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On irrational numbers
Around 370 BC, Plato's student eudoxus (about 408-355 BC) solved the problem of irrational numbers. He created a new proportion theory by purely axiomatic method, and skillfully handled commensurability and incommensurability. His method of dealing with incommensurability was included in the second volume of Euclid's Elements of Geometry (Proportional Theory). [5] and it is basically consistent with the modern explanation of irrational numbers obtained by Dedekind in 1872. The treatment of similar triangles in China middle school geometry textbooks after 2 1 century still reflects some difficulties and slight speculation caused by incommensurable measurement.
On Zeno Paradox
Zhi Nuo's four paradoxes were later successfully explained by Aristotle and others.
The first paradox: Burnett explained Zhi Nuo's "dichotomy": that is, it is impossible to pass through an infinite number of points in a limited time, and it is necessary to pass half of a given distance to complete the journey, so it is necessary to pass half, and so on until infinity. Aristotle criticized Zhi Nuo for making a mistake here: "He thinks that it is impossible for a thing to pass through infinite things in a limited time, or to contact infinite things alone. It should be noted that length and time are said to be "infinite" with two meanings. Generally speaking, all continuous things are said to be "infinite", which has two meanings: either divided infinity or extended infinity. Therefore, on the one hand, it is impossible for things to contact with infinite things in a limited time; On the other hand, it can contact with things that are divided into infinity, because time itself is divided into infinity. Therefore, things that pass through infinity are carried out in infinite time instead of finite time, and contact with infinite things is carried out in infinite times instead of finite times.
The second paradox: Aristotle pointed out that this argument is the same as the previous dichotomy, and the conclusion of this argument is that people who run slowly can't catch up. So the solution of this argument must be the same method. It is wrong to think that the leading things in sports can't catch up, because the leading time can't catch up. However, if Zhi Nuo allows it to cross a limited distance, it can catch up. [4]
The third paradox: Aristotle thinks that Zhi Nuo's statement is wrong, because time is not composed of inseparable' present', just as any other quantity is not composed of inseparable parts. Aristotle believes that this conclusion is caused by taking time as' now'. If we are not sure about this premise, this conclusion will not appear.
The fourth paradox: Aristotle believes that the mistake here is that he regards the time when a moving object passes through another moving object as the time when it passes through a stationary object of the same size at the same speed, but in fact they are not equal.