The first mathematical crisis, an important event in the history of mathematics, occurred around 400 BCancient GreekPeriod, sinceRadicalFrom the discovery of II to about 370 BCIrrational numberThe definition of appears as an end flag.
The emergence of this crisis has impacted thePythagorean school, also markswestern worldThe beginning of the study of irrational numbers.
In a sense, mathematics in the modern sense, that is, pure mathematics as a deductive system, originated from ancient GreecePythagorean school。It is aidealismThe school flourished around 500 BC.They believe that "all things are counted"integer）Mathematical knowledge is reliable and accurate, and can be applied to the real world. Mathematical knowledge is obtained by pure thinking, without observation, intuition and daily experience.[1]
Definition of rational number
The first mathematical crisis
Integer is an abstract concept generated in the process of computing the limited integration of objects.In daily life, we not only need to calculate individual objects, but also measure various quantities, such as length, weight and time.In order to meet these simple measurement needs, scores are used.Therefore, if the rational number is defined as the quotient of two integers, thenRational numberThe system includes all integers and fractions, so it is sufficient for actual measurement.
Rational numbers have a simple geometric explanation.On a horizontal line, mark a segment of line as the unit length. If its fixed end point and right end point represent numbers 0 and 1 respectively, then the integer can be represented by the collection of points on the line with the interval of unit length. The positive integer is on the right of 0, and the negative integer is on the left of 0.Fractions with q as the denominator can be expressed by points divided into q equal parts every unit interval.Therefore, every rational number corresponds to a point on the line.[1]
Crisis outbreak
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The discovery of irrational numbers
Ancient mathematicians believed that this could use up all the points on the line.However, about the 5th century BC, Hippersos of Pythagoras School found that the right side of an isosceles right triangle cannot be contracted with its hypotenuse.The newly discovered number is called "rational number" because it is opposed to the so-called "reasonable number", namely rational number, within the schoolIrrational number。It was precisely because of this mathematical discovery that Hippersos was thrown into the sea by the Pythagorean school and punished by "drowning".[2]
The right side of a right triangle and its hypotenuse cannotConcordanceThe discovery of this simple mathematical fact puzzled the people of Pythagoras School.It not only violated the Pythagorean creed, but also impacted the Greek belief that "all quantities can be expressed by rational numbers" at that time.Therefore, people usually call the contradiction discovered by Hippersos the Hippersos paradox.[1]
However, there is another saying that it is said that the ratio of the side of the regular pentagon to the diagonal
Is the first irrational number to be discovered.[3]
Zeno paradox
Famous philosopher of ancient GreeceZeno(about 490-425 BC)paradox, has also been recognized as one of the important incentives for the first mathematical crisis by the current mathematical history.
First, "dichotomy".
Moving things must complete half of the journey before reaching the destination, and after completing half of the journey, they must also complete half of the journey... So divided, even infinite, so the distance between them and the destination is infinite, and they will never reach the destination.
Second, "Achilles will never catch up with the tortoise".
Achilles is the fastest hero in Greece, while the tortoise climbs the slowest.However, Zeno proved that the fastest can never catch up with the slowest in the race, because the pursuer and the pursued start to move at the same time, and the pursuer must first reach the starting point of the pursued. By analogy, there is an infinite distance between them, so the pursued must always lead.
Third, "Flying arrow does not move".
Any object must possess certainspaceTo leave one's own space means to lose its existence.The flying arrow can be divided into countless moments when it passes a distance. At each moment, the flying arrow occupies a space of the same size as itself. Since the flying arrow is always in its own space, it is stationary.
Fourth, the "playground".
There are two rows of objects of the same size and number, one row from the end point to the middle point, and the other row from the middle point to the starting point. When they move in opposite directions at the same speed, there will be a contradiction in time.Zeno believes that this can prove that half the time is equal to twice the time.
The above four paradoxes fundamentally challenge the measurement and calculation methods that Pythagoras School has been implementing.[4]
Crisis resolution
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On irrational numbers
In about 370 BC, Eudoxus, Plato's student, solved the problem of irrational numbers.He created a new theory of proportion with purely axiomatic methods, and dealt with it subtlyCommensurabilityAnd incommensurable.His way of dealing with injustice was taken by Euclid《Geometric primitives》Included in Volume II (Theory of Proportions).And it is basically consistent with the modern interpretation of irrational numbers drawn by Didkin in 1872.After the 21st century, the treatment of similar triangles in Chinese middle school geometry textbooks still reflects some difficulties and subtleties caused by incommensurability.[4]
On Zeno Paradox
Zeno's four paradoxes were laterAristotleThe explanation was completed successfully.
The first paradox: Burnett explained Zeno's "dichotomy": that is, it is impossible to pass through an infinite number of points in a limited time. Before you complete the whole journey, you must first go through half of the given distance, so you must go through half of the half, and so on, until infinity.Aristotle criticized Zeno for making a mistake here: "He advocated that a thing could not pass through the infinite thing in a limited time, or contact with the infinite thing separately. Note that length and time are said to be" infinite "with two meanings.Generally speaking, all continuous things are said to be "infinite" with two meanings: either separated infinity or extended infinity.Therefore, on the one hand, things cannot contact with the infinite in number in the limited time;On the other hand, it can contact with things that are divided into infinity, because time itself is divided into infinity.Therefore, through an infinite thing is carried out in infinite time rather than in limited time, and contact with infinite things is carried out in infinite number rather than in the range of finite number.[4]
The second paradox: Aristotle pointed out that this argument is the same as the previous dichotomy. The conclusion of this argument is that slow runners cannot be caught up.Therefore, the solution to this argument must also be the same method. It is wrong to think that what is leading in sports cannot be caught up, because it cannot be caught up during its lead time. However, if Zeno allows it to cross the specified limited distance, it can also be caught up.[4]
The third paradox:AristotleIt is wrong to think Zeno's statement, because time is not composed of indivisible 'present', just as any other quantity is not composed of indivisible parts.Aristotle believes that this conclusion is due to the fact that time is composed of "now". If this premise is not sure, this conclusion will not appear.[4]
The fourth paradox:AristotleHe believes that the mistake here is that he regards the time spent by a moving object passing through another moving object as equal to the time spent by a stationary object of the same size passing through at the same speed. In fact, the two are not equal.[4]
Subsequent impacts
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The first mathematical crisis showed that some truths in geometry had nothing to do with arithmetic, and geometric quantities could not be completely represented by integers and their ratios.On the contrary, numbers can be expressed by geometric quantities.The reverence of integers was challenged, and the mathematical viewpoint of ancient Greece was greatly impacted.Thus, geometry began to occupy a special position in Greek mathematics.At the same time, it also reflects that intuition and experience are not necessarily reliable, but reasoning is reliable.From then on, the Greeks began to set out from the axiom of "self-knowledge", through deductive reasoning, and thus established a geometry system.This is a revolution in mathematical thought and a natural product of the first mathematical crisis.
Looking back at all kinds of mathematics before that, it was nothing more than "calculation", that is, providing algorithms.Even in ancient Greece, mathematics started from reality and applied to practical problems.For example, Thales predicts solar eclipses, calculates pyramid height by using shadows, and measures the distance of ships off the coast, all of which fall within the scope of computing technology.As for mathematics in Egypt, Babylon, China, India and other countries, they have never experienced such crises and revolutions, and they continue to take the road of giving priority to calculation and giving priority to use.However, due to the occurrence and solution of the first mathematical crisis, Greek mathematics embarked on a completely different development path, formingEuclidThe axiom system of The Original and the logic system of Aristotle have made another outstanding contribution to world mathematics.According to historical records, irrational numbers were discovered very early in ancient Greece and China.However, the East and the West have different ways to understand and develop the theory of irrational numbers: the Greeks focused on the length relationship of geometric quantities, starting from the geometric point of view of the incommensurability of line segments, and used logical methods to explore;The Chinese people focus on the calculation of numbers in practical applications, starting with the calculation process of inexhaustible square root, and understanding and establishing its rules through calculation methods.[5]
However, since then, the Greeks regarded geometry as the basis of all mathematics, subordinated the study of numbers to the study of shapes, and cut off the close relationship between them.The biggest misfortune of doing so is to give up the study of irrational numbers themselves, which greatly limits the development of arithmetic and algebra, and the basic theory is very weak.This abnormal development has lasted for more than 2000 years in Europe.[1]
