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, together with the establishment of real number theory and * * * 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.
"Ask this sentence is true or not? Mathematically, this is a concrete example of Russell's paradox.
The *** R defined by Russell in this paradox is considered by almost all * * theorists as a * * * thing that can legally exist in a simple * * * theory.
Even so, what is the reason? This is because r is * * *. If R contains itself as an element, there is R, then there is R from the perspective of * * *.
A * * * really contains itself, and such a * * * 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 * * * must follow the basic principles of R R, otherwise it is illegal.
In this way, the * * * of all R R defined in Russell's paradox should be the * * * of all legal * * * that is, the * * * of all * * *, that is to say, the same thing contains all similar things, which will inevitably lead to the biggest such thing.
In the final analysis, R is the "maximum * * *" that contains everything.
Therefore, it is clear that, in essence, Russell paradox is the biggest paradox stated in negative form.
Since then, mathematicians have been looking for ways to solve this crisis, one of which is to build the theory of * * * on a set of axioms to avoid paradox.
First of all, the German mathematician Zermero put forward seven axioms, established a kind of * * * theory that won't produce paradox, and formed an axiomatic system of * * * theory without contradiction (the so-called ZF axiomatic system) through the improvement of another German mathematician Friedrich Kerr, and the mathematical crisis was alleviated here.
Now through the study of discrete mathematics, we know that * * * theory is mainly divided into Cantor * * * theory and axiomatic * * * theory, and * * * is obtained by defining complete set I and empty set first, and then through a series of unary and binary operations.
The theoretical system of * * * based on the seven axioms has avoided Russell's paradox and enabled modern mathematics to develop.