Russell and Whitehead’s attempt to provide logicist foundations for the whole of mathematics in Mathematica Principia was undermined by the discovery of various paradoxes which led to contradictions. A meta-mathematical approach led by Hilbert attempted to sidestep this impasse. He was willing to abandon the Platonist viewpoint―implied in ill-fated logicism―in favor of a Formalist stance which conceded the tautological nature of mathematics. Davis and Hersh cite the following passage from Hilbert:
The goal of my theory is to establish once and for all the certitude of mathematical methods… The present state of affairs where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches and uses in mathematics , the paragon of proof and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?
Hilbert’s formalism was defeated in 1930 by Gödel’s famous Incompleteness Theorem.
Davis, Philip J. and Hersh, Reuben (1981) The Mathematical Experience. Houghton Mifflin, Boston and New York.
Russell, Bertrand (1919) Mysticism and Logic: And Other Essays. Longman, London.