The First Mathematical Crisis
The first mathematical crisis happened 400 years ago. In ancient Greece, the Pythagorean school opposed it. Count? It is defined that any number can be written as the quotient of two integers, that is, all numbers are rational numbers.
But hippasus, a disciple of this school, found that the side length is? 1? Square, its diagonal is 2? Unable to write the quotient of two integers, the first irrational number was found.
When other disciples of Pythagoras knew about it, they killed hippasus and threw him into the sea in order to maintain the legitimacy of the sect. It seems that when the ancients couldn't solve the problem, they solved the person who asked the question first.
Even so, the discovery of irrational numbers quickly triggered a mathematical revolution, which was called the first mathematical crisis and affected the history of mathematics for nearly two thousand years.
The second mathematical crisis
Calculus is a great invention. Newton and Leibniz were both inventors of calculus, and their ideas of discovery were completely different. However, their definition of the basic concept of calculus is ambiguous, which has been strongly opposed and attacked by some people, among which the strongest attack is the British Archbishop Becquerel, who put forward a paradox:
From the derivation of calculus, we can see that △x is not zero as the denominator, but it is equal to zero in the final formula. This contradiction is disastrous, and mathematicians can't find a solution for a long time. It was not until 100 years after the invention of calculus that French mathematician Cauchy defined infinitesimal with limit that this problem was completely solved.
the third mathematical crisis
Mathematicians always have a dream, trying to establish some basic axioms, and then prove all the theorems of mathematics with strict mathematical logic deduction; After Cantor invented set theory, mathematicians saw the dawn. French scientist Poincare believes that we can build the whole mathematical building with the help of combinatorial theory.
While mathematicians were happy, Russell, a British philosopher and logician, put forward an amazing paradox Russell paradox:
Russell's paradox is popularly described as: in a certain city, a barber whose reputation spread all over the city said: I want to shave all the people in this city who don't shave themselves, and I only shave these people. ? Who should shave the barber's own face?
The proposition of Russell's paradox triggered another crisis in mathematics. Mathematicians worked hard to build a mathematical building, and finally found that the foundation was actually defective. Mathematicians put forward their own solutions. It was not until 1908 that the first axiomatic set theory system was established, which made up for the defects of set theory.
Although three mathematical crises have been solved, the influence on the history of mathematics is very far-reaching. Mathematicians try to establish a strict mathematical system, but no matter how careful they are, there will be defects, including Godel's incomplete theorem later discovered.