The key contradict of the dilemma of mathematical truth is that Platonist ontology could not comply with empiricist epistemology, which was the boundary Frege maintained all the time. Inherited from Frege’s theories, Neo-Fregean take linguistic analysis as the guide of ontology. They insist that mathematics should be reduced to logic, emphasize especially the importance of abstraction principle (on ground of contextual principle) in introducing numerical singular, which offers a linguistic solution to the dilemma of mathematical truth. However, they had no reasonable justification for the legitimate position of abstraction principle, which led them to a double attitude to the nature of logic. As a result,if Neo-Fregan insist on the truth of first order logic, they have to face the problem that choice axiom could not reconcile with first order logic,or if they insist on abstraction principle, they have to answer the Caesar problem.
In recent 50 years,the debate between mathematical realism and anti-realism has been dominating the mainstream development in the contemporary philosophy of mathematics. Penelope Maddy proposed a naturalistic set theoretic realism in 1990. This project brings the philosophy of mathematics a new research idea,that is,philosophy should attach importance to mathematical practice. This article will critically analyze Maddy’s naturalistic set theoretic realism on the basis of research paradigm background belief. By analyzing Maddy’s strategy,this article will try to make the task of the philosophy of mathematics clearer and put forward some necessary conditions needed by a reasonable research paradigm for the philosophy of mathematics.
This article discusses the meaning of the under-determination thesis raised by the scientific anti-realists,and takes the under-determinative problems in space-time theories and the reply of space-time realists for examples,analyzes deeply the new features of the defense to scientific realism.All these work are based on the semantical analysis,epistemological analysis and the ontological analysis of scientific realism.It indicates that the existence of under-determination thesis has certain grounds,and realists are trying to give a reasonable reply so that they can get out of dilemma of traditional scientific realism.This kind of reply has the feature of contextual realism.
