published his famous Grundlagen der Geometrie
(Foundations of geometry), in which he provided a
completely rigorous axiomatic foundation of the sub-
ject clarifying the hidden assumptions that E
UCLID
had made in his development of the subject two mil-
lennia earlier. Moreover, Hilbert advanced the topic of
FORMAL LOGIC
and used his results to prove that his
approach to geometry is
CONSISTENT
given that the
arithmetic of the real numbers is free of contradic-
tions. In 1900 Hilbert posed 23 problems to the math-
ematicians of the 20th century that he felt lay at the
heart of vital mathematical research. Two of his prob-
lems were solved almost immediately, but the remain-
ing 21 challenges did indeed stimulate important
mathematical thinking. Many of his challenges are
still unsolved today. Hilbert also greatly influenced
the development of quantum theory in theoretical
physics: his notion of a “Hilbert space” provided the
right conceptual framework for the subject. Hilbert
also made important contributions to the fields of spe-
cial relativity and general relativity.
Hilbert received a doctorate of mathematics from
the University of Königsberg in 1885 after writing a
thesis in
ABSTRACT ALGEBRA
. He was appointed to a
faculty position at the university, where he remained
for 10 years before accepting the position as chair of
mathematics at the University of Göttingen in 1895.
Hilbert taught and worked at Göttingen for the
remainder of his career.
His 1897 text Zahlbericht (Number theory) was
hailed as a brilliant synthesis of current thinking in
algebraic number theory, and the original results it con-
tained were acknowledged as outstanding. Hilbert’s
abilities to grasp the subtleties of a sophisticated math-
ematical theory, develop penetrating insights, and pro-
vide new innovative and stimulating perspectives on a
subject were apparent. Throughout his career Hilbert
worked on a wide variety of disparate subjects, making
groundbreaking contributions to each before moving
on to the next. He published his famous work on
E
UCLIDEAN GEOMETRY
in 1899.
In 1900 Hilbert was invited to address the Paris
meeting of the International Congress of Mathemati-
cians. During his speech he detailed 10 mathematical
problems that he felt were of great importance. (He
expanded the list to 23 problems when he published his
address.) These problems include the
CONTINUUM
HYPOTHESIS
, G
OLDBACH
’
S CONJECTURE
, a search for
the axiomatization of physics, and a search for a gen-
eral algorithm for solving D
IOPHANTINE EQUATION
s.
Some important progress, and in many cases, complete
solution, has been made on all the challenges posed
except for one, the so-called Riemann hypothesis,
which asks for the locations of the roots of the
ZETA
FUNCTION
. This remains, perhaps, the most famous
unsolved problem of today.
Later in life Hilbert worked on formal logic and on
the foundations of theoretical physics. Between 1934
and 1939 he published two volumes of Grundlagen der
Mathematik (Foundations of mathematics), cowritten
with Paul Bernays (1888–1977), which were intended
to develop a proof of the consistency of mathematics.
(G
ÖDEL
’
S INCOMPLETENESS THEOREMS
showed, how-
ever, that such a goal is unattainable.) His development
of functional analysis provided the correct mathemati-
cal framework for the theory of quantum mechanics.
Hilbert received many honors throughout his
career, including a special citation from the Hungarian
Academy of Sciences in 1905. Upon his retirement in
1929, the city of Göttingen named a street after him,
and the city of Königsberg, his birthplace, declared him
an honorary citizen. He died in Göttingen on February
14, 1943. He is remembered for shaping the very nature
of 20th-century research in the pure mathematics.
Hilbert’s infinite hotel (Hilbert’s paradox) German
mathematician D
AVID
H
ILBERT
(1862–1943) observed
that studies of the infinite can often lead to nonintu-
itive and surprising conclusions. His famous infinite-
hotel paradox illustrates some of his ideas:
Imagine a hotel with an infinite number of
rooms. Suppose every room is occupied. Is it
possible for the hotel to accommodate one
more guest?
If the rooms are numbered 1, 2, 3, and so on, Hilbert
pointed out that, despite the inconvenience, each exist-
ing guest can be moved from room nto room n+ 1,
thereby leaving room 1 free for a latecomer. There is
indeed room for another single guest. One can go fur-
ther: suppose a tourist bus, with an infinite number of
tourists on board, arrives at the hotel. Moving each
hotel guest from room nto room 2n, instead, leaves all
the odd-numbered rooms vacant for the infinite num-
ber of new arrivals.
250 Hilbert’s infinite hotel