From a philosophical point of view, contradictions exist everywhere, even in mathematics, which is known for sure. There are many contradictions in mathematics, big and small, such as positive and negative, addition and subtraction, differentiation and integration, rational and irrational numbers, real and imaginary numbers, and so on. In the whole process of mathematics development, there are still many profound contradictions, such as finite and infinite, continuous and discrete, existence and construction, logic and intuition, concrete object and abstract object, concept and calculation and so on.

The struggle and resolution of contradictions run through the history of mathematics. Mathematical crisis occurs when the contradiction intensifies to the basis of the whole mathematics. The resolution of the crisis can often bring new content, new development, and even revolutionary changes to mathematics.

The development of mathematics has experienced three crises about the basic theory.

The first math crisis

In a sense, mathematics in the modern sense, that is, pure mathematics as a deductive system, originates from the ancient Greek school of Pythagoras. It is an idealist school that flourished around 500 BC. They believe that “everything is number” (referring to integers), that mathematical knowledge is reliable, accurate, and can be applied to the real world, and that mathematical knowledge is obtained through pure thinking, without the need for observation, intuition, and everyday experience.

Integers are abstractions that arise during computations on finite integrations of objects. In daily life, not only individual objects are counted, but various quantities such as length, weight and time are measured. To meet these simple metric needs, fractions are used. Thus, if a rational number is defined as the quotient of two integers, then since the rational number system includes all integers and fractions, it is sufficient for practical measurements.

Rational numbers have a simple geometric interpretation. On a horizontal straight line, mark a line segment as the unit length. If the fixed endpoint and the right endpoint of it are set to represent the numbers 0 and 1, respectively, the set of points on the straight line with intervals of unit length can be used to represent the integer. Integers are to the right of 0, and negative integers are to the left of 0. Fractions with q as the denominator can be represented by points where each unit interval is divided into q equal parts. Thus, every rational number corresponds to a point on the line.

Ancient mathematicians believed that this would use up all the points on the line. However, the Pythagorean school discovered around 400 BC that there are points on a straight line that do not correspond to any rational numbers. In particular, they proved that there is a point p on this line that does not correspond to a rational number, where the distance op is equal to the diagonal of a square with sides of unit length. So new numbers had to be invented to correspond to such points, and since these numbers could not be rational, they had to be called irrational. The discovery of irrational numbers is one of the greatest achievements of the Pythagorean school and an important milestone in the history of mathematics.

The discovery of irrational numbers caused the first mathematical crisis. First, it was a fatal blow to the Pythagorean philosophy, which relied entirely on integers. Second, irrational numbers seem to contradict common sense. The geometrical correspondence is also surprising because, contrary to intuition, there are incommensurable segments, ie segments with no common unit of measure. Since the Pythagorean definition of proportionality assumes that any two quantities of the same kind are commensurable, all propositions in the Pythagorean theory of proportionality are limited to commensurable quantities, so that their general theory of similarity also failed.

The “logical inconsistency” was so great that for a while they took great pains to keep the matter a secret from the outside world. But it soon became apparent that incommensurability is not an uncommon phenomenon. Theodorus points out that the sides of squares with areas equal to 3, 5, 6, . Over time, the existence of irrational numbers has gradually become a well-known fact.

An indirect factor that triggered the first mathematical crisis was the emergence of the “Zeno’s Paradox”, which increased the worries of mathematicians: Is mathematics still possible as an exact science? Does the harmony of the universe still exist?

In about 370 BC, this contradiction was resolved by Eudox of the Pythagorean school by means of a new definition of proportion. His method of dealing with incommensurable quantities appears in Euclid’s Elements, Book 5, and is basically consistent with the modern interpretation of irrational numbers drawn by Dididekind in 1872. The treatment of similar triangles in today’s middle school geometry textbooks still reflects some difficulties and subtleties brought about by incommensurable quantities.

The first mathematical crisis showed that some truths of geometry had nothing to do with arithmetic, and that geometric quantities could not be fully represented by integers and their ratios. Conversely, numbers can be represented by geometric quantities. The supremacy of integers was challenged, and the ancient Greek mathematical view was greatly impacted. Thus, geometry began to occupy a special place in Greek mathematics. At the same time, it also reflects that intuition and experience are not necessarily reliable, but reasoning is reliable. From then on, the Greeks started from the “self-evident” axioms, through deductive reasoning, and thus established a geometric system. It was a revolution in mathematical thought, a natural product of the first mathematical crisis.

Looking back at the various mathematics before this, it is nothing more than “calculation”, that is, to provide algorithms. Even in ancient Greece, mathematics started from reality and applied it to practical problems. For example, Thales predicting solar eclipses, using shadows to calculate the height of pyramids, measuring the distance of ships from shore, etc., are all within the scope of computing technology. As for mathematics in Egypt, Babylon, China, India and other countries, they have not experienced such crises and revolutions, so they continue to take the road of focusing on calculation and focusing on application. However, due to the occurrence and resolution of the first mathematical crisis, Greek mathematics embarked on a completely different development path, forming the axiom system of Euclid’s “Original” and the logic system of Aristotle, which made a great contribution to world mathematics. Another outstanding contribution.

However, since then, the Greeks have regarded geometry as the basis of all mathematics, subordinate the study of numbers to the study of form, and cut off the close relationship between them. The biggest misfortune of doing so is to give up the study of irrational numbers, which greatly restricts the development of arithmetic and algebra, and the basic theory is very thin. This abnormal development has continued in Europe for more than 2,000 years.

The second math crisis

The heated debate about calculus in the seventeenth and eighteenth centuries was called the Second Mathematical Crisis. From a historical or logical point of view, its occurrence is also inevitable.

The seeds of this crisis appeared around 450 BC. Zeno noticed the contradiction arising from the problem of understanding infinity, and proposed four paradoxes about the finitude and infinity of space and time:

”Dichotomy”: towards An object moving at a destination must first pass through the midpoint of the journey, but to pass through this point, it must first pass through the 1/4 point of the journey… and so on and so forth. ——The conclusion is: infinity is an inexhaustible process, and motion is impossible.

”Achilles (the runner in Homer) can’t catch up with the tortoise”: Achilles always has to reach the tortoise’s starting point first, so the tortoise must always run ahead. This argument is the same as the dichotomy paradox, except that the distance to be traveled does not have to be divided equally.

”Flying arrow does not move”: It means that the arrow must be in a certain position at any instant during the movement process, so it is stationary, so the arrow cannot be in motion.

”Playground or parade”: Two objects A and B move in opposite directions with equal velocity. From the standpoint of c, for example, A and B both move 2 kilometers in 1 hour, but from A’s point of view, B moves 4 kilometers in 1 hour. Movement is contradictory, so movement is impossible.

The contradictions revealed by Zeno are profound and complex. The first two paradoxes challenge the view that time and space are infinitely divisible and therefore motion is continuous, while the latter two paradoxes challenge the view that time and space are not infinitely separable and therefore motion is discontinuous. Zeno’s paradoxes may have a deeper background, not necessarily specifically for mathematics, but they have caused an uproar in the realm of mathematics. They illustrate that the Greeks had seen the contradictions between “infinitesimal” and “very small”, but they could not resolve them. As a consequence, infinitesimals have since been excluded from the proof of Greek geometry.

After many years of hard work, finally in the late 17th century, the subject of infinitesimal calculus, calculus, was formed. Newton and Leibniz are recognized as the founders of calculus. Their achievements are mainly in: unifying the solutions of various related problems into differential and integral methods; there are clear calculation steps; differential and integral methods are inverse to each other operation. Due to the completeness of operations and the wide range of applications, calculus became an important tool for solving problems at that time. At the same time, questions about the foundations of calculus are becoming more and more serious. The key question is whether the infinitesimal competition is zero? Are infinitesimals and their analysis reasonable? As a result, it has caused a century and a half of debates in the mathematics community and even the philosophical community, resulting in the second mathematical crisis.

Are infinitesimals really zero? Both answers lead to contradictions. Newton explained it in three different ways: in 1669 it was a constant; in 1671 it was a variable tending to zero; in 1676 it was replaced by “the final ratio of two vanishing quantities” . However, he has never been able to resolve the above contradictions. Leibniz tried to replace infinitesimals with finite differences proportional to infinitesimals, but he also failed to find a bridge from finite to infinitesimal.

The British Archbishop Berkeley wrote an article in 1734, attacking the flow (derivative) “is the ghost of the disappearing quantity… People who can digest the second-order and third-order flow will not vomit because they swallow the theological arguments. .” He said that by ignoring higher-order infinitesimals and eliminating the original error, “it is by means of a double error to obtain a correct result although unscientific.” Although Berkeley also grasped some unclear and illogical problems in calculus and infinitesimal methods at that time, he came out of his disgust for science and the maintenance of religion, rather than from the pursuit and exploration of science.

At that time, some mathematicians and other scholars also criticized some problems of calculus, pointing out that it lacked the necessary logical foundation. For example, Rolle once said, “Calculus is a collection of ingenious fallacies.” In the early days of the age of daring, the problems of logic in science were not isolated phenomena.

Mathematical thought in the eighteenth century was indeed loose and intuitive, emphasizing formal calculation without regard to the soundness of the foundations. Among them in particular: there is no clear concept of infinitesimal, so concepts such as derivative, differentiation, integral, etc. are unclear; the concept of infinity is unclear; the arbitrariness of summation of divergent series, etc.; , regardless of the existence of derivatives and integrals, and whether the function can be developed into a power series, etc.

It wasn’t until the 1820s that some mathematicians focused more on the rigorous foundations of calculus. From the beginning of the work of Bolzano, Abel, Cauchy, Dirichley and others to the end of the work of Willstras, Dedekind and Cantor, more than half a century has passed in between, basically The contradiction is resolved and a rigorous foundation is laid for mathematical analysis.

Bolzano gave the correct definition of continuity; Abel pointed out that the abuse of series expansion and summation should be strictly limited; Cauchy started from the definition of variables in the 1821 Course of Algebraic Analysis, realizing that functions do not necessarily have to be There are analytic expressions; he grasps the concept of limit, pointing out that infinitesimals and infinitesimals are not fixed quantities but variables, and infinitesimals are variables whose limit is zero; and defines derivatives and integrals; Dirichley gives A modern definition of a function. On the basis of these works, Willstras eliminates the inaccuracies, gives the definition of the limit in general, the definition of continuity, and establishes the derivative and integral strictly on the basis of the limit.

In the early 1870s, Willstras, Dedekind, Cantor and others independently established the theory of real numbers, and on the basis of the theory of real numbers, established the basic theorem of limit theory, so that mathematical analysis was established in On the strict basis of real number theory.

The third math crisis

The third crisis in the foundations of mathematics emerged from the sudden shock of 1897, and as a whole has not been resolved to a satisfactory degree. The crisis was caused by the discovery of paradoxes on the fringes of Cantor’s general set theory. Since the concept of sets has permeated numerous branches of mathematics, and since set theory has in fact become the foundation of mathematics, the discovery of paradoxes in set theory naturally raises doubts about the validity of the entire fundamental structure of mathematics.

In 1897 Forty revealed the first paradox of set theory; two years later, Cantor discovered very similar paradoxes involving results in set theory. In 1902, Russell discovered a paradox that involved no concept other than the concept of set itself.

Russell, British, philosopher, logician, mathematician. In 1902, he wrote “Principles of Mathematics”, and then co-authored “Principles of Mathematics” (1910-1913) with Whitehead, summarizing mathematics into an axiom system, which is one of the epoch-making works. He has written extensively in many fields and was awarded the Nobel Prize for Literature in 1950. He cared about social phenomena, participated in peace movements, and opened schools. His autobiography was published in 1968-1969.

Russell’s paradox has been popularized in many forms, the most famous of which was given by Rosso in 1919, which deals with the plight of a village barber. The barber proclaimed the principle that he shaves only those who do not shave himself. The paradoxical nature of the situation is recognized when one attempts to answer the following question: “Can a barber shave himself?” If he shaves himself, then he does not conform to his principles; if he does not give If he shaves himself, then in principle he should shave himself.

Russell’s paradox has shaken the entire mathematical edifice, and it is no wonder that Frege, after receiving Russell’s letter, wrote at the end of the second volume of his just-published Fundamental Laws of Arithmetic: “A scientist will not touch To something more embarrassing than that, when the work is done, its foundations crumble. A letter from Mr. Russell put me in this position while the book was waiting to go to press.” Dedekind had originally planned to put the third edition of “Continuity and Irrational Numbers” into print, but at this time he also pulled the manuscript back. Braun, who discovered the “fixed point principle” in topology, also thought that his past work was “nonsense” and claimed to abandon the fixed point principle.

Since the above-mentioned paradoxes were discovered in Cantor’s set theory and found, many additional paradoxes have arisen. The modern paradoxes of set theory are related to several ancient paradoxes of logic. For example, the fourth century BC Oberlidian paradox: “This statement I am making now is false”. If the statement is true, it is false; however, if the statement is false, it is true again. Thus, the statement can neither be true nor false, and there is no way to escape the contradiction. Even earlier is the Epimenides (6th century BC, Crete) paradox: “The Cretans were always liars”. A simple analysis shows that this statement is also self-contradictory.

The existence of paradoxes in set theory makes it clear that something is wrong. Since their discovery, numerous articles have been published on the subject and numerous attempts have been made to address them. As far as mathematics is concerned, there seems to be an easy way out: one just has to base set theory on axiomatic foundations, with sufficient restrictions to rule out known contradictions.

The first such attempt was made by Zermelo in 1908, and many others have since processed it. However, this procedure has been criticized for avoiding certain paradoxes rather than accounting for them; furthermore, it does not guarantee that other paradoxes will not arise in the future.

Another procedure can both explain and rule out known paradoxes. If you examine it carefully, you will find that each of the above paradoxes involves a set S and a member M of S (that is, M is defined by S). Such a definition is said to be “non-assertive”, and a non-assertive definition is circular in a sense. Consider, for example, Russell’s barber paradox: denoting the barber with M and the set of all members with S, then M is non-assertive defined as “S gives and only gives to that member of those who do not shave themselves. “. The nature of the cycle of this definition is obvious – the definition of barber involves all members, and the barber itself is a member here. Thus, disallowing non-assertive definitions may be a solution to the known paradox of set theory. There is, however, a serious reproach to this solution, namely those parts of mathematics that include non-assertive definitions that mathematicians are very reluctant to discard, such as the theorem “Every non-empty set of real numbers with an upper bound has a least upper bound. (Supreme Realm)”.

Other attempts to resolve the paradoxes of set theory have been to find the crux of the problem logically, which has led to a comprehensive study of the basis of logic.

From 1900 to about 1930, the crisis of mathematics involved many mathematicians in a great debate. They saw that the crisis involved the fundamentals of mathematics, and that the philosophical underpinnings of mathematics must be examined closely. In this great debate, what had been previously inconspicuous differences of opinion expanded into scholastic debates. The three major schools of mathematical philosophy came into being: logicism represented by Russell, intuitionism represented by Brouwe, and formalism represented by Hilbert. They are both schools of idealism, and they each propose their own way of dealing with the paradoxes in general set theory. Although they were snarky in their arguments, they seemed to be at odds with each other.

In 1931, the proof of Gödel’s incompleteness theorem exposed the weaknesses of the factions, and the philosophical debate dimmed. Since then, each faction has developed and evolved along its own path. Although the issues at issue are far from settled, most mathematicians do not care much about philosophical issues. Only in recent years have questions in the philosophy of mathematics sparked renewed interest.

Recognizing infinite sets and infinite cardinal numbers, it is as if all disasters have come out. This is the essence of the third mathematical crisis. Although paradoxes can be eliminated and contradictions can be resolved, the certainty of mathematics is being lost step by step. There are a lot of axioms in modern axiomatic set theory. It is difficult to say which ones are true and which are false, but they cannot be eliminated. They are connected with the whole mathematics of flesh and blood. So, the third mathematical crisis has been resolved on the surface, but in essence it continues in other forms more profoundly.

Since the contradiction in mathematics is inherent, its fierce conflict-crisis is inevitable. The resolution of the crisis has brought many new understandings, new contents, and sometimes revolutionary changes to mathematics. Comparing the mathematics of the 20th century with all previous mathematics, the content is much richer and the understanding is much deeper. On the basis of set theory, abstract algebra, topology, functional analysis and measure theory were born, and mathematical logic also flourished and became a part of the mathematical organism. Ancient algebraic geometry, differential geometry, and complex analysis have now been extended to higher dimensions. The face of algebraic number theory has also changed many times, becoming more and more beautiful and complete. A series of classic problems have been solved satisfactorily, and more new problems have been generated at the same time. Especially after the Second World War, new achievements emerged in an endless stream, never interrupted. Mathematics presents an extremely prosperous scene, and this is the product of people’s struggle with contradictions and crises in mathematics.