the Chen Dynasty
Abstract: The development of mathematics is never completely linear, but paradoxes often appear.
A series of historical events
The paradox of mathematics has shaken people's belief in the reliability of mathematics, and there have been three mathematical crises in the history of mathematics.
The appearance of mathematical paradox and crisis not only brings trouble and disappointment to mathematics, but also brings new vitality and hope to its development and promotes its prosperity.
The endless and repeated process of crisis generation, solution and generation constantly promotes the development of mathematics, which is also an important development process of mathematical thought.
Keywords: mathematical paradox; Mathematical crisis; Pythagoras paradox; Becker paradox; Russell's paradox
Mathematics has always been regarded as a rigorous, harmonious and accurate subject. Throughout the history of mathematics development, the development of mathematics has never been completely linear, and its system is not always harmonious, but paradoxes often appear.
Paradox refers to the derivation of two contradictory propositions on the basis of a certain theoretical system and reasonable reasoning principles, or the proof of such a compound proposition, which is expressed as the equivalence of two contradictory propositions [1].
The development of mathematical paradox in mathematical theory is a serious matter, because it directly leads people to doubt the corresponding theory, and if a paradox involves a wide range, even the basis of the whole discipline, this doubt may develop into a general sense of crisis, especially the emergence of some important paradoxes, which will naturally cause people to doubt the basis of mathematics and shake their belief in the reliability of mathematics.
There have been three mathematical crises in the history of mathematics, each caused by one or two typical mathematical paradoxes.
Three mathematical crises in history are reviewed, and their important roles in the development of mathematics are emphatically introduced.
1 Pythagoras Paradox and the First Mathematical Crisis
1. 1 the content of the first mathematical crisis
In the 6th century BC, the Pythagorean school, which ruled the ancient Greek academic circles, was regarded as the absolute authoritative truth at that time. The Pythagorean school advocates a philosophical view called "numerology", which holds that the essence of the universe is the harmony of numbers [2].
They think that everything is a number, there are only two kinds of numbers, that is, positive integers and reducible numbers (that is, fractions, the ratio of two integers), and there are no other numbers, that is to say, there are only integers or fractions in the world.
One of the great contributions of Pythagorean school in mathematics is to prove Pythagorean theorem [3], which is what we call Pythagorean theorem.
Pythagorean theorem points out that the three sides of a right triangle should have the following relationship, namely a2=b2+c2, A and B represent the two right sides of the right triangle respectively, and C represents the hypotenuse.
However, it didn't take long for Herbs, a student of Pythagoras School, to discover the problem of this assertion.
He found that the diagonal length of an equilateral square could not be expressed by integers or the ratio of integers.
Assuming that the side length of a square is 1 and the diagonal length is d, according to Pythagorean theorem, d2= 12+ 12=2, that is, d2=2, then what is D? Obviously d is not an integer, so it must be the ratio of two integers.
Hebers spent a lot of time looking for the ratio of these two integers, but he couldn't find it. Instead, he found a proof that two numbers are incommensurable [4], which is proved by reduction to absurdity as follows: Let Rt△ABC and two right-angled sides be a=b, then c2=2a2. Starting from Pythagorean theorem, let the common divisors in A and C be reduced, that is, A and C are prime numbers, so C is even and A.
This discovery is called Pythagoras paradox in history.
1.2 the influence of the first mathematical crisis
The Pythagorean paradox has dealt a heavy blow to the Pythagorean school. The world outlook of "number is everything" has been greatly shaken, and the revered position of rational numbers has also been challenged, which has affected the foundation of the whole mathematics and caused extreme ideological confusion in the mathematics field, which is called the first mathematical crisis in history.
The influence of the first mathematical crisis was enormous, which greatly promoted the development of mathematics and its related disciplines.
First, the first mathematical crisis made people realize the existence of irrational numbers for the first time, and irrational numbers were born. Later, many mathematicians formally studied irrational numbers, gave a strict definition of irrational numbers, put forward a new category of numbers-real numbers, and established a complete real number theory [5], which laid the foundation for the development of mathematical analysis.
Furthermore, the first mathematical crisis shows that intuition and experience are not necessarily reliable, but reasoning is reliable. Since then, the Greeks began to attach importance to deductive reasoning, and thus established a geometric axiom system.
In order to eliminate contradictions and alleviate the crisis, Euclidean geometry came into being at this time [6].
The first mathematical crisis greatly promoted the development of geometry, and in the next two thousand years, geometry became the basis of almost all rigorous mathematics. This is a great revolution in the history of mathematical thought.
2 Becker Paradox and the Second Mathematical Crisis
2. 1 Contents of the Second Mathematical Crisis
/kloc-In the 7th century, Newton and Leibniz founded calculus. Calculus can prompt and explain many natural phenomena, and its important role in theoretical research and practical application of natural science has aroused great concern.
However, because calculus has just been established, calculus at this time has only methods, without strict theory as the basis, and there are loopholes in many places, which cannot be justified.
For example, Newton found the derivative of the function y = xn [7]: (x+△ x) n = xn+n? xn- 1？ △x+[n(n+ 1)/2]？ xn-2？ (△x) 2+…+(△x) n, and then divide the increment △ x of the independent variable by the increment △y of the function, △ y/△ x = [(x+△ x) n-xn]/△ x = n? xn- 1+[n(n- 1)/2]？ xn-2？ △x+……+n？ x？ (△x) n-2+(△x) n- 1 Finally, throw away the term containing infinitesimal △ x, that is, the derivative of the function y=xn is y'=nxn- 1.
Becquerel, a philosopher, soon found a problem in Newton's exposition on the process of derivative derivation. He hit the nail on the head and pointed out that dividing △x by △y first means that △x is not equal to zero, and then throwing away the item containing △x means that △x is equal to zero. Isn't this a contradiction? Therefore, Becker mocked infinitesimal as "the ghost of losing quantity", and he thought that calculus got the correct result through double mistakes, and called the derivation of calculus "obvious sophistry".
[8] This is the famous "Becquerel Paradox".
Indeed, in the discussion of the same problem, the so-called infinitesimal is sometimes regarded as 0, and sometimes it is different from 0, which is doubtful.
Is infinitesimal zero or not? Is infinitesimal and its analysis reasonable? The appearance of Becquerel's paradox endangered the foundation of calculus, and caused a debate in the field of mathematics for more than two centuries, thus forming the second crisis in the history of mathematics development.
2.2 the impact of the second mathematical crisis [8]
The emergence of the second mathematical crisis forced mathematicians to take infinitesimal △x seriously. In order to overcome the confusion and solve this crisis, countless people have invested a lot of labor.
At first, through the efforts of Euler, Lagrange and others, calculus made some progress; Since19th century, Cauchy, Wilstras and others have made the theory of calculus rigorous in order to thoroughly solve the basic problems of calculus.
The fundamental contradiction in calculus is how to express infinitesimal by mathematical and logical methods, so as to express the essence of calculus closely related to infinitesimal.
When solving the mathematical problem of infinitesimal, Robida's axiom appeared: if one quantity increases or decreases compared with another, it is infinitesimal, which can be considered as unchangeable.
Cauchy's ε-δ method describes infinitesimal and defines infinitesimal as a variable with 0 as the limit. Up to now, infinitesimal has been replaced by limit.
Later, Wilstrass clarified it, gave a strict definition of limit, and established a limit theory, thus making calculus based on limit.
The ε-δ definition of limit is to describe the dynamic limit with static ε-δ and the infinite process with * * * *. It is a bridge and signpost from finite to infinite, which shows the relationship between finite and infinite, and makes calculus a big step towards science and mathematics.
The establishment of limit theory accelerates the development of calculus, which is of great significance not only in mathematics but also in epistemology.
Later, on the basis of investigating the limit theory, through the efforts of Dydykin, Cantor, Heine, Wilstras and Bamen Harmo, the real number theory came into being. When investigating the basis of real number theory, Cantor founded the theory of * * *.
In this way, with the three theories of limit theory, real number theory and * * * theory, calculus can be established on a relatively stable and perfect basis, thus ending the chaotic debate situation for more than 200 years and opening up the development path of function theory in the next century.
Russell Paradox and the Third Mathematical Crisis
3. 1 Contents of the Third Mathematical Crisis
Less than 30 years after the first two mathematical crises were solved, that is, in the 1970s of 19, German mathematician Cantor founded the most revolutionary theory in mathematics, with the original intention of laying a solid foundation for the whole mathematical building.
1900, at the international congress of mathematicians held in Paris, the great French mathematician Poincare excitedly announced [9]: "We can now say that mathematics has reached absolute rigor." However, while people rejoiced at the birth of * * * * theory, a series of mathematical paradoxes also appeared, which puzzled mathematicians. Among them, the paradox put forward by British mathematician Russell 1902 has the greatest influence. The content of "Russell Paradox" is as follows: Let *** B be a * * composed of all * * that are not its own elements. Q: If B belongs to B, then B is an element of B, so B does not belong to itself, that is, B does not belong to B; On the other hand, if B does not belong to B, then B is not an element of B, so B belongs to itself, that is, B belongs to B.
In this way, using the concept of * * *, Russell deduces the paradox that *** B does not belong to B if and only if * * * B belongs to B.
Later, Russell himself put forward a popular version of Russell's paradox, namely Barber's paradox [10].
The barber announced a principle that he would only shave people in the village who didn't shave themselves.
So the question now is, who should shave the barber's beard? .
If he shaves himself, then he is the one who shaves in the village. According to his principle, he shouldn't shave himself. If he doesn't shave himself, he is the one who doesn't shave in the village, and then he will shave according to his own principles.
There is also a paradox: a barber shaves if and only if he doesn't shave.
This is the famous Russell paradox in history.
The appearance of Russell's paradox shook the foundation of mathematics, shocked the whole field of mathematics and led to the third mathematical crisis.
3.2 the impact of the third mathematical crisis
The appearance of Russell's paradox shook the * *, which was originally the cornerstone of the whole mathematics building, and naturally aroused people's doubts about the validity of the basic structure of mathematics.
The genius of Russell's paradox is that it only uses the concept of * * * itself, and does not involve other concepts, which is even more puzzling.
The third mathematical crisis caused by Russell's paradox made mathematicians face great difficulties.
Mathematician Frege wrote at the end of the second volume of Basic Mathematics [1 1]: "For a scientist, nothing is more disappointing than when his work was just finished, one of its cornerstones collapsed.
When the printing of this book was almost finished, a letter from Mr. Russell put me in this position. "Visible, the third mathematical crisis makes people face an embarrassing situation.
However, no one will avoid science, and mathematicians immediately devote themselves to eliminating paradoxes. Fortunately, the root of Russell's paradox was soon found. It turns out that when Cantor put forward the theory of * * * *, he didn't restrict the concept of * * *, which enabled him to construct a "collective of all * * * *", which was so great that a paradox arose.
Many mathematicians have made unremitting efforts to eliminate various paradoxes in * * * * theory, especially Russell paradox.
For example, the school of logicism, represented by Russell [12], put forward type theory and later theories of twists and turns, finite size, classlessness, bifurcation, etc., which all played a certain role in eliminating paradoxes; The most important thing is the axiomatization of the * * * theory put forward by the German mathematician Zermero. Zermelo believes that an appropriate axiomatic system can limit the concept of * * * and logically ensure the purity of * * *. He first put forward the axiomatic system of * * * *, and later by Rankel and Feng? The supplement of Neumann et al. forms a complete axiomatic system of * * * (ZFC system) [5]. In the ZFC system, * * * and "belonging" are two undefined original concepts with ten axioms.
With the establishment of ZFC system, various contradictions have been avoided, thus eliminating a series of * * * paradoxes represented by Russell's paradox, and the third mathematical crisis has disappeared.
Although the paradox is eliminated, the mathematical certainty is lost step by step. It is difficult to say which axioms are true and which are false in modern axioms, but it cannot be ruled out. They are closely related to the whole mathematics, so the third crisis was solved on the surface, but continued in other forms in essence [7].
In order to eliminate the third mathematical crisis, mathematical logic has also made great progress. Proof theory, model theory and recursion theory were born one after another, and basic mathematical theory, type theory and multivalued logic appeared.
It can be said that the third mathematical crisis greatly promoted the modernity of basic mathematical research and mathematical logic, which directly caused the "golden age" of mathematical philosophy research.
4 conclusion
Three mathematical crises in history have brought great troubles to people. The emergence of the crisis made people realize the defects of the existing theory. The appearance of paradox in science often indicates that human understanding will enter a new stage, so paradox is the product of scientific development and one of the sources of scientific development.
The first mathematical crisis made people discover irrational numbers and established a complete real number theory. Euclidean geometry also came into being, and a system of geometric axioms was established. The emergence of the second mathematical crisis directly led to the emergence and perfection of limit theory, real number theory and * * * theory, which made calculus on a solid and perfect basis. The third mathematical crisis made the theory of * * * a complete axiom system of * * * * (ZFC system), which promoted the basic research of mathematics and the modernity of mathematical logic.
The history of mathematics development shows that there is a close relationship between the in-depth study of mathematics foundation, the emergence of paradox and the relative solution of crisis. Every time the crisis is eliminated, it will bring many new contents, new understandings and even revolutionary changes to mathematics, so that the mathematical system will achieve new harmony and the mathematical theory will be further deepened and developed.
The existence of paradox reflects that in a certain historical stage, there will be many contradictions between mathematical concepts and principles, which will lead to people's doubts and sense of crisis. But everything is gradually developed and perfected in the process of constantly generating contradictions and solving contradictions. When the old contradictions are solved, new contradictions will arise. In this process, people will accumulate new knowledge and develop new theories.
Mathematicians' research and solution to paradox promoted the prosperity and development of mathematics. The appearance of paradox and crisis in mathematics not only brings trouble and disappointment to mathematics, but also brings new vitality and hope to its development.
The history of paradoxes and crises in mathematics also illustrates this point: the existing paradoxes and crises have been eliminated, and new paradoxes and crises have emerged.
But people's understanding is developing, and paradox or crisis can be solved sooner or later.
"produce paradoxes and crises, then try to solve them, and then produce new paradoxes and crises." This is an endless and repeated process, which constantly promotes the development of mathematics, and this process is also an important development process of mathematical thought.
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