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 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.