Mathematics three times crisis (four) the first mathematical crisis _ Mathematics

Pythagoras School believes that any two segment A and b can be public, meaning that there is a small segment d as a common unit of measure between A and B, making a=nd,b=md. This means b/a=m/n, where m and n are all integers. Therefore, when the Pythagoras School believes that two segments A and b can be public, the expression in our present language means that the ratio of any two segments is an integer or a fraction. In short, it is a rational number. Thus the discovery of the Hippasos metric means that the ratio of square diagonal to edge length is neither an integer nor a fraction, in short, not a rational number, but a completely new number that people do not understand at the time. These numbers are later called irrational numbers. By the way, the ancient Greeks used the term "reasonable" and "unreasonable", which were intended to be "comparable" and "not comparable". Later in the translation process, in the "comparable" meaning, derived from the "rational (reasonable)" and "unreasonable" meaning. Later, the former righteousness gradually forgotten, only later meaning. Thus, the "comparable number" and "the number of" can be converted into: The former is a reasonable number, the latter is unreasonable. Finally, when translated into Chinese, there is a "rational number" and "unreasonable number" of the method.

We now know that the √2 discovered by Hippasos is the first irrational number born in human history. The discovery of irreducible or irrational numbers is perhaps the greatest contribution of the Pythagoras school, in the eyes of the present man. However, at the time it was discovered why the ancient Greeks thought it was a paradox and caused such a serious problem. It is necessary for us to explain this further.

First of all, the discovery shook the Pythagorean school's mathematical and philosophical roots, which would overturn the Pythagoras School's basic philosophy of "everything counts." The discovery of irreducible amounts indicates that some quantities cannot be expressed in numbers. This declares them that "all things and phenomena can be summed up as the number of integers and integers" of the harmony theory of the bankruptcy; and their understanding of the nature of the universe in the harmonious theory of number is vain.

Second, the finding destroys the mathematical idea behind the idea that "arbitrary segments are accessible". In particular, the Pythagoras school accepts the idea of a mathematical atom theory. The simple idea is that the lines are connected by atomic sequence, like necklaces are made up of a string of beads. Atoms may be very down, but they are of the same texture, in size, they can be the last unit of measure. This understanding constitutes the geometric foundation of Pythagoras School.

Note: The mathematical and philosophical foundations of "All things are counted" are shaken; the mathematical idea behind the idea of any two-digit can be destroyed.

In addition, the early Greek mathematicians thought that any amount of the public was also based on another reason. At that time, there is a comparison of the number of methods, that is, today's Euclidean method. If A and B are two lines long, according to the mathematical atom theory, they believe that by doing so, they will always encounter a positive integer so that A and B are multiple integers of this positive integer.

More importantly, this discovery destroys some common sense that people gain through experience and intuition. According to experience and all kinds of experiments, any quantity can be expressed as a rational number in the range of any precision. This is not only a universally accepted belief of the ancient Greeks, but it is also true that today, when measuring technology has developed at a high level, this assertion is no exception. For daily life, rational numbers are enough. For scientific research, only rational numbers are sufficient. For all practical purposes of measurement, rational numbers are fully sufficient.

and the existence of irreducible amounts means that when we compare the lengths of two segments using the Euclidean method, this process will go on indefinitely, endlessly, meaning that even if we have an ideal ruler with very very fine scales, we cannot measure all the lengths, because when we face irreducible amounts, We need to see the scale on the ruler infinitely, and never see it through, meaning that when you compare the lengths of two segments, sometimes you never find a common unit of measure, meaning that the rational numbers that are sufficient for all the practical purposes of the measurement are not enough for math ...

In short, this means that many of the assertions that have been convinced by people's experience, fully consistent with common sense, are overthrown by the presence of small √2. This should be how contrary to common sense, how absurd thing. How difficult it is to admit this "absurd" thing. It simply overthrew what was previously known. In fact, the discovery of irreducible quantity is not only a fatal blow to the Pythagoras school, it is a great shock to the idea of all the ancient Greeks at that time.

The influence of the discovery of irreducible quantity not only manifests in the violent impact and destroys the idea that many traditional viewpoints and Pythagoras school insist, but also expresses in its negation to concrete mathematics result. In fact, many of the Pythagorean theorem proofs are based on any amount of covenant. As they argue about similar geometric theorems, they are based on this hypothesis. As an example, they have proved the theorem that the ratio of the area of the equal-height triangle equals the ratio of their bottom. As shown in Fig. Two triangular abc and ADE, their bottom side BC and De are on the same line MN. Thus the two are equal in height. Then the Pythagoras school proved the ratio of the area to the corresponding bottom by the following way.

Because all the amount can be public degree, so according to the definition of the public, can be set BC as a unit of the M-Times, and de for the public degree of n times. The BC is divided into m, and vertex a connection, and then get m a small triangle, the de and so into N, and then get n small triangles. These small triangles are equal in height, and the area equals. And the area of ABC equals m this small triangle, the area of Ade equals n such small triangle. Therefore, can be introduced: triangular ABC area: The area of the triangle ade = M:n = Bc:de.

However, this proved to be completely ineffective due to the discovery of the not-so-common metric. Because the foundation that has been established has collapsed. As a result, mathematical conclusions based on "any two line segments can be reached" were lost, and all the proofs based on that assumption were shattered, and many theorems of the established geometry had to disintegrate. And most embarrassing of all, people believe the correctness of these theorems, but with the discovery of the not-so-measurable, they cannot give strong evidence to support their views. This is what people sometimes call the "logic shame" of Greek geometry.

In the face of the measure of the public, the ancient Greeks fell into confusion and confusion. Worse, there can be no one to deal with the multiple destructive blows that are caused by the measure of the public. At that time, it directly aroused people's understanding of the crisis, which led to a big controversy in the history of Western mathematics, known as the "first mathematical crisis."