's deep-seated results, but it doesn't matter whether it is 15 or 20 that people choose this axiom instead of that one. In fact, even some outstanding mathematicians have spent time studying various variants, which are dismissed as "trivial assumptions".
In the first few decades of this century, so much time and energy were spent on axiomatization that Weil complained in 1935 that the results of axiomatization had been exhausted. Although he is well aware of the value of axiomatization, he implores people to return to practical problems. He put forward that axiomatization only endows real mathematics with accuracy and order, and it is a classification function.
All abstract, generalized, specialized and axiomatic problems cannot be regarded as pure mathematics. We have pointed out the value of such work and basic research, and we must understand the motivation of this work. The characteristic of pure mathematics is that it has no direct or indirect application significance. The essence of pure mathematics is that problems are problems. Some pure mathematicians believe that any mathematical development has potential practical value, but no one can foresee its future application. However, a mathematical theme is like a land containing oil, and the black potholes on the surface may indicate the specific location of oil exploitation. If oil is found, the value of this land will be determined. The proven value ensures that drilling not too far from it is expected to find more oil. Of course, you can also choose a place far away from it, because drilling here is easier and oil is still promising. But people's energy and intelligence are limited, so we should devote ourselves to grasping greater risks. If the goal is potential application, then, as the outstanding physical chemist Josiah Willard Gibbs said, pure mathematicians can do whatever they want, while applied mathematicians should at least keep a clear head.
Criticism of pure mathematics, which exists for its own meaning, can be traced back to Bacon's academic progress (1620). He opposes pure, mysterious and self-sufficient mathematics, thinking that it is "completely divorced from the principles of reality and natural philosophy and only satisfies the appetite of those who want to explain and understand things that are not important to people's minds." He understood applied mathematics like this:
Many parts of nature, including perspective, music, astronomy, cosmology, architecture, mechanics and so on, cannot be invented without the help or intervention of mathematics, or must be displayed skillfully or skillfully enough to help their application. Because with the rapid development of physics and the introduction of new axioms, it will seek new help from mathematics in many aspects. Therefore, the mixed part of applied mathematics becomes more beneficial.
In Bacon's time, mathematicians paid little attention to physical research, but today the fact is that they escaped from natural science. In the past 100 years, there has been a split between those who adhere to the ancient and elegant purpose of mathematical activities and those who engage in research according to their interests. Before that, these mathematical activities provided substantive and rich themes. Now mathematicians and scientists go their separate ways, and relatively new mathematical inventions are of little practical value. In addition, mathematicians and scientists no longer understand each other. Disturbingly, with the deepening of specialization, mathematicians no longer even know other mathematicians.
Mathematics, which studies its own reasons without "reality", has aroused opposition almost from the beginning. In Fourier's classic work "Analysis Theory of Heat" (1822), he enthusiastically praised the application of mathematics in physical problems:
The in-depth study of nature is the richest source of mathematical discovery. The advantage of this kind of research lies not only in having a completely clear purpose, but also in eliminating ambiguous problems and useless calculations. It is not only a means to establish analysis itself, but also a method to discover the most important ideas that natural science must always maintain. Basic ideas are those that represent natural phenomena. ……
Its main feature is clarity and no confusing symbols. It puts completely different phenomena together and finds their implicit similarities. If substances bypass us, such as air and light, it is because they are particularly thin; If objects are fixed in the infinite universe far away from us, if human beings want to know the operation of celestial bodies for a long time, and if gravity and heat energy act forever in the unfathomable place inside the solid sphere, mathematical analysis can still grasp the laws of these phenomena and make them superficial and measurable, just as human reasoning ability is destined to make up for the shortness of life and the imperfection of senses; What is even more amazing is that it follows the same method when studying all phenomena. In order to verify the unity and simplicity of the design of the universe and make the eternal order governing all natural things clearer, it explains everything in the same language.
Although jacoby has done first-class work in mechanics and astronomy, he questioned his argument that this is at most a one-sided story. 1on July 2, 830, he wrote to Legendre and said, "Fourier really believes that the main goal of mathematics is public interest and the explanation of natural phenomena; But scientists like him should know that the only goal of natural science is the glory of human spirit, and accordingly, the problem of number theory is as important as that of planetary systems. "
Of course, mathematical physicists are not partial to jacoby's point of view. Willian Thomson and PeterGuthrieTait said in 1867 that the best mathematics is put forward by application, which will produce surprising pure mathematical theory, but mathematicians who confine themselves to pure analysis or geometry cannot reach the rich and beautiful land of mathematical truth.
Many mathematicians are also worried about the new trend of pure research. 1888, Kroneck wrote to Helmholtz, who has made great achievements in mathematics, physics and medicine, and said, "The wealth brought by your reasonable practical experience and interesting questions will point out a new direction for mathematicians and inject new impetus. ...... One-sided and over-introspective mathematical thinking has brought people to the barren land. "
1895, the then mathematical leader F klein also felt the need to oppose this abstract pure mathematical trend:
With the rapid development of modern thinking, we cannot help worrying that our science is facing the danger of becoming more and more independent. Since the rise of modern analysis, the close relationship between mathematics and natural science, which is beneficial to both sides, is in danger of being destroyed.
In his mathematical theory of gyroscope (1897), Klein returned to this question:
The greatest need in mathematical science today is that all branches of pure mathematics and natural science-which will find their most important applications in the future-should establish close relations again, which has been proved to be extremely fruitful in the work of Lagrange and Gauss.
In Science and Method, Poincare admitted that axiomatization, unusual geometry and peculiar functions showed us what miracles would be created when people's intelligence was more and more liberated from outside rule, although he complained about some purely logical creations in the late19th century (see Chapter 8). But he insisted that "we must devote most of our energy to the other direction, that is, the natural side." In The Value of Science, he said:
If we don't remember the most important and pleasant influence of the desire to know nature in the development of mathematics, we will completely forget the history of science. ..... A pure mathematician who forgets the existence of the outside world will be like a painter who knows how to mix colors and composition harmoniously, but has no model. His creativity will soon dry up.
Later, in 1908, F klein worried that the freedom to create arbitrary structures would be abused, and once again stressed that arbitrary structures are "the death of all sciences" and geometric axioms are "not arbitrary, but statements of practice. They usually come from the perception of space, and their exact content depends on convenience. " In order to give a fair evaluation of non-Euclidean geometry, he pointed out that vision only verified Euclid's parallel axiom to some extent. On the other hand, he pointed out that "anyone with freedom and privilege must bear the responsibility". The responsibility here, Klein refers to the exploration of nature.
In his later years, Klein was once a master of mathematics at the University of G? ttingen, the holy city of mathematics. He felt it necessary to lodge a stronger protest. In his book/kloc-Mathematical Development in the 9th Century (1925), he recalled Fourier's interest in solving practical problems with the best existing mathematical methods, and compared this with the meticulous carving of pure mathematics and the abstraction of specific concepts. He wrote:
Mathematics in our time is like a huge arsenal in peacetime. The window is full of ingenious, exquisite and beautiful things, attracting experts. Their real motives and goals-fighting and conquering the enemy-have almost been completely forgotten.
Courand, the leader of Gottingen mathematics after Klein and later the head of Courand Institute of Mathematics of new york University, also lamented the excessive emphasis on pure mathematics. 1924, in the preface of the first edition of Methods of Mathematical Physics by Courand and Hilbert, Courand began with the following comments:
In the past, mathematics got strong stimulation from the close connection between analyzing problems and methods and physical intuitive thoughts, but in recent years, this connection has shown a loose trend. Mathematical research has left the intuitive starting point of mathematics, especially focusing on the exquisiteness of its methods and the accuracy of concepts in analysis. Many analytical leaders have lost their knowledge of the links between their disciplines and physics and other fields. On the other hand, physicists no longer understand mathematicians' problems and methods, even their language and interests. The torrent of scientific development may gradually split into smaller and smaller streams and canals, or even dry up. In order to get rid of this bad luck, we must link mathematical research with natural science. Only in this way can scholars lay a foundation for further research work.
1939 Courand wrote again:
Mathematics is only a conclusion system extracted from definitions and assumptions, and consistency must be guaranteed. In addition, mathematicians can create at will. Such an assertion contains a serious threat to the vitality of science. If this description is accurate, mathematics will not attract anyone with knowledge. This will be a game of definition rules and reasoning without motivation and goals. Intelligence can arbitrarily deduce a meaningful hypothesis system, which is just a false truth. Only under the guidance of the constraint of the organic whole and the inherent inevitability can free thinking obtain scientific results.
Boekhoff, a leading figure in American mathematics, put forward the same view in Scientific American 1943:
We hope that in the future, more and more theoretical physicists will have a deeper understanding of mathematical principles; Mathematicians are no longer limited to the aesthetic development of mathematical abstraction.
John. The mathematical physicist Singh described the situation of 1944 in the preface of a technical article in the style of Bernard Shaw:
Most mathematicians are engaged in the study of ideas that are unanimously regarded as absolute mathematics. They have formed a closed guild, and they must swear not to break the rules when they first join. They usually keep their vows, and only a few mathematicians wander around looking for motivation directly from problems in other scientific fields. In 1744 or 1844, the second kind of person includes almost all mathematicians. In 1944, this is only a small part. What needs to remind most people is that there are still so many people and explain this point.
These few people don't want to be called "physicists" or "engineers". Because they followed a mathematical tradition that lasted for more than 20 centuries, including Euclid, Archimedes, Newton, Lagrange, Hamilton, Gauss, Poincare and others, these few people don't want to belittle the work of most people, but they do worry that relying entirely on their own mathematics will lose its meaning.
Isolated mathematicians not only devote their energy to the future of mathematics, but also deprive other sciences of a support that they have always relied on. It is in the study of nature that problems that are much more complicated than those created by mathematicians behind closed doors appear (and may continue to appear). Scientists always rely on mathematicians to solve these problems. They know that mathematicians are not only skilled users of built tools-they can use these tools quite skillfully themselves; They rely on the unique qualities of mathematicians-his logical insight and ability to see the special from the general and find the general from the special.
In all these, mathematicians are both instructors and constraints. He gave scientific calculation methods-logarithm, calculus, differential equation and so on. -But he gave me more than that. He gave a blueprint and adhered to the logic of thinking. Every time a new discipline appeared, he gave it-or tried to give it-a solid logical structure, just as Euclid did with the land survey in Egypt. When a subject first came into his hands, it was as ugly as a rough stone, but when it left his hands, it was already a shining gem.
Now, science is more active than ever, with no obvious signs of decline. Only the most careful observer noticed that the janitor was absent without leave and he didn't sleep. He works hard as usual, but he is working for himself. ...
In short, the alliance has broken down-how exciting it was when it existed. Nature will ask powerful questions, but these questions will never touch mathematicians. A mathematician may be sitting in an ivory tower waiting for the enemy's bullets, but the enemy will never come to him. Naturally, he will not be provided with ready-made questions that are only formulated. They must dig with shovels and pickaxes, and those who don't want to get their hands dirty will never find them.
The change and decline of thinking are as inevitable as human beings. A mathematician who really loves truth will not hide this. It is impossible for human power to inspire such rich intellectual motives. Some things are imaginative and some are not; If not, they have no passion. If mathematicians really lose the universal connection they once had, and if they see God's hand more truly in the correction of precise logic than in the movement of stars, then any attempt to lure them back to the original point is not only futile, but also a denial of individual intellectual freedom rights. But every young mathematician, if he has his own philosophy-everyone does-should fully grasp the facts before making a decision. He should realize that if he follows the model of modern mathematics, then he will be the heir of a great tradition-but only part of it. Other inheritance will fall into the hands of others, and he will never get it again. ...
Our science begins with mathematics, and it will end soon after mathematics quits (if we want to quit). A century later, there will be bigger and better large-scale laboratories. Whether these experimental results are simple facts or become science depends on their close relationship with the essence of mathematics.
Von Neumann warned nervously. In the frequently cited paper Mathematicians (1947), he said:
When a mathematics discipline continues to develop far away from its empirical origin, or further, if it is the second or third generation, it will face serious difficulties if it is only indirectly inspired by the thought from "reality". Will become more and more pure.