Formalism is a school of thought founded by David Hilbert in the philosophy of mathematics concerning the foundations of mathematics.
Central to this is the question that is to be regarded as a mathematical inference or implication. The aim was to prove only in shape the completeness and consistency of the axiom systems of mathematics.
In the 1920s, the formalism ( Göttingen mathematician ) was at the basis of the dispute mathematics intuitionism ( Brouwer and Berlin mathematician ) and the logicism ( Frege and Bertrand Russell) against.
As of Gödel's incompleteness theorem showed that there is no axiom system that meets the formalist task ( Hilbert scheme ), the formalism suffered a heavy defeat. On the other hand, one can say that today almost all mathematicians are formalistic Axiomatiker.