Euclidean geometry had the greatest influence in ancient mathematics. It was praised and imitated by scholars of all ages for its logical axiom-deduction, and Descartes was the most outstanding and representative among these scholars. Descartes believes that the truth of mathematical axioms is certain. To know this, we only need to resort to "second sight"-an intuitive investigation, and the truth of axioms can be made clear to us. Because, on the one hand, axiom itself is the most common and simple thing, which is easier to understand than anything else; On the other hand, intuition, which is used to discover and prove the axiom's self-evident, "is not the evidence of repeated senses, nor is it a deceptive judgment wrongly constructed from imagination, but the concept given to us by a pure and dedicated mind so quickly and clearly that there is no need to doubt what we understand." "(1) That is to say, intuition belongs to pure reason, and the supreme rational intuition is deterministic. Descartes found that mathematical research follows the cognitive order from simple to complex, just as geometry goes from the simplest axiom to the most complex conclusion through deduction. He once said: "Geometrists are used to drawing conclusions with simple and easy reasoning chains in extremely difficult proofs. This makes me imagine that all things that people can know are so interrelated. There is nothing that we can't reach because we are far away, and there is nothing that we can't find because we are hidden, as long as we try our best to keep the order necessary to deduce another truth from one truth in our minds. "

(2) It can be seen that Descartes put forward the idea of understanding the order from simple to complex in a deductive way. Intuitively determining the truth of the simplest things, and then gradually deducing the most complex things from the simplest things is the methodological essence of Descartes' cognitive order thought. He pointed out: "The whole method is just to arrange those things that the mind should observe in order to discover some truth."

(3) It should be noted that the deductive method advocated by Descartes here is fundamentally different from the deductive logic of syllogism. Deduction of syllogism is a purely logical method, which can not be used to explore the unknown, but can only be used to convey the known. (4) It is an independent method, which does not require whether the premise is correct or not, nor can it guarantee whether the conclusion is correct. Deduction advocated by Descartes is not only a logical method, but also can be used to convey the known and explore the unknown. It is not a method used alone, but it is used with intuition to ensure the correctness of the premise first, and then deduce the correct conclusion from the correct premise. Here, deduction is an intuitive deduction method and an intuitive expansion. Even a scholar who skillfully uses intuition, a simple or even more complicated deductive process can be intuitive. Therefore, the deduction advocated by Descartes is actually an intuition-deduction method with intuition as the core. It is a proof method with discovery function (Descartes called it "synthesis" method), which can be used not only to explore the unknown and obtain knowledge about the unknown, but also to determine the truth of this knowledge.

Intuition for the simplest things and the cognitive order from which complex conclusions are derived are the essence of Descartes' mathematical methodology. However, the simplest things that can be put into intuition are limited after all, and the objects we know are extremely complicated. It is not enough to rely on intuition-deduction and follow the cognitive order from simple to complex. When the object of cognition is not the simplest thing that can be clearly and intuitively understood, nor something that can be deduced from a self-evident axiom through deduction, our research must follow an appropriate cognitive procedure: "We must gradually simplify the easily confused and ambiguous proposition (that is, repeated quotation) into other simpler propositions (that is, simple quotation), and then proceed from the simplest proposition in all intuitive propositions and try to rise to know all others step by step. (5) That is, add a cognitive process from complex to simple in front of the cognitive process from simple to complex-the "analysis" process, and add an "analysis" method that Descartes thinks is the discovery method in front of the "synthesis" method as a proof method. Descartes pointed out: "If we want to understand a problem thoroughly, we must abstract it from any redundant ideas, boil it down to a very simple problem, and divide it into as small parts as possible, without forgetting to list them one by one." (6) Through this discussion, we can not only know that the method of "analysis" is actually the simultaneous application of analysis and induction, but also know that the process of "analysis" is actually a cognitive process from concrete to abstract and from individual to general. Descartes believes that understanding things related to form can not be separated from the help of sensory experience, and geometric objects and propositions are related to geometric figures, so it is impossible to grasp geometric objects and propositions without experience. To study the proposition of geometry, we need to analyze, summarize (or enumerate) our experience about geometry with the help of geometry. But experience is only a more important rational supplement than oneself, and the analysis and induction of experience is only a more important intuitive and deductive supplement than oneself. Because, on the one hand, the certainty of analytical induction is not as high as that of intuitive deduction, and analytical induction is only used when intuitive deduction is not feasible. Descartes pointed out that induction "only refers to the kind that can reach the truth more definitely than any other kind of proof within the scope of simple intuition;" Whenever we can't attribute some knowledge to simple intuition, such as when we give up all the connections of syllogism, then this is the only way to trust completely. "(7) On the other hand, the purpose of analysis and induction is only to find an object order suitable for intuitive-deductive reasoning.

The above research shows that Descartes' mathematical methodology has obvious rationalism characteristics because of its extreme respect for intuition-deduction. This feature is characterized by cutting off the dialectical relationship between sensibility and rationality, opposing them and making sensibility subordinate to rationality. Descartes believes that empirical induction (or discovery) is only a supplement to rational deduction (or proof). In this regard, people can't help asking why Descartes' mathematical methodology has the characteristics of rationalism. The answer to this question is naturally helpful to grasp Descartes' mathematical methodology in a deeper level. The author believes that there are at least two interrelated reasons for the formation of rationalism.