David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
and
philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental ideas including
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...
, the
calculus of variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
,
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
,
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the
foundations of geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometry, non-Euclidean geometries. These are fundamental to the study and of hist ...
,
spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
of operators and its application to
integral equations
In mathematical analysis, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3 ...
,
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, and the
foundations of mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
(particularly
proof theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
). He adopted and defended
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
's set theory and
transfinite number
In mathematics, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of i ...
s. In 1900, he presented a
collection of problems that set a course for mathematical research of the 20th century.
Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
.
Life
Early life and education
Hilbert, the first of two children and only son of Otto, a county judge, and Maria Therese Hilbert (
née
The birth name is the name of the person given upon their birth. The term may be applied to the surname, the given name or to the entire name. Where births are required to be officially registered, the entire name entered onto a births registe ...
Erdtmann), the daughter of a merchant, was born in the
Province of Prussia
The Province of Prussia (; ; ; ) was a province of the Kingdom of Prussia from 1824 to 1878. The province was established in 1824 from the provinces of East Prussia and West Prussia, and was dissolved in 1878 when the merger was reversed.
König ...
,
Kingdom of Prussia
The Kingdom of Prussia (, ) was a German state that existed from 1701 to 1918.Marriott, J. A. R., and Charles Grant Robertson. ''The Evolution of Prussia, the Making of an Empire''. Rev. ed. Oxford: Clarendon Press, 1946. It played a signif ...
, either in
Königsberg
Königsberg (; ; ; ; ; ; , ) is the historic Germany, German and Prussian name of the city now called Kaliningrad, Russia. The city was founded in 1255 on the site of the small Old Prussians, Old Prussian settlement ''Twangste'' by the Teuton ...
(according to Hilbert's own statement) or in Wehlau (known since 1946 as
Znamensk) near Königsberg where his father worked at the time of his birth. His paternal grandfather was David Hilbert, a judge and ''
Geheimrat
was the title of the highest advising officials at the imperial, royal, or princely courts of the Holy Roman Empire, who jointly formed the ''Geheimer Rat'' reporting to the ruler. The term remained in use during subsequent monarchic reigns in Ge ...
''. His mother Maria had an interest in philosophy, astronomy and
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, while his father Otto taught him
Prussian virtues
Prussian virtues ( German: ) are the virtues associated with the historical Kingdom of Prussia (1701–1918). They were derived from Prussia's militarism and the ethical code of the Prussian Army as well as from bourgeois values such as honesty a ...
. After his father became a city judge, the family moved to Königsberg. David's sister, Elise, was born when he was six. He began his schooling aged eight, two years later than the usual starting age.
In late 1872, Hilbert entered the
Friedrichskolleg
The Collegium Fridericianum (also known as the Friedrichskolleg, Friedrichskollegium, and Friedrichs-Kollegium) was a prestigious gymnasium in Königsberg, Prussia. Alumni were known as ''Friderizianer''.Gause, p. 716
History
Postcard ca. 1930
...
Gymnasium (''Collegium fridericianum'', the same school that
Immanuel Kant
Immanuel Kant (born Emanuel Kant; 22 April 1724 – 12 February 1804) was a German Philosophy, philosopher and one of the central Age of Enlightenment, Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works ...
had attended 140 years before); but, after an unhappy period, he transferred to (late 1879) and graduated from (early 1880) the more science-oriented
Wilhelm Gymnasium. Upon graduation, in autumn 1880, Hilbert enrolled at the
University of Königsberg
The University of Königsberg () was the university of Königsberg in Duchy of Prussia, which was a fief of Poland. It was founded in 1544 as the world's second Protestant Reformation, Protestant academy (after the University of Marburg) by Duke A ...
, the "Albertina". In early 1882,
Hermann Minkowski
Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...
(two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters), returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.
Career
In 1884,
Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...
arrived from Göttingen as an
Extraordinarius (i.e., an associate professor). An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert especially 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
Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
,
titled ''Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen'' ("On the invariant properties of special
binary forms, in particular the
spherical harmonic functions").
Hilbert remained at the University of Königsberg as a ''Privatdozent'' (
senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf by
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
, he obtained the position of Professor of Mathematics at the
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. He remained there for the rest of his life.
Göttingen school
Among Hilbert's students were
Hermann Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
,
chess
Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...
champion
Emanuel Lasker
Emanuel Lasker (; December 24, 1868 – January 11, 1941) was a German chess player, mathematician, and philosopher. He was the second World Chess Champion, holding the title for 27 years, from 1894 to 1921, the longest reign of any officially ...
,
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel set theory, Z ...
, and
Carl Gustav Hempel
Carl Gustav "Peter" Hempel (; ; January 8, 1905 – November 9, 1997) was a German writer, philosopher, logician, and epistemologist. He was a major figure in Logical positivism, logical empiricism, a 20th-century movement in the philosophy ...
.
John von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...
was his assistant. At the
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
and
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
.
Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis):
Otto Blumenthal
Ludwig Otto Blumenthal (20 July 1876 – 12 November 1944) was a German mathematician and professor at RWTH Aachen University.
Biography
He was born in Frankfurt, Hesse-Nassau. A student of David Hilbert, Blumenthal was an editor of ''Mathemati ...
(1898),
Felix Bernstein (1901),
Hermann Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
(1908),
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German-American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
(1910),
Erich Hecke
Erich Hecke (; 20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms.
Biography
Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He ...
(1910),
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Univers ...
(1911), and
Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation.
Biograph ...
(1925). Between 1902 and 1939 Hilbert was editor of the ''
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
'', the leading mathematical journal of the time. He was elected an International Member of the United States
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
in 1907.
Personal life

In 1892, Hilbert married Käthe Jerosch (1864–1945), who was the daughter of a Königsberg merchant, "an outspoken young lady with an independence of mind that matched
ilbert's" While at Königsberg, they had their one child, Franz Hilbert (1893–1969).
Franz suffered throughout his life from mental illness, and after he was admitted into a psychiatric clinic, Hilbert said, "From now on, I must consider myself as not having a son." His attitude toward Franz brought Käthe considerable sorrow.
Hilbert considered the mathematician
Hermann Minkowski
Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...
to be his "best and truest friend".
Hilbert was baptized and raised a
Calvinist
Reformed Christianity, also called Calvinism, is a major branch of Protestantism that began during the 16th-century Protestant Reformation. In the modern day, it is largely represented by the Continental Reformed Protestantism, Continenta ...
in the
Prussian Evangelical Church.
[The Hilberts had, by this time, left the Calvinist Protestant church in which they had been baptized and married. – Reid 1996, p.91] He later left the Church and became an
agnostic
Agnosticism is the view or belief that the existence of God, the divine, or the supernatural is either unknowable in principle or unknown in fact. (page 56 in 1967 edition) It can also mean an apathy towards such religious belief and refer to ...
.
[
David Hilbert seemed to be agnostic and had nothing to do with theology proper or even religion. Constance Reid tells a story on the subject:]The Hilberts had by this time round 1902
Round or rounds may refer to:
Mathematics and science
* Having no sharp corners, as an ellipse, circle, or sphere
* Rounding, reducing the number of significant figures in a number
* Round number, ending with one or more zeroes
* Round (crypt ...
left the Reformed Protestant Church in which they had been baptized and married. It was told in Göttingen that when avid Hilbert's sonFranz had started to school he could not answer the question, "What religion are you?" (1970, p. 91)
In the 1927 Hamburg address, Hilbert asserted: "mathematics is pre-suppositionless science (die Mathematik ist eine voraussetzungslose Wissenschaft)" and "to found it I do not need a good God ( ihrer Begründung brauche ich weder den lieben Gott)" (1928, S. 85; van Heijenoort, 1967, p. 479). However, from Mathematische Probleme (1900) to Naturerkennen und Logik (1930) he placed his quasi-religious faith in the human spirit and in the power of pure thought with its beloved child– mathematics. He was deeply convinced that every mathematical problem could be solved by pure reason: in both mathematics and any part of natural science (through mathematics) there was "no ignorabimus" (Hilbert, 1900, S. 262; 1930, S. 963; Ewald, 1996, pp. 1102, 1165). That is why finding an inner absolute grounding for mathematics turned into Hilbert's life-work. He never gave up this position, and it is symbolic that his words "wir müssen wissen, wir werden wissen" ("we must know, we shall know") from his 1930 Königsberg address were engraved on his tombstone. Here, we meet a ghost of departed theology (to modify George Berkeley's words), for to absolutize human cognition means to identify it tacitly with a divine one. —
He also argued that mathematical truth was independent of the existence of God or other ''
a priori
('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, Justification (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...
'' assumptions.
["Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs." David Hilbert, ''Die Grundlagen der Mathematik'']
Hilbert's program, 22C:096, University of Iowa
When
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...
was criticized for failing to stand up for his convictions on the
Heliocentric theory
Heliocentrism (also known as the heliocentric model) is a superseded astronomical model in which the Earth and planets orbit around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed th ...
, Hilbert objected: "But
alileowas not an idiot. Only an idiot could believe that scientific truth needs martyrdom; that may be necessary in religion, but scientific results prove themselves in due time."
Later years
Like
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, Hilbert had closest contacts with the
Berlin Group, whose leading founders had studied under Hilbert in Göttingen (
Kurt Grelling
Kurt Grelling (2 March 1886 – September 1942) was a German logician and philosopher, member of the Berlin Circle.
Life and work
Kurt Grelling was born on 2 March 1886 in Berlin. His father, the Doctor of Jurisprudence Richard Grelling ...
,
Hans Reichenbach
Hans Reichenbach (; ; September 26, 1891 – April 9, 1953) was a leading philosopher of science, educator, and proponent of logical empiricism. He was influential in the areas of science, education, and of logical empiricism. He founded the ''G ...
, and
Walter Dubislav
Walter Dubislav (20 September 1895 – 17 September 1937) was a German logician and philosopher of science (''Wissenschaftstheoretiker'').
Biography
After studying mathematics and philosophy, Dubislav attained a doctorate in 1922 with "Contribut ...
).
Around 1925, Hilbert developed
pernicious anemia
Pernicious anemia is a disease where not enough red blood cells are produced due to a deficiency of Vitamin B12, vitamin B12. Those affected often have a gradual onset. The most common initial symptoms are Fatigue, feeling tired and weak. Other ...
, a then-untreatable vitamin deficiency of which the primary symptom is exhaustion; his assistant
Eugene Wigner
Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
described him as subject to "enormous fatigue" and how he "seemed quite old", and that even after eventually being diagnosed and treated, he "was hardly a scientist after 1925, and certainly not a Hilbert".
Hilbert was elected to the
American Philosophical Society
The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
in 1932.
Hilbert lived to see the
Nazis purge many of the prominent faculty members at
University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
in 1933. Those forced out included
Hermann Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
(who had taken Hilbert's chair when he retired in 1930),
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
, and
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...
. One who had to leave Germany,
Paul Bernays
Paul Isaac Bernays ( ; ; 17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator ...
, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book ''
Grundlagen der Mathematik
''Grundlagen der Mathematik'' (English: ''Foundations of Mathematics'') is a two-volume work by David Hilbert and Paul Bernays. Originally published in 1934 and 1939, it presents fundamental mathematical ideas and introduced second-order arithme ...
'' (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert–
Ackermann book ''
Principles of Mathematical Logic
''Principles of Mathematical Logic'' is the 1950 American translation of the 1938 second edition of David Hilbert's and Wilhelm Ackermann's classic text ''Grundzüge der theoretischen Logik'', on elementary mathematical logic. The 1928 first edit ...
'' (1928). Hermann Weyl's successor was
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
.
About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education,
Bernhard Rust
Bernhard Rust (30 September 1883 – 8 May 1945) was Minister of Science, Education and National Culture ('' Reichserziehungsminister'') in Nazi Germany. Claudia Koonz, ''The Nazi Conscience'', p 134 A combination of school administrator and ze ...
. Rust asked whether "the Mathematical Institute really suffered so much because of the departure of the
Jews
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, rel ...
". Hilbert replied: "Suffered? It doesn't exist any longer, does it?"
Death
By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, as many of the former faculty had either been 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
Arnold Johannes Wilhelm Sommerfeld (; 5 December 1868 – 26 April 1951) was a German Theoretical physics, theoretical physicist who pioneered developments in Atomic physics, atomic and Quantum mechanics, quantum physics, and also educated and ...
, a theoretical physicist and also a native of Königsberg. News of his death only became known to the wider world several months after he died.
The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930. The words were given in response to the Latin maxim: "''
Ignoramus et ignorabimus
The Latin maxim , meaning "we do not know and will not know", represents the idea that scientific knowledge is limited. It was popularized by Emil du Bois-Reymond, a German physiologist, in his 1872 address ("The Limits of Science").
Seven ...
''" or "We do not know and we shall not know":
The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians,
Kurt Gödel
Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
—in a round table discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.
[
"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.... Gödel's contributed talk took place on Saturday, 6 September ]930
Year 930 ( CMXXX) was a common year starting on Friday of the Julian calendar.
Events
By place
Europe
* The Althing, the parliament of Iceland, is established at þingvellir ("Thing Fields"). Chieftains from various tribes gather for ...
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, Gödel quietly announced that "one can even give examples of propositions (and in fact of those of the type of Goldbach or Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
) that, while contentually true, are unprovable in the formal system of classical mathematics 53 (Dawson:69) "... As it happened, Hilbert himself was present at Königsberg, 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 o-onehas 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 59"(Dawson:71). Gödel'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 Gödel had given a talk about it beforehand... "An abstract had been presented in 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. Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
show that even
elementary
Elementary may refer to:
Arts, entertainment, and media Music
* ''Elementary'' (Cindy Morgan album), 2001
* ''Elementary'' (The End album), 2007
* ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977
Other uses in arts, entertainment, an ...
axiomatic systems such as
Peano arithmetic
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nea ...
are either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system.
Contributions to mathematics and physics
Solving Gordan's Problem
Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous ''finiteness theorem''. Twenty years earlier,
Paul Gordan
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory". His most famous ...
had demonstrated the
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as ''Gordan's Problem'', Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated ''
Hilbert's basis theorem
In mathematics Hilbert's basis theorem asserts that every ideal (ring theory), ideal of a polynomial ring over a field (mathematics), field has a finite generating set of an ideal, generating set (a finite ''basis'' in Hilbert's terminology).
In ...
'', showing the existence of a finite set of generators, for the invariants of
quantics 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
In mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also ...
—it did not display "an object"—but rather, it was an
existence proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existen ...
and relied on use of the
law of excluded middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...
in an infinite extension.
Hilbert sent his results to the ''
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
''. Gordan, the house expert on the theory of invariants for the ''Mathematische Annalen'', could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:
Klein
Klein may refer to:
People
*Klein (surname)
*Klein (musician)
Places
* Klein (crater), a lunar feature
*Klein, Montana, United States
* Klein, Texas, United States
* Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm
* Klein River, a r ...
, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, 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:
Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:
For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although
Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea"—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 afterwards, his
constructivist philosophy would continue with the young
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'brewer'.
Brouwer
* Adriaen Brouwer (1605–1638), Flemish painter
* Alexander Brouwer (b. 1989), Dutch beach volleyball player
* Andries Brouwer ( ...
and his developing
intuitionist
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of ...
"school", much to Hilbert's torment in his later years. Indeed, Hilbert would lose his "gifted pupil"
Weyl
Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
to intuitionism—"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 responded:
Nullstellensatz
In the subject of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is called ''
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...
'' if and only if every polynomial over it has a root in it. Under this condition, Hilbert gave a criterion for when a collection of polynomials
of
variables has a ''common'' root: This is the case if and only if there do not exist polynomials
and indices
such that
:
.
This result is known as the Hilbert root theorem, or "Hilberts Nullstellensatz" in German. He also proved that the correspondence between vanishing ideals and their vanishing sets is bijective between
affine varieties
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.
More formally, an affine algebraic set is the set of the common zeros over an algeb ...
and
radical ideal
Radical (from Latin: ', root) may refer to:
Politics and ideology Politics
*Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century
*Radical politics ...
s in