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111-561: During his tenure at the University of Göttingen , Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. His seminars covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction, and promoted mathematics education reform at all grade levels in Germany and abroad. He became

222-580: A common root: This is the case if and only if there do not exist polynomials q 1 , … , q k {\displaystyle q_{1},\ldots ,q_{k}} and indices λ 1 , … , λ k {\displaystyle \lambda _{1},\ldots ,\lambda _{k}} such that This result is known as the Hilbert root theorem , or "Hilberts Nullstellensatz" in German. He also proved that

333-475: A century. The subject had attracted students such as Klemens Wenzel Lothar von Metternich , later diplomat and Prime Minister of Austria, and Wilhelm von Humboldt , who later established the University of Berlin . In the 19th century, Gustav Hugo , Karl Friedrich Eichhorn , and Georg Friedrich Sartorius , who taught law here, became the pioneers of the German Historical School of Jurisprudence. At

444-494: A curve, which is now called Hilbert curve . Approximations to this curve are constructed iteratively according to the replacement rules in the first picture of this section. The curve itself is then the pointwise limit. The text Grundlagen der Geometrie (tr.: Foundations of Geometry ) published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditional axioms of Euclid . They avoid weaknesses identified in those of Euclid , whose works at

555-514: A cylinder looped back through itself to join with its other end from the "inside". It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional Möbius strip , with one method of construction being the attachment of the edges of two Möbius strips . During the 1890s, Klein began studying mathematical physics more intensively, writing on

666-822: A diplomat and historian, had personal friendship with Otto von Bismarck during his two-year-long study in Göttingen. George Bancroft , a politician and historian, received his PhD from the University of Göttingen in 1820. In the 1930s, the university became a focal point for the Nazi crackdown on "Jewish physics", as represented by the work of Albert Einstein . In what was later called the "great purge" of 1933 , academics including Max Born , Victor Goldschmidt , James Franck , Eugene Wigner , Leó Szilárd , Edward Teller , Edmund Landau , Emmy Noether , and Richard Courant were expelled or fled. Most of them fled Nazi Germany for places like

777-457: A dissertation, written under Ferdinand von Lindemann , 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 , he obtained

888-530: A famous philosopher known as the father of phenomenology , moved to Göttingen to teach. Ludwig Prandtl joined the University of Göttingen in 1904, and developed it into a leader in fluid mechanics and in aerodynamics over the next two decades. In 1925, Prandtl was appointed as the director of the Kaiser Wilhelm Institute for Fluid Mechanics. He introduced the concept of boundary layer and founded mathematical aerodynamics by calculating air flow in

999-503: A foreign member of the Royal Netherlands Academy of Arts and Sciences . Around 1900, Klein began to become interested in mathematical instruction in schools. In 1905, he was instrumental in formulating a plan recommending that analytic geometry , the rudiments of differential and integral calculus , and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around

1110-424: A geometric way, connecting potential theory and conformal mappings . This work drew on notions from fluid dynamics . Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on methods of Charles Hermite and Leopold Kronecker , he produced similar results to those of Brioschi and later completely solved

1221-516: A grand uniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. He came up with his proof during an asthma attack at 2:30 A.M. on 23 March 1882. Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over a period of about 20 years. University of G%C3%B6ttingen The University of Göttingen , officially

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1332-539: A highly influential list consisting of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. After reworking the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed from

1443-521: A judge and Geheimrat . His mother Maria had an interest in philosophy, astronomy and prime numbers , while his father Otto taught him Prussian virtues . 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 Gymnasium ( Collegium fridericianum ,

1554-418: A library with over 200,000 volumes. This period marked Göttingen's ascendancy in academic circles, emphasizing its role in fostering an environment conducive to scientific inquiry and innovation. In the first years of the University of Göttingen, it became known especially for its Faculty of Law . In the 18th century Johann Stephan Pütter , a scholar of public law at that time, taught jus publicum for half

1665-662: 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 " or "We do not know and we shall not know": Wir müssen wissen. Wir werden wissen. We must know. We shall know. The day before Hilbert pronounced these phrases at

1776-537: A professorship at the University of Göttingen in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's prime center for mathematics research. However, he never managed to transfer from Leipzig to Göttingen his own leading role as developer of geometry . He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular, mechanics and potential theory . The research facility Klein established at Göttingen served as model for

1887-444: 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 and his developing intuitionist "school", much to Hilbert's torment in his later years. Indeed, Hilbert would lose his "gifted pupil" Weyl to intuitionism—"Hilbert

1998-621: A scientist after 1925, and certainly not a Hilbert." Hilbert was elected to the American Philosophical Society in 1932. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933. Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired in 1930), Emmy Noether and Edmund Landau . One who had to leave Germany, Paul Bernays , had collaborated with Hilbert in mathematical logic, and co-authored with him

2109-534: 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 continue to remain challenges. The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society . In an account that had become standard by the mid-century, Hilbert's problem set

2220-404: A solid and complete logical foundation. He believed that in principle this could be done by showing that: 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

2331-572: A total of 38 additional centers and institutes (including associated centers and institutes but excluding institutes or departments within the faculties themselves). Natural Sciences, Mathematics, and Informatics Law, Economic Sciences, and Social Sciences Associated Institutes In the QS World University Rankings 2024, it was placed the 232nd internationally and the 13th nationally. The Times Higher Education World University Rankings for 2024 ranked it as 111th globally and 10th at

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2442-701: Is a member of the European Union , university students have the opportunity to participate in the Erasmus Programme . The university also has exchange programs and partnerships with reputable universities outside Europe such as Tsinghua University , Peking University and Fudan University in China, Kyoto University and Waseda University in Japan and Amherst College in the United States. The most famous tradition of

2553-410: Is difficult to appreciate their novelty when first presented, and understand the fact that they were not immediately accepted by all his contemporaries. Klein saw his work on complex analysis as his major contribution to mathematics, specifically his work on: Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate the plane. In 1879, he examined

2664-500: Is no life outside Göttingen. Even if it is life, it is no life like here ). "Ancient university towns are wonderfully alike. Göttingen is like Cambridge in England or Yale in America: very provincial, not on the way to anywhere – no one comes to these backwaters except for the company of professors. And the professors are sure that this is the centre of the world. There is an inscription in

2775-497: 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, 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: Without doubt this

2886-556: 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

2997-563: 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 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

3108-416: Is their defined relationships that are discussed. Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points ( line segments ), and congruence of angles . The axioms unify both the plane geometry and solid geometry of Euclid in a single system. Hilbert put forth

3219-508: The Berlin Group whose leading founders had studied under Hilbert in Göttingen ( Kurt Grelling , Hans Reichenbach and Walter Dubislav ). Around 1925, Hilbert developed pernicious anemia , a then-untreatable vitamin deficiency whose primary symptom is exhaustion; his assistant Eugene Wigner 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

3330-573: The Faculty of Law in Göttingen . Ursula von der Leyen , President of the European Commission , studied economics in Göttingen, where she met her husband Heiko von der Leyen . The university is spread out in several locations around the city. The central university complex with the Central Library and Mensa (student refectory/dining hall) is located right next to the inner city and comprises

3441-576: The Georg August University of Göttingen ( German : Georg-August-Universität Göttingen , commonly referred to as Georgia Augusta ), is a public research university in the city of Göttingen , Lower Saxony , Germany . Founded in 1734 by George II , King of Great Britain and Elector of Hanover , it began instruction in 1737 and is recognized as the oldest university in Lower Saxony. Recognized for its historic and traditional significance,

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3552-576: The Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn , 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868. Plücker died in 1868, leaving his book concerning

3663-551: The Göttingen school of history . Later, Max Weber , one of the most important theorists of the development of modern Western society, also studied history in Göttingen. The Brothers Grimm , the best-known storytellers of folktales like " Cinderella ", " The Frog Prince ", " Little Red Riding Hood ", " Sleeping Beauty ", and " Snow White ", taught here and compiled the first German dictionary. However, political disturbances, in which both professors and students were implicated, lowered

3774-773: The Max Planck Society , the Leibniz Association , the Fraunhofer Society , and the Helmholtz Association . With its extensive collection, the Göttingen State and University Library stands among Germany's largest libraries. In 1734, King George II of Great Britain and Elector of Hanover , commanded his Prime Minister in Hanover, Gerlach Adolph von Münchhausen, to establish a university in Göttingen to spread

3885-866: The Royal Society in 1885, and was awarded its Copley Medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for several further years. Klein was one of ninety-three signatories of the Manifesto of the Ninety-Three , a document penned in support of the German invasion of Belgium in the early stages of World War I . He died in Göttingen in 1925. Klein's dissertation, on line geometry and its applications to mechanics , classified second degree line complexes using Weierstrass 's theory of elementary divisors. Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie , he discovered

3996-459: The foundations of mathematics (particularly proof theory ). He adopted and defended Georg Cantor 's set theory and transfinite numbers . 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 . Hilbert,

4107-464: The gyroscope with Arnold Sommerfeld . During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften . This enterprise, which endured until 1935, provided an important standard reference of enduring value. In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On

4218-562: 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 , 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: Das ist nicht Mathematik. Das ist Theologie. This

4329-643: The 'Royal Society for Sciences'. The university maintains three botanical gardens: the Alter Botanischer Garten der Universität Göttingen, the Neuer Botanischer Garten der Universität Göttingen, and the Forstbotanischer Garten und Pflanzengeographisches Arboretum der Universität Göttingen. As of 2023, the university consists of 13 faculties and around 22,484 students are enrolled. 535 professors and over 4,000 academic staff work at

4440-452: The 18th century. Thereafter, Dirichlet and Riemann took over the chair successively and made significant contributions in the fields of algebra, geometry, and number theory. By 1900, David Hilbert and Felix Klein had attracted mathematicians from around the world to Göttingen, which made it a leading center of mathematics by the turn of the 20th century. In 1903, its teaching staff numbered 121 and its students 1529. Edmund Husserl ,

4551-608: The 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel —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. Gödel's incompleteness theorems show that even elementary axiomatic systems such as Peano arithmetic are either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system. Hilbert's first work on invariant functions led him to

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4662-441: The 19th century. Throughout the 18th century, the University of Göttingen was renowned among German universities for its commitment to the free spirit and scientific exploration. Georg Christoph Lichtenberg , a prominent scholar, held one of the first professorships dedicated to experimental physics in Germany from 1769 to 1799. By 1812, Göttingen had established itself as a modern, internationally recognized university, boasting

4773-830: The 6th in the world on "2D rank" combining both categories of data. As of 2002, the University of Göttingen was associated with 44 Nobel laureates according to an official count released by the University of Göttingen in that year. By this number alone, the University of Göttingen ranked among the global top 15 universities. Recent Nobel laureates associated with the university are Klaus Hasselmann ( Nobel Prize in Physics , 2021), Stefan Hell ( Nobel Prize in Chemistry , 2014), Thomas C. Südhof ( Nobel Prize in Physiology or Medicine , 2013), and Thomas Arthur Steitz ( Nobel Prize in Chemistry , 2009). Klaus Hasselmann received his PhD in physics from

4884-548: The 8th nationally. Within the framework of the 2006–07 German Universities Excellence Initiative , it won funding for its future concept "Tradition, Innovation, Autonomy," its graduate school "Neurosciences and Molecular Biosciences," and its research cluster "Microscopy at the Nanometer Range." In the 2012 Excellence Initiative, Göttingen succeeded in obtaining funds for its graduate school "Neurosciences and Molecular Biosciences" and its research cluster "Microscopy at

4995-477: The American banker J. P. Morgan , the seismologist Beno Gutenberg , the endocrinologist Hakaru Hashimoto , who studied there before World War I , and several notable Nobel laureates like Max Planck and Werner Heisenberg . Anthropologist Marlina Flassy earned her doctorate in Göttingen, before becoming the first woman and indigenous Papuan to be appointed Dean at Cenderawasih University . The German inventor of

5106-404: The German philosopher best known for his work The World as Will and Representation , became a student at the university, where he studied metaphysics and psychology under Gottlob Ernst Schulze , who advised him to concentrate on Plato and Kant . During this time, the University of Göttingen achieved renown for its critical work on history as well. An Enlightenment institution, it produced

5217-740: The International Court of Justice on behalf of the Federal Republic of Germany . Hsu Tzong-li , a Taiwanese judge who has served as the President of the Judicial Yuan (Taiwan's constitutional court) since 2016, earned his doctorate in law from University of Göttingen in 1986. Within the Göttingen Campus the university is organizationally and personally interlinked with the following independent and semi-independent institutions: As Germany

5328-430: The Nanometer Range". In September 2018, Göttingen succeeded in gaining funds for its research cluster "Multiscale Bioimaging". The University of Göttingen is a traditional institution with a significant historical standing. This is substantiated by a scientific study on the historical development of world university rankings starting in 2013, which examined a web dataset from 24 Misplaced Pages language versions, covering 59% of

5439-546: The Ratskeller there which reads 'Extra Gottingam non est vita', 'Outside Göttingen there is no life'. This epigram, or should I call it epitaph, is not taken as seriously by the undergraduates as by the professors." The university offers eight snack shops and six Mensas serving lunch at low prices for the students. One Mensa also provides dinner for students. Apart from those celebrities mentioned above, notable people that have studied or taught at Georg-August University include

5550-594: The So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric . This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, believing it to be circular. Klein's synthesis of geometry as

5661-575: The Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig University . His colleagues included Walther von Dyck , Rohn, Eduard Study and Friedrich Engel . Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed and he battled with depression for the next two years. Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period. Klein accepted

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5772-572: The United States, Canada, and the United Kingdom. Following the great purge, in 1934 David Hilbert , by then a symbol of German mathematics, was dining with Bernhard Rust , the Nazi minister of education. Rust asked, "How is mathematics at Göttingen, now that it is free from the Jewish influence?" Hilbert replied, "There is no mathematics in Göttingen anymore." After World War II , the University of Göttingen

5883-463: The University of Göttingen had profound impacts on the US. A number of American politicians, lawyers, historians and writers received their education from both Harvard and Göttingen. For example, Edward Everett , once Secretary of State and President of Harvard University , stayed in Göttingen for two years of study. George Ticknor spent two years studying classics in Göttingen. Even John Lothrop Motley ,

5994-732: The University of Göttingen in 1957. Stefan Hell has been a lecturer (in Privatdozent capacity) at the University of Göttingen since 2004 and the director of the Max Planck Institute for Biophysical Chemistry in the Göttingen Campus since 2002. Thomas Südhof , currently a professor at Stanford University , worked on his doctoral thesis at the Max Planck Institute for Biophysical Chemistry in the lab of British biochemist Victor P. Whittaker and received his PhD in medical science from

6105-553: The University of Göttingen in 1982. Thomas Arthur Steitz was a Macy Fellow at the University of Göttingen during 1976–1977. Today, a number of judges in national and international courts of the highest level are still affiliated with the Faculty of Law . As of 2021, four out of sixteen in-office Justices of the Federal Constitutional Court ( German : Bundesverfassungsgericht ; abbreviated: BVerfG ), Germany's supreme constitutional court , are affiliated with

6216-471: The University of Göttingen: two of them ( Andreas Paulus & Christine Langenfeld) are, currently, professors at the Faculty of Law of the University of Göttingen, while two others (Ines Härtel & Henning Radtke) obtained their PhD in law (Dr.iur) from the University of Göttingen. Also in 2021, Georg Nolte , a former professor of public international law at the University of Göttingen, took office as Judge of

6327-415: The accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry. Currently, the significance of Klein's contributions to geometry is evident. They have become so much part of mathematical thinking that it

6438-536: The action of PSL(2,7) , considered as an image of the modular group , and obtained an explicit representation of a Riemann surface now termed the Klein quartic . He showed that it was a complex curve in projective space , that its equation was x y  +  y z  +  z x  = 0, and that its group of symmetries was PSL(2,7) of order 168. His Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale (1882) treats complex analysis in

6549-575: The attendance to 860 in 1834. The expulsion in 1837 of the seven professors – the so-called Die Göttinger Sieben (the Germanist Wilhelm Eduard Albrecht (1800–1876), the historian Friedrich Christoph Dahlmann (1785–1860), the orientalist Georg Heinrich August Ewald (1803–1875), the historian Georg Gottfried Gervinus (1805–1875), the physicist Wilhelm Eduard Weber (1804–1891), and the philologist brothers Jakob (1785–1863) and Wilhelm Grimm (1786–1859)) – for protesting against

6660-510: The basis of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes , and thus became acquainted with Alfred Clebsch , who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits to Berlin and Paris. In July 1870, at the beginning of the Franco-Prussian War , he was in Paris and had to leave

6771-454: The best mathematical journals in the world. Founded by Clebsch, it grew under Klein's management, to rival, and eventually surpass Crelle's Journal , based at the University of Berlin . Klein established a small team of editors who met regularly, making decisions in a democratic spirit. The journal first specialized in complex analysis , algebraic geometry , and invariant theory . It also provided an important outlet for real analysis and

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6882-430: The best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein recruited David Hilbert from the University of Königsberg . This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932. Under Klein's editorship, Mathematische Annalen became one of

6993-521: The changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised. Hilbert's approach signaled the shift to the modern axiomatic method . In this, Hilbert

7104-539: The city are the departments of Anthropology and Educational Sciences as well as the Medical Faculty with its associated hospitals. Just north of the city a new scientific center has been built in which most of the natural sciences (chemistry, biology, plant pathology, agronomy, forestry, geology, physics, computer science) are now located, including the GZMB. Other institutes are set around the inner city. Closely linked with

7215-471: The city walls. By the university's centenary in 1837, it was known as the "university of law", as the students enrolled by the faculty of law often made up more than half of the university's students. At the end of the 19th century, the famous civil law scholar Rudolf von Jhering , who created the theory of "culpa in contrahendo" (fault in conclusion of a contract), remained a law professor in Göttingen until he died. Lassa Francis Lawrence Oppenheim , known as

7326-484: The correspondence between vanishing ideals and their vanishing sets is bijective between affine varieties and radical ideals in C [ x 1 , … , x n ] {\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]} . In 1890, Giuseppe Peano had published an article in the Mathematische Annalen describing the historically first space-filling curve . In response, Hilbert designed his own construction of such

7437-551: The country. For a brief time he served as a medical orderly in the Prussian army before being appointed Privatdozent (lecturer) at Göttingen in early 1871. The University of Erlangen appointed Klein professor in 1872, when he was only 23 years old. For this, he was endorsed by Clebsch, who regarded him as likely to become the best mathematician of his time. Klein did not wish to remain in Erlangen, where there were very few students, and

7548-460: 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. 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

7659-524: The departure of the Jews." Hilbert replied, "Suffered? It doesn't exist any longer, does it?" 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 , a theoretical physicist and also

7770-407: The down wind direction. Many of Prandtl's students went on to make fundamental contributions to aerodynamics. Between 1921 and 1933, the physics theory group was led by Max Born , who, during this time, became one of the three discoverers of the non-relativistic theory of quantum mechanics . He may also have been the first to propose its probabilistic relationship with classical physics. It was one of

7881-605: The faculties for Theology, Social sciences, Law, Economics/Business Administration and Linguistics. The departments of Ancient History, Classics, various languages, Psychology and Philosophy are nearby. Located to the south of the city is the Faculty of Mathematics and Computer Science with its main building, the Mathematisches Institut, on the same street as the German Aerospace Center and the Max Planck Institute for Dynamics and Self-Organisation. In other parts of

7992-424: The father of the modern discipline of international law and author of the famous two-volume "International Law: A Treatise", earned his doctorate in law from the University of Göttingen in 1881. Likewise, the Faculty of Theology in conjunction with other orientalists and ancient historians across the university became an international center for the study of religion and antiquity. In 1809, Arthur Schopenhauer ,

8103-565: The first of two children and only son of Otto, a county judge, and Maria Therese Hilbert ( née Erdtmann), the daughter of a merchant, was born in the Province of Prussia , Kingdom of Prussia , either in Königsberg (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,

8214-719: The first president of the International Commission on Mathematical Instruction in 1908 at the Fourth International Congress of Mathematicians in Rome. Felix Klein was born on 25 April 1849 in Düsseldorf , to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province . His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended

8325-557: The fundamental properties of the asymptotic lines on the Kummer surface . They later investigated W-curves , curves invariant under a group of projective transformations . It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan . Klein devised the " Klein bottle " named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space , but it may be immersed as

8436-511: The ideals of the Enlightenment . Napoleon famously remarked, "Göttingen belongs neither to a State, nor to Germany, and is the University of Europe". The initial university infrastructure was modest, comprising only a riding hall and a fencing house, with lectures held in the Paulinerkirche , Dominican monastery, or professors' homes. A university auditorium wasn't constructed until

8547-582: 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. Hermann Weyl's successor was Helmut Hasse . About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust . Rust asked whether "the Mathematical Institute really suffered so much because of

8658-492: The jet engine, Pabst von Ohain , also studied aerodynamics under Ludwig Prandtl . Professor Gunther Heinrich von Berg (Doctor of Law), taught at the University of Göttingen between 1794 and 1800 before he entered politics. 51°32′31″N 9°56′04″E  /  51.54194°N 9.93444°E  / 51.54194; 9.93444 David Hilbert David Hilbert ( / ˈ h ɪ l b ər t / ; German: [ˈdaːvɪt ˈhɪlbɐt] ; 23 January 1862 – 14 February 1943)

8769-535: The later "foundationalist" Russell–Whitehead or "encyclopedist" Nicolas Bourbaki , and from his contemporary Giuseppe Peano . The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics. 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. The introduction of

8880-576: The main centers of the development of modern physics . Oppenheimer , the American scientist and "father of the atomic bomb", was one of Max Born 's most famous students and received his doctorate here. During this time, the German language became an international academic language. A number of dissertations in the UK and the US had German titles. One might be considered having had a complete academic training only when one had studied in Germany. Thus, many American students were proud of having studied in Germany, and

8991-420: The national level. According to the 2023 ARWU World Rankings , the university holds a position within the 151–200 range internationally and between the 6th and 9th domestically. In the 2023 Center for World University Rankings (CWUR), the University of Göttingen was ranked the 97th worldwide and the 6th nationwide. In the 2022-2023 US News Best Global Universities Rankings , it was ranked the 163rd globally and

9102-609: The new group theory . In 1893, Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the World's Columbian Exposition . Due partly to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman, by Grace Chisholm Young , an English student of Arthur Cayley 's, whom Klein admired. In 1897, Klein became

9213-480: The position of Professor of Mathematics at the University of Göttingen . During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. He remained there for the rest of his life. Among Hilbert's students were Hermann Weyl , chess champion Emanuel Lasker , Ernst Zermelo , and Carl Gustav Hempel . John von Neumann was his assistant. At the University of Göttingen , Hilbert

9324-434: The problem by means of the icosahedral group . This work enabled him to write a series of papers on elliptic modular functions . In his 1884 book on the icosahedron , Klein established a theory of automorphic functions , associating algebra and geometry. Poincaré had published an outline of his theory of automorphic functions in 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove

9435-496: 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. See also Hilbert's twenty-fourth problem . 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

9546-503: The revocation by Ernest Augustus, King of Hanover , of the liberal constitution of 1833 hurt the reputation of the city and the university. At the end of the 19th and beginning of the 20th century, the University of Göttingen achieved its academic peak. Göttingen maintained a strong focus on natural science, especially mathematics. The tradition began with Carl Friedrich Gauss , who was known as "the Prince of Mathematicians" and taught here in

9657-490: The same school that Immanuel Kant 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 "Albertina". In early 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters), returned to Königsberg and entered

9768-471: The speech that Hilbert gave said: 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

9879-575: The statue of Gänseliesel (a poor princess in an old fairy tale who was compelled to keep geese by a wicked woman and later regained her identity), kiss the Gänseliesel and give bouquets to her. There is an old saying about life in Göttingen, still inscribed in Latin nowadays on the wall of the entrance to the Ratskeller (the restaurant located in the basement of the old town hall): Latin : Extra Gottingam non est vita, si est vita, non est ita ( There

9990-452: The study of the properties of a space that is invariant under a given group of transformations , known as the Erlangen program (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become

10101-435: The subject of algebra , a field is called algebraically closed 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 ( p λ ) λ ∈ Λ {\displaystyle (p_{\lambda })_{\lambda \in \Lambda }} of n {\displaystyle n} variables has

10212-580: The time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert without referring to the publication history of the Grundlagen since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated

10323-538: The time, Göttingen was a very popular place for the study of law in Germany: Even the great German poet Heinrich Heine obtained a doctorate in law here in 1825. Otto von Bismarck , the main creator and the first Chancellor of the second German Empire , also studied law in Göttingen in 1833: he lived in a tiny house on the "Wall", now known as "Bismarck Cottage". According to oral tradition, he lived there because his rowdiness had caused him to be banned from living within

10434-639: The university has affiliations with 47 Nobel Prize winners by its own count. Previously backed by the German Universities Excellence Initiative , the University of Göttingen is a member of the U15 Group of major German research universities and the Coimbra Group of major European research universities, denoting its research prominence. The university is also closely linked with a number of leading Göttingen-based research institutions like

10545-462: The university is that PhD students who have just passed their Rigorosum (oral doctoral examination) or dissertation defense sit in a wagon – decorated with flowers and balloons and accompanied by relatives and friends, drive around the inner city and arrive at the Marktplatz – the central square where the old town hall and the Gänseliesel statue are located. The "newly born doctor" shall climb up to

10656-528: The university is the Göttingen State and University Library (German: Niedersächsische Staats- und Universitätsbibliothek Göttingen, or SUB Göttingen). With around 9 million media units and precious manuscripts, the library is designed for Göttingen University as well as the central library for the German State of Lower Saxony (with its central catalogue) and for the Göttingen Academy of Sciences, founded as

10767-527: The university, assisted by a technical and administrative staff of over 7,000. The post-war expansion of the university led to the establishment of a new, modern "university quarter" in the north of the city. The architecture of the old university can still be seen in the Auditorium Maximum (1826/1865) and the Great Hall (1835/1837) at Wilhelmsplatz. The University of Göttingen encompasses 13 faculties and

10878-444: The university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski. In 1884, Adolf Hurwitz 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

10989-441: The world's population and accounting for 68% of all Misplaced Pages articles across 287 languages. This comprehensive analysis identified the most influential universities globally over the past ten centuries, resulting in the "2017 Misplaced Pages World University Rankings", where the University of Göttingen ranked the 20th in terms of "web page rank" (search engine data), the 8th on account of "chei rank" (interactive citation data), and

11100-530: The world. In 1908, Klein was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians . Under his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany. The London Mathematical Society awarded Klein its De Morgan Medal in 1893. He was elected a member of

11211-436: Was a German mathematician 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 , the calculus of variations , commutative algebra , algebraic number theory , the foundations of geometry , spectral theory of operators and its application to integral equations , mathematical physics , and

11322-566: 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 to be his "best and truest friend". Hilbert was baptized and raised a Calvinist in the Prussian Evangelical Church . He later left the Church and became an agnostic . He also argued that mathematical truth

11433-465: 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. In 1920, Hilbert proposed a research project in metamathematics that became known as Hilbert's program. He wanted mathematics to be formulated on

11544-461: Was anticipated by Moritz Pasch 's work from 1882. 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 is reported to have said to Schoenflies and Kötter , by tables, chairs, glasses of beer and other such objects. It

11655-574: 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: Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists. In

11766-640: Was editor of the Mathematische Annalen , the leading mathematical journal of the time. He was elected an International Member of the United States National Academy of Sciences in 1907. 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 [Hilbert's]." While at Königsberg, they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from mental illness, and after he

11877-510: Was independent of the existence of God or other a priori assumptions. When Galileo Galilei was criticized for failing to stand up for his convictions on the Heliocentric theory , Hilbert objected: "But [Galileo] was 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." Like Albert Einstein , Hilbert had closest contacts with

11988-432: 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 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 —it did not display "an object"—but rather, it was an existence proof and relied on use of

12099-518: Was pleased to be offered a professorship at the Technische Hochschule München in 1875. There he and Alexander von Brill taught advanced courses to many excellent students, including Adolf Hurwitz , Walther von Dyck , Karl Rohn , Carl Runge , Max Planck , Luigi Bianchi , and Gregorio Ricci-Curbastro . In 1875, Klein married Anne Hegel, granddaughter of the philosopher Georg Wilhelm Friedrich Hegel . After spending five years at

12210-480: 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 Göttingen 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), and Wilhelm Ackermann (1925). Between 1902 and 1939 Hilbert

12321-443: Was the first university in the western Zones to be re-opened under British control in 1945. Göttingen's glorious history of affiliation with numerous celebrities continued: Jürgen Habermas , a famous German philosopher and sociologist, pursued his study in Göttingen. Later, Richard von Weizsäcker , a late President of Germany , earned his doctorate in law here. Gerhard Schröder , a former Chancellor of Germany , also graduated from

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