A Philosophical Analysis Of The Truth Of Arithmetic Propositions

From the perspective of the development of knowledge,people accept arithmetic propositions on the premise of their truth.When examining the source of this truth,one tends to believe its priori universality.Human knowledge should be based on such general and self-evident propositions.Hence we find that thinking only from an individual's point of view can lose something that is related to the nature of reason.As a matter of fact,when we turn our eyes to the history of human thought,we find that human beings have always insisted on questioning the basis of reason,and the exploration of the truth of arithmetic propositions belongs to this scope.Kant clearly distinguished between the concepts of analysis and synthesis,and believed that mathematical propositions,including arithmetic and geometry propositions,were synthetic.The truth of arithmetic proposition is based on the subject's comprehensive application of pure intuition.Since then,many mathematicians or philosophers have expressed their views on this question,trying to find an ultimate answer for the basis of the truth of arithmetic and even mathematics.In arithmetic,Frege exhibited completely different mathematical philosophic thoughts from Kant.In The Foundations of Arithmetic,he tried to show that arithmetic propositions were both priori and logical and the truth of arithmetic propositions is based on definition and logical proof.However,the emergence of Russell's paradox hindered his plans.From a different perspective,Frege's desire to explain arithmetic entirely in terms of logic is an attempt to construct a new language,and the interpretation of one language may not be entirely dependent on the construction of another.Hilbert,as a supporter of Kant's arithmetic thought,tried to incorporate the infinite into the finite frame by resorting to the consistency of the system.Godel's incompleteness theorem totally negates Hilbert's plan,and to some extent hinders logical positivists from revising the idea that "mathematics is a grand tautology".At the same time,it suggests another source of the truth about arithmetic propositions—— mathematical intuition.This paper attempts to clarify Kant's and Frege's thoughts on the truth of arithmetic propositions,to trace back the possible doubts in their argumentation,and to further analyze the exploration of the answers on this basis,thus trying to clarify the source of reason for arithmetic propositions.Therefore,this paper is divided intofour parts: The first part is the introduction of the question "the basis of the truth of arithmetical propositions",which is a very important question in the history of mathematical philosophy.The second part and the third part respectively explore the answers of Kant and Frege and the possible doubts.The fourth part is the continuation of the question and Godel's answer.This part mainly criticizes the exploration of Kant and Frege in the sense of Godel's incompleteness theorem and its meaning,and introduces "mathematical intuition" into the discussion of the truth of mathematics.