Thinking is an indirect and generalized reflection process of the generality and regularity of things, and it is also a complex and advanced psychological process. According to whether it can be programmed, thinking can be divided into two basic types: logical thinking and illogical thinking. Since the birth of mathematics, mathematics and logic are inseparable. Logical thinking method is the most commonly used and basic thinking method in mathematics. The so-called logical reasoning refers to the thinking process of deducing new judgments by observing logical laws and rules according to known judgments.
Inductive reasoning is a kind of reasoning method that logically deduces the causal relationship between various phenomena based on the experience of many individual things through various means (observation, experiment, analysis, comparison, etc.). ), and gradually transition to a generalized general rule.
Inductive reasoning can be divided into complete induction and incomplete induction according to whether the object it examines is complete or not.
First, complete induction
Complete induction is a method of reasoning according to the attributes of all objects of a certain kind of thing. In mathematics, it can be divided into exhaustive induction and classification.
1. Exhaustive induction
Exhaustive induction is a complete induction commonly used in mathematics. It is a kind of inductive reasoning. When a limited number of objects are used to study a certain kind of things, the attributes of all the objects are discussed separately. When it is affirmed that they all have certain attributes (making special judgments), the general conclusion (full name judgment) that these things all have such attributes is drawn.
The objects investigated in mathematics are mostly infinite, and this method is not applicable in many cases. But for some infinite objects, if they can be divided into a limited number of categories to study separately, this is the classification method.
2. Classification method
The so-called classification can be defined in set language as follows:
There are many problems in middle school mathematics that need to be proved by complete induction. In the proof, according to all possible situations in the premise, the research objects are classified as mentioned above, and then proved separately by class. If each category is proved, the full name judgment (conclusion) will be obtained, which is the classification method. For example, in sine theorem, the ratio of sine of side to diagonal is equal to the diameter of circumscribed circle, which is proved in three cases: acute angle, right angle and obtuse angle. If every premise of complete induction is true, then the conclusion must be true, so it is a strict reasoning method. It can be used to prove mathematically.
Second, incomplete induction.
It is often difficult to use complete induction in mathematics, not only because some things we examine contain infinite objects and cannot be classified by limited methods, so we cannot use exhaustive method; Moreover, it is not easy to list those limited, but many things, so people often only summarize them according to some objects with certain attributes. This reasoning method is called incomplete induction, which is based on the fact that some objects of a class of things have certain attributes and make a general conclusion that all such things have such attributes.
From the history of mathematics development, we can clearly see that both the emergence of a new branch of mathematics and the definition of a concept have experienced a period of accumulating empirical materials. It is the most preliminary but basic work in mathematics to discover its laws and summarize mathematical theorems or principles from a large number of materials obtained from observation and experiments. Gauss said that many of his discoveries were made by induction. Although incomplete induction can not be used as a rigorous demonstration method, it can enable us to quickly discover some laws of quantitative relations and provide us with research directions. Many famous theorems in the theory of prime number distribution, such as prime number theorem, Bertrand theorem, Dirichlet theorem, etc. All of them are first summed up from experience through incomplete induction and become conjectures, and then proved by strict mathematical derivation. There are many conjectures obtained by incomplete induction, which initially reveals the distribution law of prime numbers, but it has not been proved so far. So mathematicians attach great importance to the role of incomplete induction. In middle school textbooks, the algorithm is summed up from the calculation of specific numbers, which is not completely summarized. In mathematics, incomplete induction can be divided into enumeration induction and causal induction.
1. Enumeration induction
Enumeration induction is to find a few special objects to experiment, then summarize the characteristics of * * *, and finally put forward a more reasonable guessing method. The steps can be summarized as "experiment-induction-conjecture". As for the number of special objects to be investigated, it depends on the specific situation.
2. Causal induction
The characteristics of causality, in some successive phenomena, through the related changes of some phenomena, summarize the causal relationship between phenomena. This method is called causal induction. It can be roughly divided into the following five categories.
(1) seeking common ground method: looking for the same elements from different occasions, that is, finding that there is only one factor in various conditions, then A is the reason for A. ..
(2) Difference method: Find out the causal relationship from the differences between the two occasions.
(3) Seeking common ground while reserving differences * * * The same method: exploring the combination of seeking common ground while reserving differences and looking for causal links.
(4)*** Reform: From the change of one phenomenon to the change of another phenomenon, find out the causal relationship between the two phenomena.
(5) Residual method: In a group of complex phenomena, subtract the phenomena with known causality, and explore the causes of other phenomena.
Among the five methods, the most basic ones are 1 and 2, both of which are methods to find causality.
The objective basis of incomplete induction is individuality and * * *, and individuality contains * * *, through which we can know * * *; Some phenomena in personality reflect the essence, some do not, some attributes are shared by the whole people, and some attributes only exist in some objects, which determines that the conclusions summarized from personality are not necessarily the * * * essence of things, nor do they necessarily grasp the essence of things. The objective basis of incomplete induction determines the logical characteristics of this reasoning: although it is a method to expand knowledge and discover truth, it is often an inaccurate and probabilistic reasoning. The conclusion put forward by incomplete induction is only a predictive hypothesis, and whether it is correct or not depends on strict proof or counterexample.