Abstract

David Hilbert proposed his well-known Hilbert Program in the early 1920s for foundations of mathematics. The purpose of his program was to prove the consistency of mathematics by using the finitary methods and relying on axiomatic system. Thus, riddles and paradoxes related with the foundations of mathematics could be solved. Hilbert considers, formalizing whole mathematics in a consistent finite way depending on axioms, as an effort to develop a proof theory. So much so that any problems which may occur in a mathematical system, including those related to infinity, will be solved. Hilbert begins with finite number of signs and rules and then proceeds to develop various proved consistent statements. From there, he continues to a higher-order mathematics that includes ideal objects. Hilbert's development of the program started in 1899 with the Foundations of Geometry, and completed by The Grounding of Elementary Number Theory in 1931. In this paper, we will exhibit the various stages and attempts of Hilbert to found the numbers. Our thesis states that Hilbert has always maintained an intuitive apprehension in these foundational attempts i.e. Hilbert advocates intuitional insight prior to any logical inference that makes it possible to found the numbers. Although Hilbert did not define himself as a formalist in any of his works, he was named as a formalist because of Brouwer's criticism of him. Hilbert’s emphasis on intuition reveals that a kind of formalism which is a mere play of signs can be associated with him. In this paper, we will also deal with the relations Hilbert established between signs, axioms, logic and intuition.