David Hilbert (January 23,
1862 â€“ February 14, 1943) was a German mathematician, recognized as one of the most
influential and universal mathematicians of the 19th and early 20th
centuries. He discovered and developed a broad range of
fundamental ideas in many areas, including
invariant theory and the
axiomatization of geometry. He also
formulated the theory of
Hilbert
spaces, one of the foundations of
functional analysis.
Hilbert adopted and warmly defended
Georg
Cantor's set theory and
transfinite numbers. A famous example of
his leadership in
mathematics is his
1900 presentation of a
collection of
problems that set the course for much of the mathematical
research of the 20th century.
Hilbert and his students contributed significantly to establishing
rigor and some tools to the mathematics used in modern physics. He
is also known as one of the founders of
proof theory,
mathematical logic and the distinction
between mathematics and
metamathematics.
Life
Hilbert,
the first of two children and only son of Otto and Maria Therese
(Erdtmann) Hilbert, was born in either KĂ¶nigsberg (according to Hilbert's own statement) or in Wehlau
(today Znamensk, Kaliningrad Oblast)) near KĂ¶nigsberg where his father was occupied at
the time of his birth in the Province of Prussia. In the fall of 1872, he entered the
Friedrichskolleg
Gymnasium (the
same school that
Immanuel Kant had
attended 140 years before), but after an unhappy duration he
transferred (fall 1879) to and graduated from (spring 1880) the
more science-oriented Wilhelm Gymnasium. Upon graduation he
enrolled (autumn 1880) at the
University of KĂ¶nigsberg, the
"Albertina". In the spring of 1882,
Hermann Minkowski (two years younger than
Hilbert and also a native of KĂ¶nigsberg but so talented he had
graduated early from his gymnasium and gone to Berlin for three
semesters), returned to KĂ¶nigsberg and entered the university.
"Hilbert knew his luck when he saw it. In spite of his father's
disapproval, he soon became friends with the shy, gifted
Minkowski." In 1884,
Adolf Hurwitz
arrived from GĂ¶ttingen as an
Extraordinarius, i.e., an associate
professor. An intense and fruitful scientific exchange between the
three began and especially Minkowski and Hilbert would exercise a
reciprocal influence over each other at various times in their
scientific careers. Hilbert obtained his doctorate in 1885, with a
dissertation, written under
Ferdinand von Lindemann, titled
Ăśber invariante Eigenschaften spezieller binĂ¤rer Formen,
insbesondere der Kugelfunktionen ("On the invariant properties
of special
binary forms, in particular
the spherical harmonic functions").
Hilbert remained at the University of KĂ¶nigsberg as a professor
from 1886 to 1895. In 1892, Hilbert married KĂ¤the Jerosch
(1864â€“1945), "the daughter of a Konigsberg merchant, an outspoken
young lady with an independence of mind that matched his own".
While at KĂ¶nigsberg they had their one child Franz Hilbert
(1893â€“1969).
In 1895, as a result of intervention on his
behalf by Felix Klein he obtained the
position of Chairman of Mathematics at the University of
GĂ¶ttingen, at that time the best research center for
mathematics in the world and where he remained for the rest of his
life.
His son Franz would suffer his entire life from an (undiagnosed)
mental illness, his inferior intellect a terrible disappointment to
his father and this tragedy a matter of distress to the
mathematicians and students at GĂ¶ttingen. Sadly, Minkowski â€”
Hilbert's "best and truest friend" â€” would die prematurely of a
ruptured appendix in 1909.
Math department in GĂ¶ttingen where
Hilbert worked from 1895 until his retirement in 1930
The GĂ¶ttingen school
Among the students of Hilbert were:
Hermann
Weyl, the champion of chess
Emanuel
Lasker,
Ernst Zermelo, and
Carl Gustav Hempel.
John von Neumann was his assistant. At the
University of GĂ¶ttingen, Hilbert was surrounded by a social circle
of some of the most important mathematicians of the 20th century,
such as
Emmy Noether and
Alonzo Church.
Among his 69 Ph.D. students in GĂ¶ttingen were many who later became
famous mathematicians, including (with date of thesis):
Otto Blumenthal (1898),
Felix Bernstein (1901),
Hermann Weyl (1908),
Richard Courant (1910),
Erich Hecke (1910),
Hugo Steinhaus (1911),
Wilhelm Ackermann (1925). Between 1902 and
1939 Hilbert was editor of the
Mathematische Annalen, the
leading mathematical journal of the time.
Later years
Hilbert
lived to see the Nazis purge many of the
prominent faculty members at University of
GĂ¶ttingen, in 1933. Among those forced out were
Hermann Weyl, who had taken Hilbert's
chair when he retired in 1930,
Emmy
Noether and
Edmund Landau. One of
those who had to leave Germany was
Paul
Bernays, Hilbert's collaborator in
mathematical logic, and co-author with
him of the important book
Grundlagen der Mathematik (which
eventually appeared in two volumes, in 1934 and 1939). This was a
sequel to the Hilbert-
Ackermann
book
Principles of
Mathematical Logic from 1928.
About a year later, he attended a banquet, and was seated next to
the new Minister of Education,
Bernhard
Rust. Rust asked, "How is mathematics in GĂ¶ttingen now that it
has been freed of the Jewish influence?" Hilbert replied,
"Mathematics in GĂ¶ttingen? There is really none any more."
Hilbert's tomb:
Wir mĂĽssen wissen
Wir werden wissen
By the time Hilbert died in 1943, the Nazis had nearly completely
restructured the university, many of the former faculty being
either Jewish or married to Jews. Hilbert's funeral was attended by
fewer than a dozen people, only two of whom were fellow academics,
among them
Arnold Sommerfeld, a
theoretical physicist and also a native of KĂ¶nigsberg. News of his
death only became known to the wider world six months after he had
died.
On his tombstone, at GĂ¶ttingen, one can read his epitaph, the
famous lines he had spoken at the end of his retirement address to
the Society of German Scientists and Physicians in the fall of
1930:
- Wir mĂĽssen wissen.
- Wir werden wissen.
As translated into English the inscriptions read:
- We must know.
- We will know.
The day before Hilbert pronounced this phrase at the 1930 annual
meeting of the Society of German Scientists and Physicians,
Kurt GĂ¶delâ€”in a roundtable
discussion during the Conference on Epistemology held jointly with
the Society meetingsâ€”tentatively announced the first expression of
his
incompleteness theorem,
the news of which would make Hilbert "somewhat angry".
The finiteness theorem
Hilbert's first work on invariant functions led him to the
demonstration in 1888 of his famous
finiteness theorem.
Twenty years earlier,
Paul Gordan had
demonstrated the
theorem of the finiteness
of generators for binary forms using a complex computational
approach. The attempts to generalize his method to functions with
more than two variables failed because of the enormous difficulty
of the calculations involved. Hilbert realized that it was
necessary to take a completely different path. As a result, he
demonstrated
Hilbert's basis
theorem: showing the existence of a finite set of
generators, for the invariants of
quantic in any number of variables, but in an
abstract form. That is, while demonstrating the existence of such a
set, it was not a
constructive
proof â€” it did not display "an object" â€” but rather, it was an
existence proof and relied on use of
the
Law of Excluded Middle in
an infinite extension.
Hilbert sent his results to the
Mathematische Annalen. Gordan,
the house expert on the theory of invariants for the
Mathematische Annalen, was not able to appreciate the
revolutionary nature of Hilbert's theorem and rejected the article,
criticizing the exposition because it was insufficiently
comprehensive. His comment was:
- Das ist nicht Mathematik. Das ist
Theologie.
- :(This is not Mathematics. This is
Theology.)
Klein, on the other hand, recognized the importance of the work,
and guaranteed that it would be published without any alterations.
Encouraged by Klein and by the comments of Gordan, Hilbert in a
second article extended his method, providing estimations on the
maximum degree of the minimum set of generators, and he sent it
once more to the
Annalen. After having read the
manuscript, Klein wrote to him, saying:
- Without doubt this is the most important work on general
algebra that the Annalen has ever published.
Later, after the usefulness of Hilbert's method was universally
recognized, Gordan himself would say:
- I have convinced myself that even theology has its
merits.
For all his successes, the nature of his proof stirred up more
trouble than Hilbert could imagine at the time. Although Kronecker
had conceded, Hilbert would later respond to others' similar
criticisms that "many different constructions are subsumed under
one fundamental idea" â€” in other words (to quote Reid): "Through a
proof of existence, Hilbert had been able to obtain a
construction"; "the proof" (i.e. the symbols on the page)
was "the object". Not all were convinced. While
Kronecker would die soon after, his
constructivist philosophy would
be carried forward by the young
Brouwer and his developing
intuitionist "school", much to
Hilbert's torment in his later years. Indeed Hilbert would lose his
"gifted pupil"
Weyl to intuitionism â€” "Hilbert
was disturbed by his former student's fascination with the ideas of
Brouwer, which aroused in Hilbert the memory of Kronecker". Brouwer
the intuitionist in particular opposed the use of the Law of
Excluded Middle over infinite sets (as Hilbert had used it).
Hilbert would respond:
- : 'Taking the Principle of the Excluded Middle from the
mathematician ... is the same as ... prohibiting the boxer the use
of his fists.'
The possible loss did not seem to bother Weyl.
Axiomatization of geometry
The text
Grundlagen der
Geometrie (tr.:
Foundations of Geometry)
published by Hilbert in 1899 proposes a formal set, the
Hilbert's axioms, substituting the
traditional
axioms of Euclid. They
avoid weaknesses identified in those of
Euclid, whose works at the time were still used
textbook-fashion. Independently and contemporaneously, a
19-year-old American student named
Robert Lee Moore published an equivalent
set of axioms. Some of the axioms coincide, while some of the
axioms in Moore's system are theorems in Hilbert's and
vice-versa.
Hilbert's approach signaled the shift to the modern
axiomatic method. Axioms are not taken as
self-evident truths. Geometry may treat
things, about
which we have powerful intuitions, but it is not necessary to
assign any explicit meaning to the undefined concepts. The
elements, such as
point,
line,
plane,
and others, could be substituted, as Hilbert says, by tables,
chairs, glasses of beer and other such objects. It is their defined
relationships that are discussed.
Hilbert first enumerates the undefined concepts: point, line,
plane, lying on (a relation between points and planes),
betweenness, congruence of pairs of points, and
congruence of
angles. The axioms unify both the
plane geometry and
solid geometry of Euclid in a single
system.
The 23 Problems
He put
forth a most influential list of 23 unsolved problems at the
International
Congress of Mathematicians in Paris in
1900. This is generally reckoned the most successful and
deeply considered compilation of open problems ever to be produced
by an individual mathematician.
After re-working the foundations of classical geometry, Hilbert
could have extrapolated to the rest of mathematics. His approach
differed, however, from the later 'foundationalist'
Russell-Whitehead or 'encyclopedist'
Nicolas Bourbaki, and from his contemporary
Giuseppe Peano. The mathematical
community as a whole could enlist in problems, which he had
identified as crucial aspects of the areas of mathematics he took
to be key.
The problem set was launched as a talk "The Problems of
Mathematics" presented during the course of the Second
International Congress of Mathematicians held in Paris. Here is the
introduction of the speech that Hilbert gave:
- Who among us would not be happy to lift the veil behind
which is hidden the future; to gaze at the coming developments of
our science and at the secrets of its development in the centuries
to come? What will be the ends toward which the spirit of
future generations of mathematicians will tend? What
methods, what new facts will the new century reveal in the vast and
rich field of mathematical thought?
He presented fewer than half the problems at the Congress, which
were published in the acts of the Congress. In a subsequent
publication, he extended the panorama, and arrived at the
formulation of the now-canonical 23 Problems of Hilbert. The full
text is important, since the exegesis of the questions still can be
a matter of inevitable debate, whenever it is asked how many have
been solved.
Some of these were solved within a short time. Others have been
discussed throughout the 20th century, with a few now taken to be
unsuitably open-ended to come to closure. Some even continue to
this day to remain a challenge for mathematicians.
Formalism
In an account that had become standard by the mid-century,
Hilbert's problem set was also a kind of manifesto, that opened the
way for the development of the
formalist school, one of three major
schools of mathematics of the 20th century. According to the
formalist, mathematics is manipulation of symbols according to
agreed upon formal rules. It is therefore an autonomous activity of
thought. There is, however, room to doubt whether Hilbert's own
views were simplistically formalist in this sense.
Hilbert's program
In 1920 he proposed explicitly a research project (in
metamathematics, as it was then termed)
that became known as
Hilbert's
program. He wanted
mathematics to be
formulated on a solid and complete logical foundation. He believed
that in principle this could be done, by showing that:
- all of mathematics follows from a correctly-chosen finite
system of axioms; and
- that some such axiom system is provably consistent through some
means such as the epsilon
calculus.
He seems to have had both technical and philosophical reasons for
formulating 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 recognizable in the most popular
philosophy of mathematics, where
it is usually called
formalism. For example, the
Bourbaki group adopted a watered-down and selective
version of it as adequate to the requirements of their twin
projects of (a) writing encyclopedic foundational works, and (b)
supporting the
axiomatic method as
a research tool. This approach has been successful and influential
in relation with Hilbert's work in algebra and functional analysis,
but has failed to engage in the same way with his interests in
physics and logic.
GĂ¶del's work
Hilbert and the mathematicians who worked with him in his
enterprise were committed to the project. His attempt to support
axiomatized mathematics with definitive principles, which could
banish theoretical uncertainties, was however to end in
failure.
GĂ¶del demonstrated that any
non-contradictory formal system, which was comprehensive enough to
include at least arithmetic, cannot demonstrate its completeness by
way of its own axioms. In 1931 his
incompleteness theorem
showed that Hilbert's grand plan was impossible as stated. The
second point cannot in any reasonable way be combined with the
first point, as long as the axiom system is genuinely
finitary.
Nevertheless, the subsequent achievements of
proof theory at the very least
clarified consistency as it relates to theories of central
concern to mathematicians. Hilbert's work had started logic on this
course of clarification; the need to understand GĂ¶del's work then
led to the development of
recursion
theory and then
mathematical
logic as an autonomous discipline in the 1930s. The basis for
later
theoretical computer
science, in
Alonzo Church and
Alan Turing also grew directly out of
this 'debate'.
Functional analysis
Around 1909, Hilbert dedicated himself to the study of differential
and
integral equations; his work
had direct consequences for important parts of modern functional
analysis. In order to carry out these studies, Hilbert introduced
the concept of an infinite dimensional
Euclidean space, later called
Hilbert space. His work in this part of
analysis provided the basis for important contributions to the
mathematics of physics in the next two decades, though from an
unanticipated direction.Later on,
Stefan
Banach amplified the concept, defining
Banach spaces. Hilbert space is the most
important single idea in the area of
functional analysis, particularly of the
spectral theory of self-adjoint
linear operators, that grew up around it during the 20th
century.
Physics
Until 1912, Hilbert was almost exclusively a "pure" mathematician.
When planning a visit from Bonn, where he was immersed in studying
physics, his fellow mathematician and friend
Hermann Minkowski joked he had to spend 10
days in quarantine before being able to visit Hilbert. In fact,
Minkowski seems responsible for most of Hilbert's physics
investigations prior to 1912, including their joint seminar in the
subject in 1905.
In 1912, three years after his friend's death, Hilbert turned his
focus to the subject almost exclusively. He arranged to have a
"physics tutor" for himself. He started studying
kinetic gas theory and moved on to elementary
radiation theory and the molecular theory
of matter. Even after the war started in 1914, he continued
seminars and classes where the works of
Albert Einstein and others were followed
closely.
By 1907 Einstein had framed the fundamentals of the theory of
gravity, but then struggled for nearly 8 years with a confounding
problem of putting the theory into final form. By early summer
1915, Hilbert's interest in physics had focused him on
general relativity, and he invited
Einstein to GĂ¶ttingen to deliver a week of lectures on the subject.
Einstein received an enthusiastic reception at GĂ¶ttingen. Over the
summer Einstein learned that Hilbert was also working on the field
equations and redoubled his own efforts. During November 1915
Einstein published several papers culminating in "The Field
Equations of Gravitation" (see
Einstein field equations). Nearly
simultaneously David Hilbert published "The Foundations of
Physics", an axiomatic derivation of the field equations (see
Einsteinâ€“Hilbert
action). Hilbert fully credited Einstein as the originator of
the theory, and no public priority dispute concerning the field
equations ever arose between the two men during their lives (see
more at
priority).
Additionally, Hilbert's work anticipated and assisted several
advances in the
mathematical
formulation of quantum mechanics. His work was a key aspect of
Hermann Weyl and
John von Neumann's work on the mathematical
equivalence of
Werner Heisenberg's
matrix mechanics and
Erwin SchrĂ¶dinger's
wave equation and his namesake
Hilbert space plays an important part
in quantum theory. In 1926 von Neuman showed that if atomic states
were understood as vectors in Hilbert space, then they would
correspond with both SchrĂ¶dinger's wave function theory and
Heisenberg's matrices.
Throughout this immersion in physics, Hilbert worked on putting
rigor into the mathematics of physics. While highly dependent on
higher math, the physicist tended to be "sloppy" with it. To a
"pure" mathematician like Hilbert, this was both "ugly" and
difficult to understand. As he began to understand the physics and
how the physicists were using mathematics, he developed a coherent
mathematical theory for what he found, most importantly in the area
of
integral equations. When his
colleague
Richard Courant wrote the
now classic
Methods of
Mathematical Physics including some of Hilbert's ideas, he
added Hilbert's name as author even though Hilbert had not directly
contributed to the writing. Hilbert said "Physics is too hard for
physicists", implying that the necessary mathematics was generally
beyond them; the Courant-Hilbert book made it easier for
them.
Number theory
Hilbert unified the field of
algebraic number theory with his
1897 treatise
Zahlbericht (literally "report on numbers").
He also resolved a significant number theory
problem formulated by Waring in 1770. As
with the
the finiteness
theorem, he used an existence proof that shows there must be
solutions for the problem rather than providing a mechanism to
produce the answers. He then had little more to publish on the
subject; but the emergence of
Hilbert modular forms in the
dissertation of a student means his name is further attached to a
major area.
He made a series of conjectures on
class field theory. The concepts were
highly influential, and his own contribution is seen in the names
of the
Hilbert class field and
the
Hilbert symbol of
local class field theory. Results
on them were mostly proved by 1930, after work by
Teiji Takagi that established Tagaki as Japan's
first mathematician of international stature.
Hilbert did not work in the central areas of
analytic number theory, but his name
has become known for the
Hilbertâ€“PĂłlya
conjecture, for reasons that are anecdotal.
Miscellaneous talks, essays, and contributions
- His paradox of
the Grand Hotel, a meditation on strange properties of the
infinite, is often used in popular accounts of infinite cardinal numbers.
- His ErdĹ‘s number is (at most)
4.
- Foreign member
of the Royal Society
- He was awarded the second Bolyai
prize in 1910.
- His collected works (Gesammelte Abhandlungen) has been
published several times. The original versions of his papers
contained errors; when the collection was first published, the
errors were corrected and it was found that this could be done
without major changes in the statements of the theorems, with one
exceptionâ€”a claimed proof of the Continuum hypothesis. The errors were
nonetheless so numerous and significant that it took Olga Taussky-Todd three years to make the
corrections.
See also
Notes
- Reid 1996, pp. 1â€“2; also on p. 8, Reid notes that there is some
ambiguity of exactly where Hilbert was born. Hilbert himself stated
that he was born in KĂ¶nigsberg.
- Reid 1996, pp. 4â€“7.
- Reid 1996, p. 11.
- Reid 1996, p. 12.
- Reid 1996, p. 36.
- Reid 1996, p. 139.
- Reid 1996, p. 121.
- (Hilbert's colleagues exiled)
- Reid 1996, p. 205.
- Reid 1996, p. 213.
- Reid p. 192
- "The Conference on Epistemology of the Exact Sciences ran for
three days, from 5 to 7 September" (Dawson 1997:68). "It ... was
held in conjunction with and just before the ninety-first annual
meeting of the Society of German Scientists and Physicians ... and
the sixth Assembly of German Physicists and Mathematicians....
GĂ¶del's contributed talk took place on Saturday, 6 September
[1930], from 3 until 3:20 in the afternoon, and on Sunday the
meeting concluded with a round table discussion of the first day's
addresses. During the latter event, without warning and almost
offhandedly, GĂ¶del quietly announced that "one can even give
examples of propositions (and in fact of those of the type of
Goldbach or Fermat) that, while
contentually true, are unprovable in the formal system of classical
mathematics [153]" (Dawson:69) "... As it happened, Hilbert himself
was present at KĂ¶nigsberg, though apparently not at the Conference
on Epistemology. The day after the roundtable discussion he
delivered the opening address before the Society of German
Scientists and Physicians -- his famous lecture 'Naturerkennen und
Logik" (Logic and the knowledge of nature), at the end of which he
declared: 'For the mathematician there is no Ignorabimus, and, in
my opinion, not at all for natural science either. ... The true
reason why [no one] has succeeded in finding an unsolvable problem
is, in my opinion, that there is no unsolvable problem. In
contrast to the foolish Ignorabimus, our credo avers: We must know,
We shall know [159]'"(Dawson:71). GĂ¶del's paper was received on
November 17, 1930 (cf Reid p. 197, van Heijenoort 1976:592) and
published on 25 March 1931 (Dawson 1997:74). But GĂ¶del had given a
talk about it beforehand... "An abstract had been presented on
October 1930 to the Vienna Academy of Sciences by Hans Hahn" (van Heijenoort:592);
this abstract and the full paper both appear in van
Heijenoort:583ff.
- Reid p. 198
- Reid 1996, pp. 36â€“37.
- Reid 1996, p. 34.
- Rowe, p. 195
- Reid 1996, p. 37.
- cf. Reid 1996, pp. 148â€“149.
- Reid 1996, p. 148.
- Reid 1996, p. 150.
- Reid 1996, p. 129.
- Isaacson 2007:218
- Sauer 1999, Folsing 1998, Isaacson 2007:212
- Isaacson 2007:213
- Since 1971 there have been some spirited and scholarly
discussions about which of the two men first presented the now
accepted form of the field equations. "Hilbert freely admitted, and
frequently stated in lectures, that the great idea was
Einstein's."Every boy in the streets of Gottingen understands more
about four dimensional geometry than Einstein," he once remarked.
"Yet, in spite of that, Einstein did the work and not the
mathematicians" (Reid 1996:141-142, also Isaacson 2007:222 quoting
Thorne p. 119).
- It is of interest to note that in 1926, the year after the
matrix mechanics formulation of quantum theory by Max Born and Werner
Heisenberg, the mathematician John von Neumann became an assistant
to David Hilbert at GĂ¶ttingen. When von Neumann left in 1932, von
Neumannâ€™s book on the mathematical foundations of quantum
mechanics, based on Hilbertâ€™s mathematics, was published under the
title Mathematische Grundlagen der Quantenmechanik. See:
Norman Macrae, John von Neumann: The Scientific Genius Who
Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and
Much More (Reprinted by the American Mathematical Society,
1999) and Reid 1996.
- Reid 1996, p. 114
- Rota G.-C. (1997), " Ten lessons I wish I had been taught",
Notices of the AMS, 44: 22-25.
- Wolfram MathWorld â€“ Hilbert inequality
- Wolfram MathWorld â€“ Hilbertâ€™s constants
References
Primary literature in English translation
- Ewald, William B., ed., 1996. From Kant to Hilbert: A
Source Book in the Foundations of Mathematics, 2 vols. Oxford
Uni. Press.
- 1918. "Axiomatic thought," 1115â€“14.
- 1922. "The new grounding of mathematics: First report,"
1115â€“33.
- 1923. "The logical foundations of mathematics," 1134â€“47.
- 1930. "Logic and the knowledge of nature," 1157â€“65.
- 1931. "The grounding of elementary number theory,"
1148â€“56.
- 1904. "On the foundations of logic and arithmetic,"
129â€“38.
- 1925. "On the infinite," 367â€“92.
- 1927. "The foundations of mathematics," with comment by
Weyl and Appendix by Bernays, 464â€“89.
- Jean van Heijenoort, 1967.
From Frege to Godel: A Source Book in Mathematical Logic,
1879â€“1931. Harvard Univ. Press.
- - an accessible set of lectures originally for the citizens of
GĂ¶ttingen.
Secondary literature
- Bottazzini Umberto, 2003. Il flauto di Hilbert.
Storia della matematica. UTET, ISBN
88-7750-852-3
- Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision
in the Hilbert-Einstein Priority Dispute," Science 278:
nn-nn.
- Dawson, John W. Jr 1997. Logical Dilemmas: The Life and
Work of Kurt GĂ¶del. Wellesley MA: A. K. Peters. ISBN
1-56881-256-6.
- Folsing, Albrecht, 1998. Albert Einstein.
Penguin.
- Grattan-Guinness, Ivor,
2000. The Search for Mathematical Roots 1870-1940.
Princeton Univ. Press.
- Gray, Jeremy, 2000. The Hilbert Challenge. ISBN
0-19-850651-1
- Mehra, Jagdish, 1974. Einstein, Hilbert, and the Theory of
Gravitation. Reidel.
- Piergiorgio Odifreddi,
2003. Divertimento Geometrico - Da Euclide ad Hilbert.
Bollati Boringhieri, ISBN 88-339-5714-4. A clear exposition of the
"errors" of Euclid and of the solutions presented in the
Grundlagen der Geometrie, with reference to non-Euclidean geometry.
- Reid, Constance, 1996. Hilbert, Springer, ISBN
0-387-94674-8. The biography in English.
- Sauer, Tilman, 1999, " The
relativity of discovery: Hilbert's first note on the foundations of
physics," Arch. Hist. Exact Sci. 53:
529-75.
- Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der
Mathematik" in Grattan-Guinness,
I., ed., Landmark Writings in Western Mathematics.
Elsevier: 981-99. (in English)
- Thorne, Kip, 1995. Black Holes and Time Warps:
Einstein's Outrageous Legacy, W. W. Norton & Company;
Reprint edition. ISBN 0-393-31276-3.
External links