German mathematician, one of the most famous mathematicians of all time. In Foundations of Geometry (1899), he showed that Euclid’s five axioms were far from sufficient for the development of Euclidean geometry. Hilbert showed how Euclidean geometry could be cast in the form of a purely formal logical–mathematical axiomatic system, based on a new enlarged set of axioms. Hilbert’s breakthrough, however, consists in his point that the deductive power of an axiomatic system is independent of the meaning of its terms and predicates and dependent only on their logical relationships. Hence, when it comes to what can be deduced from the axioms, the intuitive meanings of terms such as ‘point’, ‘line’, ‘plane’ etc. play no role at all. A Hilbert-style introduction of a set of terms by means of axioms is called implicit definition. Hilbert’s approach to arithmetic has been characterised as formalism. Hilbert disagreed with Frege that mathematics is reducible to logic and agreed with Kant that mathematics has a specific subject matter. However, he took this subject matter to be not the form of intuition but rather a set of concrete extralogical objects, namely, symbols – numerals in the case of arithmetic. Hilbert thought that total infinities were illusions. But in order to accommodate the role that infinity plays in mathematics, he introduced ideal elements, along the lines of the ideal points at infinity in geometry. Given his thought that proving the consistency of a formal system is all that is required for using it, Hilbert took it that the search for truth should give its place to the search for consistency.
See Definition, implicit; Syntactic view of theories
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