Mathematics is usually regarded as the most developed subject in natural science, but it has experienced three crises in the history of mathematics development. In order to make mathematics develop forward, people introduced some new things to solve problems, which led to the emergence of irrational numbers in the first crisis. The second crisis occurred after the birth of calculus in the 17th century. It was an infinitesimal characterization problem and was finally solved by Cauchy. The third crisis occurred at the end of 19, and Russell's paradox caused an uproar in mathematics. Finally, set theory is based on a set of axioms to avoid paradox and alleviate the mathematical crisis. This paper reviews the emergence and development of three crises in mathematics, and gives my own views on these three crises, and finally draws the conclusion that certainty is lost.
Speaking of mathematics, I have a feeling that mathematics is the most basic subject in nature and the father of all sciences. Without mathematics, there can be no other science. As far as the history of human development is concerned, mathematics has played a huge role in it. No wonder some people say that mathematics is the most beautiful science in human science. However, in the history of mathematics development, it is not so smooth sailing, among which there have been three major crises in history, which have promoted the development of several students, so we should treat these three crises dialectically.
The first crisis occurred in ancient Greece from 580 to 568 BC, and the mathematician Pythagoras established the Pythagorean school. This school is a combination of religion, science and philosophy. Its quantity is fixed, its knowledge is confidential, and all inventions are attributed to its leaders. At that time, people's understanding of rational numbers was still limited, and they knew nothing about the concept of irrational numbers. The Pythagorean school said that numbers originally meant integers. They don't regard the fraction as a number, but only as the ratio of two integers. They mistakenly believe that all phenomena in the universe are attributed to integers or the ratio of integers. According to the Pythagorean Theorem (called Pythagoras Theorem in the West), Hibersos, a member of this school, found through logical reasoning that the diagonal length of a square with a side length of 1 is neither an integer nor a ratio of integers. Herbesos' discovery is considered "absurd" and contrary to common sense. It not only seriously violated the creed of Pythagoras school, but also impacted the traditional views of Greeks at that time. At that time, Greek mathematicians were deeply disturbed. According to legend, it was because of this discovery that Herbios was buried in the sea, which was the first mathematical crisis.
Finally, the concept of incommensurable metric is introduced into geometry to solve this crisis. Two geometric line segments are said to be incommensurable if a third line segment can measure them at the same time, otherwise they are said to be incommensurable. No third line segment can measure one side and diagonal of a square at the same time, so they are incommensurable. Obviously, as long as we admit that the existence of incommensurable metrics makes geometric quantities no longer limited by integers, the so-called mathematical crisis will no longer exist.
I think the greatest significance of the first crisis is that it led to the emergence of irrational numbers. For example, what we are talking about now cannot be expressed in words. Then new numbers must be introduced to describe this problem, and irrational numbers appear. It is with this idea that when we seek the roots of negative numbers, people introduce imaginary number I (the emergence of imaginary number leads to the emergence of complex variable functions and other disciplines, which has been widely used in modern engineering technology), which makes me have to admire human beings. But personally, I think the real solution of the first crisis lies in the strict definition of irrational numbers by German mathematicians in 1872, because mathematics emphasizes its strict logic and derivation.
The second mathematical crisis occurred in the seventeenth century. /kloc-after the birth of calculus in the 0/7th century, because of the theoretical basis of calculus, mathematics appeared a chaotic situation, that is, the second mathematical crisis. Actually, I looked up the information about the history of mathematics. The rudiment of calculus was formed as early as ancient Greece. Archimedes' approximation method actually grasps the basic elements of infinitesimal analysis. It was not until 2 100 years later that Newton and Leibniz opened up a new world-calculus. Newton, the main founder of calculus, used infinitesimal as the denominator of division in some typical derivation processes. Of course, infinitesimal cannot be zero. In the second step, Newton regarded infinitesimal as zero and removed the term containing it, thus obtaining the required formula. The application in mechanics and geometry proves that these formulas are correct, but their mathematical derivation process is logically contradictory. The focus is: Is infinitesimal zero or non-zero? If it is zero, how to divide it? If it is not zero, how to eliminate those items that contain infinitesimal quantity?
Until19th century, Cauchy developed the limit theory in detail and systematically. Cauchy thinks that taking infinitesimal as a definite quantity, even zero, is unreasonable and will conflict with the definition of limit. Infinitesimal should be as small as possible, so it is essentially a variable and a quantity with zero as the limit. At this point, Cauchy clarified the concept of infinitesimal of predecessors. In addition, Vesteras founded the limit theory, combined with the establishment of real number theory and set theory, thus liberating infinitesimal from the shackles of metaphysics and basically solving the second mathematical crisis.
My own understanding is infinitely small. Whether it is zero depends on whether it is moving or static. If it is static, we certainly think it can be regarded as zero. If it is moving, say 1/n, we say, but the product of n 1/n is 1, not infinitesimal. When we encounter such a situation, we can use the repeated derivation of Robida's law to examine the limit, or we can use Taylor expansion to expand the ratio step by step, and always compare the sizes in a limited order.
The third mathematical crisis occurred in 1902, and Russell's paradox shocked the whole mathematical world, claiming that it was flawless and absolutely correct mathematics was contradictory.
I have seen the "barber paradox" a long time ago, that is, the barber cuts the hair of people who can't cut their own hair. So should hairdressers cut their own hair? There is also the well-known "liar paradox", the general content of which is: a Crete said: "Everything Crete said is a lie." Is this sentence true or false? Mathematically, this is a concrete example of Russell's paradox.
The set R defined by Russell in this paradox is considered by almost all set theory researchers as a set that can legally exist in naive set theory. Even so, what is the reason? This is because r is a set. If r contains itself as an element, there will be r r, then there will be r r from the point of view of set. A set does contain itself, and such a set obviously does not exist. Because obviously, R can't have elements different from R, and R and R can't be the same. Therefore, any set must follow the basic principles of R R, otherwise it is illegal. From this point of view, the set of all R R defined in Russell's paradox should be the set of all legal sets, that is, the set of all sets, that is to say, similar things contain all similar things, which will inevitably lead to the largest such thing. In the final analysis, R is the "largest set" containing all sets. Therefore, it can be clearly seen that, in essence, Russell paradox is a maximal set paradox stated in a negative form.
Since then, mathematicians have been looking for ways to solve this crisis, one of which is to build set theory on a set of axioms to avoid paradox. The first person to do this work was the German mathematician Zermero, who put forward seven axioms, established a set theory that would not produce paradoxes, and through the improvement of another German mathematician Friedrich Kerr, formed an axiomatic system of set theory without contradictions (the so-called ZF axiomatic system), and this mathematical crisis was alleviated.
Now through the study of discrete mathematics, we know that set theory is mainly divided into Cantor set theory and axiomatic set theory. A set is first defined as a complete set I and an empty set, which are obtained through a series of unary and binary operations. The set theory system based on seven axioms avoids Russell paradox and makes modern mathematics develop.
How should we treat these three mathematical crises? I think the mathematical crisis has brought new impetus to the development of mathematics. In this crisis, set theory has developed rapidly, the foundation of mathematics has made faster progress, and mathematical logic has become more mature. However, contradictions and unexpected things continue to appear and will continue to appear in the future. Take the appearance of paradox as an example. In a sense, this is not a bad thing. It indicates new creation and light, and promotes the process of science. We should look at it dialectically.
Through the development history of mathematics and these three mathematical crises, I feel more and more about a book about the loss of certainty written by Professor M Klein, which says: Does mathematics need absolute certainty to prove itself? In particular, is it necessary for us to ensure that a theory is compatible before using it, or that it is obtained through absolutely reliable intuition in non-experience period? In other sciences, we don't require this. All theorems in physics are hypothetical, a theorem, as long as it can make useful predictions, we will adopt it. Once it is no longer applicable, we will modify or discard it. In the past, we used to treat mathematical theorems like this. The discovery of contradiction at that time would lead to the change of mathematical principles, although these mathematical principles were accepted by people before the discovery of contradiction. Therefore, our concept of looking at problems should be changed. Mathematics is uncertain.
No matter where mathematics develops in the future, mathematics is still the model of the best knowledge available. The achievement of mathematics is the achievement of human thought. As evidence of human achievements, it gives human courage and confidence to unlock the once seemingly unpredictable secrets of the universe, to subdue the deadly diseases that human beings are susceptible to, and to question and improve the political system in people's lives. So we say that mathematics is ubiquitous in this nature, and its role in human development is immeasurable.