Continued above: The Second Mathematical Crisis: The Lost Infinite Ghost and Berkeley's Paradox
03 Stricter analysis
Berkeley's attack on calculus is well-reasoned and well-founded, and mathematicians, even if they are unwilling, must come up with a theory that stands up if they want to refute it.
Since then, mathematicians have tried to establish a rigorous theoretical foundation for calculus.
In the 18th century, Euler, d'Alembert, and Lagrange all tried, but none of them succeeded.
Euler was one of the greatest mathematicians of the 18th century, and he was successful in many fields.
In the field of calculus, he wrote "Introduction to Infinite Analysis", "Principles of Differential Calculus", and "Principles of Integral", which became the "embodiment of analysis".
But Euler's interpretation of infinitesimal quantities is still vague, and he argues that 0:0 can be equal to an arbitrarily finite ratio.
In his view, an infinitesimal quantity is actually a quantity equal to zero.
The term "analysis" was first used as opposed to the synthetic approach, which refers to assuming that the conclusion is true and then working backwards.
According to Veda, algebra is a method of analysis (backwards), in which the root of an equation is derived from the conclusion of the equation and then backwards.
In the 17th century, analysis was synonymous with algebra. Both Newton and Leibniz believed that calculus is an extension of algebra, only using infinity as an object to calculate, and infinitesimal is the most important concept, so calculus is also called infinitesimal analysis.
After Euler, the term analysis became more popular, and Euler made the main object of analysis a function by defining it. In the 18th century, mathematicians did not solve the basic problems of calculus, but the application of analysis flourished, and branches such as differential equations, complex variable functions, differential geometry, analytic number theory, variational method, and infinite series appeared, which made analysis and algebra and geometry called the three major disciplines of mathematics.
The importance of analysis increased, and there was an urgent need to establish a solid foundation for it, but the movement to rigorous analysis was not finalized until the 19th century.
The pioneer of establishing a rigorous basis for analysis was the Czech mathematician Bolzano. When he tried to prove the intermediate value theorem in calculus, he found that he had to first define what a "continuous function" was. So he eliminated geometric intuition and gave a strict definition of mathematics. But Bolchano's work was buried for a long time, and it was the French mathematician Cauchy who really had a huge impact on the rigor of analysis.
Cauchy was quite prolific in mathematics during his lifetime, and his results involved almost all areas of mathematics, probably second only to Euler in terms of output. He founded his own journal and published his own papers. In addition, after the launch of the Bulletin of the Paris Academy of Sciences, he published a total of 589 articles in 20 years, so much so that the Academy had to limit the submission of papers to no more than four pages. Cauchy's Tutorial of Analysis in 1821 and Introduction to the Tutorial of Infinitesimal Calculations in 1823 made him the epitome of rigorous analysis.
He begins by redefining the limit: "When a variable successively takes a value that is infinitely close to a fixed value, and the final difference from this fixed value is as small as the hour, that value is called the limit of all other values." On the basis of the definition of limits, Cauchy established a strict definition of the concepts of infinitesimal quantities, infinitely large quantities, continuous, derivatives, differentiation, integrals, etc. For example, for Cauchy, infinitesimal quantities are variables with zero as the limit, so infinitesimal quantities are included in the category of functions, and infinitesimal quantities that have puzzled people for more than 2,000 years have been tamed by Cauchy.
Similarly, the derivative can be defined by the limit. With the derivative, it is possible to clearly explain what instantaneous velocity is, and from mathematical calculations, it can be clearly proved that there is an instantaneous velocity at every moment of the motion of an object, which also refutes Zeno's paradox that "the arrow does not move".
As for integrals, Cauchy insisted that before calculating integrals, he first proved that the integrals of continuous functions existed, which became the turning point of analysis from relying on intuition to becoming rigorous.
In addition, Cauchy also established a complete theory of infinite series, put forward the concepts of absolute convergence and conditional convergence, and solved many strange theories left over from the 18th century series problem. Overall, Cauchy's contribution to analysis was decisive, and his Analytic Course became the starting point for the movement of analytical rigor.
But there are still some minor flaws in Cauchy's theory, which will be remedied by the German mathematician Weierstrass.
Weierstrass has always been known for his rigor, and his mathematical attitude and textbooks have become rigorous models and standards.
He believed that the statement of "a variable infinitely tends to an limit" in Cauchy's concept of limit still existed in motion intuition, so in order to eliminate the ambiguity of this descriptive language, he gave a ε-δ definition of limit, and used inequality intervals to strictly express the limit, which completely freed the limit and continuity from the dependence on geometry and motion, and was based on the clear definition of numbers and functions. For example, the limit of the function x² at x→2 can be described as: "Any positive number ε always exists in a positive δ such that at 0<|x-2|<δ, there is |x²-4|<ε." In addition, Weierstrass also proposed the concept of consistent convergence, refining the theory of series.
In 1872, Weierstrass proposed a well-known counterexample in the history of analysis. He constructed a series of trigonometric functions that are continuous everywhere, but not differentiable everywhere, and shocked the entire mathematical community. This function is known as Weierstras's sick function. By this sick function, Weierstrass illustrates very fully that curves built by motion do not necessarily have tangents, so that the basis of calculus should eliminate geometric intuition, and build only on numbers. If Newton and Leibniz had discovered this sick function, they might have been so frustrated that they would have given up calculus. However, in the 19th century, Weierstrass proposed the sick function, which became a strong shot in the arm to promote the strictness of the analytical basis, and further made mathematicians realize that in order to establish a rigorous basis for analysis, it is necessary to have a strict definition of the real number system.
German mathematician Dedekin took a crucial step in defining real numbers. In 1872, in his book "Continuity and Irrational Numbers", Dedekind very skillfully applied a method called "Dedekind Segmentation" with the help of the correspondence between straight lines and real numbers, proving that density and continuity are two different properties, thus clearly defining the concept of real numbers.
Dedekind made an analogy that if a straight line is cut into two sections with a knife, then there must be a breakpoint, and this break point must be one of the two ends of the straight line. If there are only rational numbers on the line, and then the rational numbers are examined, and we will find that although the rational numbers are dense (there must be another rational number between the two rational numbers), they are not continuous√!
Therefore, if there are only rational numbers in the line, this knife will cut a gap, so the rational numbers are not continuous, and these gaps need to be filled by irrational numbers, so that irrational numbers are clearly defined.
Strictly speaking, a division of a rational number is called a real number, if there is no gap in the division it is a rational number, and if there is a gap in the division it is an irrational number.
It was also in 1872 that Weierstrass and Cantor respectively established the definition of irrational numbers, describing the properties of real numbers from different dimensions, so that a complete theory of real numbers was established.
Therefore, 1872 is known as the "year of irrational numbers". More than 2,000 years after Pythagorean Hippassos of ancient Greece was thrown into the sea, the irrational numbers that caused the first mathematical crisis were finally clearly defined, and the first mathematical crisis was finally over.
At the same time, the concept of real numbers to be used in the analysis is based on the division of rational numbers, which in turn is based on the arithmetic of natural numbers, which are fundamental and obvious to many mathematicians.
Specialized mathematical symbols play an important role in this process. More and more mathematicians are aware that mixing symbols with other concepts can cause a lot of confusion, especially the concepts of continuity such as space and time related to human intuition. This also laid the groundwork for the work of Piano, Frege, and Russell. Mathematics can only be used for independent thinking when it is freed from the intuitive dependence on optics, mechanics, and geometry that began in Newton's time.
However, in order to define irrational numbers, Dedekind and Cantor inevitably introduced infinite sets, which became the starting point for triggering the third mathematical crisis.
From this point of view, the three mathematical crises in the history of mathematics are coherent and intrinsically related, which can be described as one after the other.
The theory of computation is hidden in these three mathematical crises.
For example, Cantor's research begins with the "uniqueness problem of trigonometric series expressions of functions" and then touches on the infinite set of points.
The above is an excerpt from Wu Hanxin's "Calculation"! Transferred from the blog point of view, [Meet Mathematics] has been forwarded with permission.
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