A Study Of Intuitionism In Philosophy Of Mathematics And Philosophy Of Logic-

Intuitionistic mathematics and the corresponding philosophy of mathematics was born during the debate on the foundation of mathematics in the beginning of twentieth century.Compared to the other two main argument opponents,namely,logicism and formalism,intuitionism is more radical,which tries to make a revolutionary reconstruction of classical mathematics.In the study of intuitionistic mathematics,intuitionistic logic was born,and it has gradually become an important branch of modern logic.This paper emphasizely discusses the philosophy of mathematics and philosophy of logic belonging to Brouwer,Dummett and Heyting who made the most important contribution to the intuitionism'thought.As the founder of the school of intuitionism,Brouwer first elaborated the theory of intuitionism comprehensively.Heyting is a student of Brouwer,and he earliest constructed formal system of intuitionistic logic.Dummett is a representative of the"linguistic turn" in the field of western philosophy,and he made a defence for intuitionism which is different from his predecessors on the basis of the theory of meaning.The content of this paper includes the following four chapters:The first chapter is 'The historical review and the present research situation at home and abroad '.First,it starts from the history of the study of foundation of mathematics to introduce the theoritical background of intuitionism,then leads the problem about the relationship between foundation of mathematics and philosophy of mathematics,then tries to give an answer.The author opposes the idea that the people who study foundation of mathematics could pay no attention to the philosophy of mathematics,and tend to support the views of 'philosophy of mathematics first',namely,philosophy of mathematics provides the 'first principles'for all mathematical practices including foundation of mathematics,although this view is not so complete.The author also discusses Kant's philosophy of mathematics,logicism's and formalism's philosophy of mathematics and intuitionism's philosophy of mathematics before Brouwer.These thoughts are the theoretical background of Brouwer's philosophy of mathematics.The second chapter is ' Brouwer's philosophy of mathematics '.It discusses Brouwer's thought about philosophy of mathematics and its relationship with philosophy and epistemology.The discussion is in this order:philosophy,epistemology,philosophy of mathematics.Brouwer's continuum theory becomes an example to illustrate how to implement his philosophy of mathematics in mathematical practice.The paper also examines Heyting's and Weyl's interpretation of Brouwer's philosophy of mathematics.The author thinks that we could not understand and comment on Brouwer's philosophy of mathematics in isolation from his philosophical thoughts.His philosophy and philosophy of mathematics are the reasons for his revising mathematical practice.In the practice of mathematics,Brouwer successfully eliminated the set theory paradox.However,there are some major flaws in his philosophy,which are particularly reflected in his negative attitude towards science,therefor it also means that classical mathematics which is extremely useful to science becomes the 'evil' tool.The author believes that this view is derived from his selective disregard of the positive side of science.The third chapter is 'Intuitionism's idea of language and logic and intuitionistic logic '.It discusses Brouwer's view of language and logic and Heyting's interpretation.On this basis,it introduces popular intuitionistic propositional logic systems and Kripke semantics to show how they carry out the idea of intuitionism.The author thinks:Brouwer 's complete denial of logic's role in intuitionistic mathematics is wrong,therefore we can develop and perfect intuitionistic logics appropriate for intuitionistic mathematics.in fact,the development of contemporary intuitionistic mathematics has shown that intuitionistic logic is an indispensable tool in the study of intuitionistic mathematics.The fourth chapter is 'Dummett's defense for intuitionism'.It first discusses why Dummett turns the defense of intuitionistic mathematics into the defense of the intuitionistic logic,then discusses his specific defense.Finally,it explores the communication problem,because it is from this aspect that Dummett rejects Platonism's meaning theory which provides defense for classical logic.The author thinks:Dummett's communication condition is not sufficient,we must also consider the internal mechanism of the participants in communication,but Brouwer philosophy of mathematics labeled 'solipsism' meets the communication condition.Although the language to express continuum is not accurate,but it does not affect its communication.In my opinion,Dummett neglected the discussion of inner mental process in his defense,and mistakely set the condition of mastering linguistic meaning,and could not explain the communicablity of the classical mathematical language.The concluding remarks will list the common sense of the intuitionism,and the criticism of Dummett's theory from development of contemporary theory of meaning,cognitive philosophy and philosophy of mind.By inheriting phenomenological thought which is advocated by Weyl and Heyting,an improvement is constructing a kind of meaning theory which is built on the basis of the intentionality.As the theory of Dummett's theory of meaning,this theory of meaning also can make a defense for intuitionism,but it is more reasonable.