Hilbert was preeminent in many fields of mathematics, including
axiomatic theory, invariant theory, algebraic number theory, class
field theory and functional analysis. His examination of calculus
led him to the invention of "Hilbert space," considered one of the
key concepts of functional analysis and modern mathematical
physics. He was a founder of fields like metamathematics and modern
logic. He was also the founder of the "Formalist" school which
opposed the "Intuitionism" of Kronecker and Brouwer. He developed a
new system of definitions and axioms for geometry, replacing the
2200 year-old system of Euclid. As a young Professor he proved his
"Finiteness Theorem," now regarded as one of the most important
results of general algebra. The methods he used were so novel that,
at first, the "Finiteness Theorem" was rejected for publication as
being "theology" rather than mathematics! In number theory, he
proved Waring's famous conjecture which is now known as the
Hilbert-Waring theorem.
Any one man can only do so much, so the greatest mathematicians
should help nurture their colleagues. Hilbert provided a famous
List of 23 Unsolved Problems, which inspired and directed the
development of 20th-century mathematics. Hilbert was warmly
regarded by his colleagues and students, and contributed to the
careers of several great mathematicians and physicists including
Georg Cantor, Hermann Minkowski, Hermann Weyl, John von Neumann,
Emmy Noether, Alonzo Church, and Albert Einstein.
Eventually Hilbert turned to physics and made key contributions
to classical and quantum physics and to general relativity.
(Hilbert was a modest man: some historians believe the "Einstein
Field Equations" should carry Hilbert's name.)
