Gödel: Mathematics Is Not The Language Syntax

In Philosophy of Mathematics, one typical person of the conventionalism is Carnap, himself also is one of the important representative person of Logical Positivism. In the book of ,which is published in 1935,he expatiated systemically his viewpoint of Philosophy of Mathematics, include how to comprehend the system of language and related to this, how to comprehend the axioms and the atomic sentence of the system, this point get more system and clear illuminated in the article of:mathematics is just a symbol system which is operated by definite formal formulae, the axioms and symbol in the system has no meaning, we can't make any interpretations to them. Moreover we accept a mathematics theory just like accept a linguistic framework, accept a theory doesn't mean we accept the existence of the abstract entity, which is preinstalled by the theory, but just mean that we accept the reason rules and the inference which is the sequence of the rules, all the rules and the inference is set by the linguistic framework. So the truth of the mathematics is just our convention of the given linguistic framework, and it don't promise the existence of the abstract entity, like property, species, and natural number, for which the realist has committed.The answer of the conventionalism that Carnap insisted on will inevitable relevant to whether the object of mathematics is exist, how those objects can be grasp by our mind, the foundation on which we assert that a mathematics proposition is true, those question are very important in the philosophy of mathematics. Godel in the unpublished article refute the syntax viewpoint that Carnap held. The conclusion of the article as follows:the mathematical intuition about mathematical object can't be replaced by our convention of the symbols, convention in any sense can't limit the mathematical true; if we insist carry out the program of convention, we will clear catch the self-contradiction that provoked by the program; the mathematical sentence don't empty of content as the Logical Positivism claimed. Godel based on his important theory——the Second Incompleteness Theorem and his Platonic standpoint, strive to solve the discussion between the nominalism and realism, which has been continued in the realm of philosophy of mathematics. Even he thought his don't get success, we can find out many of the beneficial argument and promote our work on the series of the important problems in the philosophy of mathematics.On the base of the Platonism, Godel illuminated to us that mathematical sentence has content, the object of mathematics has independent existence. Through a series of analysis of the conventionalism Godel demonstrated that mathematic isn't the syntax of language, the conceive of the mathematics that mathematics is just the syntax of language is one of the outcome of Carnap's philosophy.