David Hilbert
Bio
David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Königsberg, Prussia (now Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. His own discoveries alone would have given him that honor, yet it was his leadership in the field of mathematics throughout his later life that distinguishes him.
Major contributions
Hilbert solved several important problems in the theory of invariants. Hilbert's basis theorem solved the principal problem in the 1800s invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants.
He also unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers").
Famous for his ability to make discoveries in various mathematical fields, Hilbert went on to provide the first correct and complete axiomatization of Euclidean geometry to replace Euclid's axiomatization of geometry, in his 1899 book Grundlagen der Geometrie ("Foundations of Geometry"). See Hilbert's axioms.
He also laid the foundations of functional analysis by studying integral equations and formulating a first version, in terms of quadratic forms in infinitely many variables, of what would be called Hilbert space. This work turned out in the 1920s to be foundational for quantum mechanics.
His interest in physics, in the decade 1900-1910, was not as important as later contacts with Albert Einstein and formulations of general relativity that helped its mathematical respectability.
Hilbert helped provide the basis for the theory of automata which was later built upon by computer scientist Alan Turing.
Miscellaneous talks, essays, and contributions
Hilbert presented the paradox of the Grand Hotel, a musing about strange properties of the infinite.
He put forth an influential list of 23 unsolved problems in the Paris conference of the International Congress of Mathematicians in 1900.
Additionally, Hilbert is responsible for assisting several advances in the mathematics of quantum mechanics. These include his integral calculations of Hilbert spaces and proving the mathematical equivalency of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation.
Hilbert's program
In 1920 he proposed explicitly a program (in metamathematics, as it was then termed) that became known as Hilbert's program. He wanted mathematics be formulated on a solid and complete logical foundation by showing that:
- all of mathematics follows from a finite system of axioms; and
- that that axiom system is consistent.
There seem to have been both technical and psychological reasons why he came out with this proposal. It affirmed his dislike of what had become known as the ignorabimus, still an active issue in his time in German thought, and traced back in that formulation to Emil du Bois-Reymond.
This program is still recognisable in the most popular philosophy of mathematics, amongst working mathematicians that is, usually called formalism. For example, the Bourbaki group adopted a milk-and-water version of it as adequate to the requirements of the twin projects of (a) writing encyclopedic foundational works, and (b) supporting the axiomatic method as a research tool.
Gödel's incompleteness theorem showed, however, in 1931 that Hilbert's grand plan was impossible, as stated. The point 2 cannot in any reasonable way be combined with the point 1.
Further reading
