Physics and Geometry

Abstract

To David Hilbert, one of the greatest mathematicians of the twentieth century, axiomatization signified a guiding principle of the structure of mathematics and the mathematical laws of physics. That Hilbert’s approach towards the foundations of mathematics was too optimistic, and essentially unfeasible, was established by Gödel’s theorem more than a decade before Hilbert died. In his own work, Hilbert systematized and unified numerous branches of mathematics: theory of invariants (1885–1893), theory of algebraic number fields (1893–1899), theory of analytic functions and variational problems (1899–1909), and the theory of linear integral equations (1902–1912). Hilbert’s work on linear integral equations extended the pioneering work of Fredholm, and led to the introduction of the celebrated ‘Hilbert space’, which would provide the fundamental basis of the mathematical formulation of quantum mechanics.⁸⁶