cause

Pythagoras school believes that "number" is the origin and starting basis of all things, and all phenomena in the universe can be attributed to integers or the ratio of integers. Before the discovery of hippasus's paradox, people only knew natural numbers and rational numbers, and rational number theory became the dominant mathematical norm. The irrational number discovered by hippasus exposed the limitations of the original mathematical norms. From this perspective, hippasus's paradox is caused by mistakes in subjective understanding.

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In the 5th century BC, hippasus, a member of Pythagoras School (about 470 BC). C), it is found that the hypotenuse and right-angled side of isosceles right triangle are incommensurable, and their ratio cannot be attributed to an integer or the ratio of integers. This discovery not only seriously violated the creed of Pythagoras school, but also impacted the general view of Greeks at that time, which directly led to the "crisis" of understanding at that time. Hippasus's discovery was called "hippasus Paradox" in history, which triggered the first crisis in the history of mathematics.

affect

The discovery of 00 hippasus urges people to further know and understand irrational numbers. However, based on the development level of production and technology, the Pythagorean school of ancient Greece and later mathematicians did not and could not establish a strict irrational number theory. They basically took an evasive attitude towards the problem of irrational numbers, abandoned the arithmetic processing of logarithm and replaced it with geometric processing, thus beginning the period of geometric priority development. In the next two thousand years, Greek geometry became the basis of almost all mathematics. Of course, this narrow practice of binding the whole mathematics with geometry has also had a negative impact on the development of mathematics.

The discovery of Hippasus shows that intuition and experience are not necessarily reliable, but reasoning and proof are reliable, which leads to the establishment of Aristotle's logic system and Euclidean geometry system.

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The second mathematical crisis

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/kloc-At the end of 0/7th century, the calculus theory founded by Newton and Leibniz was applied to the second mathematical crisis in practice, and most mathematicians were convinced of the reliability of this theory. However, the theory of calculus at that time was mainly based on infinitesimal analysis, which was later proved to contain logical contradictions.

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00 1734, British Archbishop Becquerel published Analytic Scholars, or To an Unbelieving Mathematician. Among them, whether the object, principle and inference of modern analysis are clearer in concept or more obvious in reasoning than the mystery and dogma of religion severely criticized the calculus theory at that time. He said that Newton first thought infinitesimal was not zero, and then made it equal to zero, which violated the antinomy, and the number of streams obtained was actually 0/0, because "you got the correct result by double mistakes, although it was unscientific", because the mistakes compensated each other. It is called "Becquerel Paradox" in the history of mathematics. The discovery of this paradox caused some confusion at that time, which led to the second crisis in the history of mathematics and the debate on the basic theory of calculus for more than 200 years.

Becker's attack on infinitesimal aims at demonstrating religious theology, but as the "Becker paradox" itself, it is a question of thinking method. Because mathematics should think according to the non-contradictory law of formal logic, it cannot be admitted that it is not equal to zero or zero in the same thinking process. But the movement of things takes its end point as the limit, and the result of movement is equal to zero in quantity and not equal to zero in starting point. These are two aspects of the movement of things and should not be included in the same thinking process. If we link them mechanically, it will inevitably lead to a paradox in thinking. The cause of Becquerel's paradox lies in the contradiction between the dialectical nature of infinitesimal quantity and the formal characteristics of mathematical methods.

affect

The product of the second mathematical crisis-the rigor of analytical basic theory and the establishment of set theory.

After Becker's Paradox was put forward in 2000, many famous mathematicians studied and explored from different angles, trying to rebuild calculus on a reliable basis. French mathematician Cauchy is a master of mathematical analysis. Through the analysis course (182 1), the lecture on infinitesimal computation (1823) and the application of infinitesimal computation in geometry (1826), Cauchy established a modern limit-oriented society. But Cauchy's system still needs to be improved. For example, his language about limit is still vague, relying on things that are intuitive in motion and geometry; Lack of real number theory. The German mathematician Wilstrass is one of the main founders of the foundation of mathematical analysis. He improved the methods of Porzano, Abel and Cauchy, described a series of important concepts such as limit, continuity, derivative and integral in calculus with "ε-δ" method for the first time, and established a strict system of the subject. The proposal and application of "ε-δ" method in calculus marks the completion of calculus operation. In order to establish the basic theorem of limit theory, many mathematicians began to define irrational numbers strictly. In 1860, Weierstrass proposed to define irrational numbers by adding bounded sequence; 1872 Dai Dejin proposed to define irrational numbers by division. 1883, Cantor proposed to define irrational numbers with basic sequence; Wait a minute. These definitions profoundly reveal the essence of irrational numbers from different aspects, thus establishing a strict real number theory, completely eliminating hippasus's paradox, establishing a limit theory based on the strict real number theory, and then leading to the birth of set theory.

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the third mathematical crisis

cause

00 Wilstrass solved Becker's paradox by excluding infinitesimal, but in 1960s, Robinson invited infinitesimal back and introduced the concept of hyperreal number, thus establishing nonstandard analysis, which can also accurately describe calculus, thus solving Becker's paradox. However, it must be noted that Becker's paradox has only been solved in a relative sense, because the contradiction of real number theory comes down to the contradiction of set theory and has not been completely solved so far.

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After the first and second mathematical crises, people attributed the non-contradiction of the basic theory of mathematics to the non-contradiction of the third mathematical crisis theory. Set theory has become the logical basis of modern mathematics, and the grand building of mathematics has been built. It seems that set theory is not contradictory, and the goal of strict mathematics has almost been achieved, and mathematicians are almost complacent about this achievement. The famous French mathematician Poincare (1854- 19 12) boasted at the international congress of mathematicians held in Paris in 1900: "Now it can be said that it has reached the absolute rigor." However, less than two years later, the famous British mathematical logician and philosopher Russell (1872- 1970) announced an amazing news: set theory is self-contradictory and has no absolute rigor! History is called "Russell Paradox". 19 18, Russell extended this paradox to become barber paradox. The discovery of Russell's paradox is tantamount to breaking the fog in a sunny day and waking people up from their dreams. Russell paradox and other paradoxes in set theory go deep into the theoretical basis of set theory, thus fundamentally endangering the certainty and rigor of the whole mathematical system. So it caused an uproar in the fields of mathematics and logic, and formed the third crisis in the history of mathematics.

The paradox of set theory lies in the contradiction between the dialectical nature of set and the formal characteristics of mathematical method or metaphysical thinking method. For example, the cause of Russell paradox lies in the contradiction between the arbitrariness of generalization principle and the non-arbitrariness of objective rules of generating set.

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The product of the third mathematical crisis-the development of mathematical logic and the emergence of a number of modern mathematics.

Mathematicians have made different efforts to solve the third mathematical crisis. Due to different starting points and different ways to solve problems, different schools of mathematical philosophy were formed at the beginning of this century, namely, the logicism school headed by Russell, the intuitionism school headed by Brouwer (188 1- 1966) and the formalism school headed by Hilbert. The formation and development of these three schools have pushed the research of basic mathematics theory to a new stage. The mathematical achievements of the three schools are first manifested in the formation of mathematical logic and its modern branch proof theory.

00 In order to eliminate the paradox of set theory, Russell put forward type theory, and Tzemero put forward the first axiom system of set theory. After frenkel's modification and supplement, the common axiom system of Tzemero-frenkel set theory was obtained, and then the axiom system of bernays-G? del set theory was further improved and simplified by bernays and G? del. Hilbert also established meta-mathematics. Godel's incompleteness theorem is the direct result of the study of set theory paradox.

Godel, an outstanding American mathematician, put forward the incompleteness theorem in the 1930s. He pointed out that a formal system containing logic and elementary number theory is incomplete if it is coordinated, that is, no contradiction can be established in this system; If the elementary arithmetic system is harmonious, then it is impossible to prove that the arithmetic system is harmonious Godel's incompleteness theorem irrefutably reveals the limitations of the formalism system, and mathematically proves the impossibility of trying to solve the paradox problem once and for all with formalism technology. In fact, it tells people that any attempt to find an absolutely reliable foundation for mathematics and thus completely avoid paradox is futile. Godel theorem is the cornerstone of mathematical logic, artificial intelligence and set theory, and a milestone in the history of mathematics. Von Neumann, a famous American mathematician, said: "Godel's achievement in modern logic is extraordinary and immortal-it is even more immortal than a monument." It is a milestone and a monument that will always exist in a visible place and in the foreseeable future. "

Today, the third mathematical crisis cannot be said to have been fundamentally eliminated, because many important topics of mathematical foundation and mathematical logic have not been fundamentally solved. However, people are gradually approaching the goal of fundamental solution. It can be expected that many new and important achievements will be produced in this process.

It is of great significance to discover, put forward and study paradox for mathematical foundation, logic and philosophy. As Taskey (190 1-) pointed out: "It must be emphasized that paradox occupies a particularly important position on the basis of establishing modern deductive science. Just as the paradox of set theory, especially Russell's paradox, became the starting point of formalization of logical and mathematical compatibility, the paradox of liar and its semantic paradox led to the development of theoretical semantics. "

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