Philosophical Inquiry On Mathematical Intuition

Intuition is an important source of knowledge in mathematics,science and philosophy,but whether it can be used as the basis of mathematical knowledge is controversial.Throughout the development of mathematics,intuition has been an important source of promoting the progress of mathematics,but it is also the cause of mistakes,and even the intuition of mathematicians will hinder mathematics.Therefore,it is very important in philosophy to study the reliability of intuition as the basis of mathematical knowledge.This article starts from the debate on whether mathematical intuition can be used as a reliable basis for mathematical knowledge,and explores the reliability,limitations and root causes of mathematical intuition from the perspectives of empirical intuition,rational intuition,logic and axiomatization,and realism.Based on this,the logic,mathematical axiomatization and realism are used as a systematic method to confirm the reliability of mathematical intuition and restrict the limitations of intuition.Finally,it attempts to clarify the characteristics of mathematical intuition from the ontological and epistemological level of philosophy,and clarify the mathematical truth which found that mathematical intuition cannot be avoided,and the reliability of mathematical intuition lies in the objective reality of mathematical truth.This thesis includes an introduction,five chapters,and conclusions.The introduction mainly examines the debate on whether mathematical intuition can be used as a reliable basis for mathematical knowledge.By analyzing the current status and differences of research at home and abroad,it shows that this problem is an important epistemological problem in mathematical philosophy,and this is the work to be done in the paper.Then based on the main problems and ideas to be solved in this paper,the logical framework and content of this paper are briefly summarized.Chapter one “Argument about Intuition as the Basis of Mathematics”mainly raises the question of whether intuition is the basis of knowledge.Many mathematicians and philosophers emphasize the importance of intuition.They all emphasize that intuition can be the basis of mathematical knowledge,but the progress of mathematics clearly shows the limitations of mathematician intuition.Whether mathematical intuition can be the basis of mathematical knowledge is an important epistemological issue in mathematical philosophy.Chapter two “Mathematical Intuition and Experience” mainly discusses the empirical intuition which is the basis of axiomatic system and its reliability in mathematics axiomatic system.Euclid's axiom system of geometry has been regarded as a very strict mathematical knowledge system for more than two thousand years.It was not until the advent of non-Euclidean geometry that mathematicians had to acknowledge the existence of unreliable sources of empirical intuition in mathematics.At the same time,the expansion of the number system also reflects the difficulty of people breaking through the constraints of traditional cognition and experience,and further concludes that the empirical intuition as a source of mathematical knowledge is unreliable.Chapter three “Mathematical Intuition and Reason” mainly explores the rational intuition as the basis of mathematical knowledge and its reliability and confirmability.Rational intuition attempts to abandon the uncertainty and ambiguity of empirical intuition to gain insight into the abstract world of mathematics.This rational intuition is an intuition that transcends experience.Whether the rational intuition after verification is better than empirical intuition and can be used as the basis of mathematical reliability needs to further confirmation.Confirmation of rational intuition comes in two ways:(1)objective reality;(2)consistency.Chapter four “Mathematical Intuition,Logic and Axiomatization”mainly discusses whether logic and mathematical axiomatization can be used as a systematic method to confirm the reliability of mathematical intuition and restrict its limitations.Both empirical intuition and rational intuitioncontain unreliable components,but if mathematical intuition is placed in a logical and axiomatic system,unreliable problems can be solved.Mathematicians initially form judgments and establish theories based on their intuitive understanding of mathematical concepts.Intuitive understanding without logical tests led to the occurrence of mathematical paradoxes,and then mathematicians use axiomatic methods to define concepts,modify or extend axioms.The reliability of mathematical intuition becomes the confirmation problem of mathematical axioms.The mathematical axiom system formed by the intuitive understanding of mathematicians is based on the objective reality of mathematical truth.Chapter five “The Philosophical Basis and Reliability of Mathematical Intuition”,this section attempts to clarify the characteristics of mathematical intuition from the perspective of philosophy's ontology and epistemology.Because mathematical axioms require intuitive insights from mathematicians,mathematical intuition is thus subjective in epistemology.The corroboration foundation of mathematical axiom is ultimately the objective reality of mathematical truth,and thus mathematical intuition has ontological objectivity.Therefore,mathematical intuition involves both mathematical ontology and epistemology.Finally,it is pointed out how to unify epistemology and ontology,unify mathematical intuition and mathematical truth,construct mathematical concepts,form mathematical axioms,and form reasonable confirmations based on the objective reality of mathematical truth are important issues in mathematical philosophy.The conclusion is a summary of the he full text work.Analysis shows that mathematical intuition plays a role that cannot be ignored when mathematicians construct mathematical knowledge and discover mathematical truths.Mathematicians form judgments and establish theories based on their intuitive understanding of mathematical concepts.This intuitive understanding has both empirical intuition and rational intuition.If it is not logically tested,intuition can sometimes be wrong or lead to mathematical paradoxes.As a result,mathematicians use axiomatic methodsto define concepts,modify or extend axioms,and thus intuition accepts the test of logic and the axioms.Finally,it is pointed out that the reliability guarantee of intuition requires the unity of rationalist epistemology and the ontology based on the truth of mathematical truth.