Three crises in the history of mathematics _ work

Author: Yang Jinzhou

There are three crises in mathematics in history, as well as in economic crisis. The first crisis occurred in ancient Greece between 580-568 BC, the mathematician Pythagoras established the Pythagoras School. This school integrates religion, science and philosophy, the number of which is fixed, the knowledge is kept secret, and all inventions are attributed to the leader of the school. At that time, people's understanding of rational numbers is still very limited, the concept of irrational numbers is even more ignorant, the Pythagorean School of Numbers, originally refers to integers, they do not think of fractions as a number, but only as the ratio of two integers, they mistakenly believe that all phenomena in the universe are attributed to the ratio of integers or integers. Hibersos, a member of the school of thought, finds by logical reasoning that the diagonal length of a square with an L is neither an integer nor a ratio of integers. Hibersos's discovery was considered "absurd" and a violation of common sense. It not only seriously violates the Pythagoras School's Creed, but also shocks the Greek's traditional ideas at that time. The Greek mathematicians at that time deeply disturbed, legend Hibersos because this discovery was thrown into the sea drowned, this is the first mathematical crisis. The crisis was solved by introducing the concept of irreducible quantities into geometry. Two geometric segments, if there is a third segment to be able to simultaneously measure them, it is said that these two segments are accessible, otherwise known as irreducible. On one side and diagonal lines of a square, there is no third segment that can simultaneously measure them, so they are irreducible. It is clear that the so-called mathematical crisis ceases to exist, as long as the existence of irreducible amounts is accepted so that the geometry is no longer limited by integers. The study of irreducible quantity began in the Eudox of 4th century BC, and its results were absorbed by Euclid, some of whom were received in his "geometrical original". The second mathematical crisis took place in 17th century. After the birth of calculus in the 17th century, due to the theoretical foundation of Calculus, The mathematical world was in chaos, that is, the second mathematical crisis. The formation of calculus has brought about revolutionary changes in the field of mathematics and has been widely used in various fields of science, but there are contradictions in theory in calculus. Infinitesimal is one of the basic concepts of calculus. The main founder of Calculus Newton in some typical derivation process, the first step is to divide the denominator by an infinitesimal, and of course the infinitesimal can not be zero, and the second step Newton takes the infinitesimal as zero, removes the items that contain it, and obtains the desired formula, which is proved to be correct in the application of mechanics and geometry, But its mathematical derivation process is logically contradictory. The focus is: whether the infinitesimal is zero or non-zero. If it's zero, how can you use it as a divisor? If it's not 0, how can you remove those items that contain a small amount of infinity? It was not until 19th century that Cauchy developed the theory of limits in detail and in a systematic way. Cauchy thinks that to use infinitesimal as a definite quantity, even zero, is not justified, it will contradict the definition of limit. The infinitesimal amount should be how small is small, so in essence it is a variable, and is the limit of the amount of zero, so KOThe West clarifies the concept of the infinitesimal of the predecessors, and frees the infinitesimal from the bondage of metaphysics, and the second mathematics crisis is solved basically.

The resolution of the second mathematical crisis made calculus more perfect.

The third mathematical crisis occurred at the end of 19th century. At that time, British mathematician Russell divided the set into two kinds.

The first collection: The collection itself is not its element, that is, a A; the second collection: The collection itself is an element of its a∈a, such as a collection of all collections. So for any one set B, not the first set is the second set.

Suppose that the totality of the first set consists of a set m, then M is either a set or a second set.

If M belongs to the first set, then m should be an element of M, i.e. M∈m, but the set that satisfies the m∈m relation should belong to the second set, and there is a contradiction.

If M belongs to the second set, then M should be satisfied with the m∈m relationship, so M is the first set of contradictions.

The theory of prisoners formed by the above reasoning process is called Russell Paradox. With the establishment of strict limit theory, the first second crisis in mathematics has been solved, but the theory of limit is based on the theory of real numbers, and the theory of real numbers is based on set theory, and now there is a Russell paradox in set theory, which has formed a bigger crisis in the history of mathematics. Since then, mathematicians have begun to find a solution to the crisis, one of which is to set theory on a set of axioms to avoid the paradox. First of all, the German mathematician Merrow, who proposed seven axioms, established a set theory that would not produce paradoxes, and after the improvement of another German mathematician, Francis Kerr, formed an axiomatic system of set axioms. The so-called ZF Axiom system. The mathematical crisis eased down. The mathematical crisis has brought new impetus to the development of mathematics. In this crisis, the theory of aggregation gets faster development, the progress of Mathematics Foundation is quicker, and the mathematical logic is more mature. However, contradictions and unexpected things continue to emerge, and they will continue to do so in the future.