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Saturday, July 13, 2019

Ernest Rutherford

From Wikipedia, the free encyclopedia


The Lord Rutherford of Nelson

Ernest Rutherford LOC.jpg

President of the Royal Society
In office
1925–1930
Preceded bySir Charles Scott Sherrington
Succeeded bySir Frederick Gowland Hopkins
Personal details
Born30 August 1871
Brightwater, New Zealand
Died19 October 1937 (aged 66)
Cambridge, England
CitizenshipBritish subject
NationalityNew Zealander
ResidenceNew Zealand, United Kingdom
Signature

Alma mater
Known for
Awards
Scientific career
FieldsPhysics and chemistry
Institutions
Academic advisors
Doctoral students
Other notable students
Influenced


Ernest Rutherford was a New Zealand physicist who came to be known as the father of nuclear physics. Encyclopædia Britannica considers him to be the greatest experimentalist since Michael Faraday (1791–1867).

In early work, Rutherford discovered the concept of radioactive half-life, the radioactive element radon, and differentiated and named alpha and beta radiation. This work was performed at McGill University in Montreal, Canada. It is the basis for the Nobel Prize in Chemistry he was awarded in 1908 "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances", for which he was the first Canadian and Oceanian Nobel laureate.

Rutherford moved in 1907 to the Victoria University of Manchester (today University of Manchester) in the UK, where he and Thomas Royds proved that alpha radiation is helium nuclei. Rutherford performed his most famous work after he became a Nobel laureate. In 1911, although he could not prove that it was positive or negative, he theorized that atoms have their charge concentrated in a very small nucleus, and thereby pioneered the Rutherford model of the atom, through his discovery and interpretation of Rutherford scattering by the gold foil experiment of Hans Geiger and Ernest Marsden. He performed the first artificially induced nuclear reaction in 1917 in experiments where nitrogen nuclei were bombarded with alpha particles. As a result, he discovered the emission of a subatomic particle which, in 1919, he called the "hydrogen atom" but, in 1920, he more accurately named the proton.

Rutherford became Director of the Cavendish Laboratory at the University of Cambridge in 1919. Under his leadership the neutron was discovered by James Chadwick in 1932 and in the same year the first experiment to split the nucleus in a fully controlled manner was performed by students working under his direction, John Cockcroft and Ernest Walton. After his death in 1937, he was honoured by being interred with the greatest scientists of the United Kingdom, near Sir Isaac Newton's tomb in Westminster Abbey. The chemical element rutherfordium (element 104) was named after him in 1997.

Biography

Early life and education

Rutherford aged 21
 
Ernest Rutherford was the son of James Rutherford, a farmer, and his wife Martha Thompson, originally from Hornchurch, Essex, England. James had emigrated to New Zealand from Perth, Scotland, "to raise a little flax and a lot of children". Ernest was born at Brightwater, near Nelson, New Zealand. His first name was mistakenly spelled 'Earnest' when his birth was registered. Rutherford's mother Martha Thompson was a schoolteacher.

He studied at Havelock School and then Nelson College and won a scholarship to study at Canterbury College, University of New Zealand, where he participated in the debating society and played rugby. After gaining his BA, MA and BSc, and doing two years of research during which he invented a new form of radio receiver, in 1895 Rutherford was awarded an 1851 Research Fellowship from the Royal Commission for the Exhibition of 1851, to travel to England for postgraduate study at the Cavendish Laboratory, University of Cambridge. He was among the first of the 'aliens' (those without a Cambridge degree) allowed to do research at the university, under the leadership of J. J. Thomson, which aroused jealousies from the more conservative members of the Cavendish fraternity. With Thomson's encouragement, he managed to detect radio waves at half a mile and briefly held the world record for the distance over which electromagnetic waves could be detected, though when he presented his results at the British Association meeting in 1896, he discovered he had been outdone by another lecturer, by the name of Marconi

In 1898, Thomson recommended Rutherford for a position at McGill University in Montreal, Canada. He was to replace Hugh Longbourne Callendar who held the chair of Macdonald Professor of physics and was coming to Cambridge. Rutherford was accepted, which meant that in 1900 he could marry Mary Georgina Newton (1876–1954) to whom he had become engaged before leaving New Zealand; they married at St Paul's Anglican Church, Papanui in Christchurch, they had one daughter, Eileen Mary (1901–1930), who married Ralph Fowler. In 1901, he gained a DSc from the University of New Zealand. In 1907, Rutherford returned to Britain to take the chair of physics at the Victoria University of Manchester.

Later years and honours

He was knighted in 1914. During World War I, he worked on a top secret project to solve the practical problems of submarine detection by sonar. In 1916, he was awarded the Hector Memorial Medal. In 1919, he returned to the Cavendish succeeding J. J. Thomson as the Cavendish professor and Director. Under him, Nobel Prizes were awarded to James Chadwick for discovering the neutron (in 1932), John Cockcroft and Ernest Walton for an experiment which was to be known as splitting the atom using a particle accelerator, and Edward Appleton for demonstrating the existence of the ionosphere. In 1925, Rutherford pushed calls to the Government of New Zealand to support education and research, which led to the formation of the Department of Scientific and Industrial Research (DSIR) in the following year. Between 1925 and 1930, he served as President of the Royal Society, and later as president of the Academic Assistance Council which helped almost 1,000 university refugees from Germany. He was appointed to the Order of Merit in the 1925 New Year Honours and raised to the peerage as Baron Rutherford of Nelson, of Cambridge in the County of Cambridge in 1931, a title that became extinct upon his unexpected death in 1937. In 1933, Rutherford was one of the two inaugural recipients of the T. K. Sidey Medal, set up by the Royal Society of New Zealand as an award for outstanding scientific research.

Lord Rutherford's grave in Westminster Abbey
 
For some time before his death, Rutherford had a small hernia, which he had neglected to have fixed, and it became strangulated, causing him to be violently ill. Despite an emergency operation in London, he died four days afterwards of what physicians termed "intestinal paralysis", at Cambridge. After cremation at Golders Green Crematorium, he was given the high honour of burial in Westminster Abbey, near Isaac Newton and other illustrious British scientists.

Scientific research

Ernest Rutherford at McGill University in 1905
 
At Cambridge, Rutherford started to work with J. J. Thomson on the conductive effects of X-rays on gases, work which led to the discovery of the electron which Thomson presented to the world in 1897. Hearing of Becquerel's experience with uranium, Rutherford started to explore its radioactivity, discovering two types that differed from X-rays in their penetrating power. Continuing his research in Canada, he coined the terms alpha ray and beta ray in 1899 to describe the two distinct types of radiation. He then discovered that thorium gave off a gas which produced an emanation which was itself radioactive and would coat other substances. He found that a sample of this radioactive material of any size invariably took the same amount of time for half the sample to decay – its "half-life" (11½ minutes in this case). 

From 1900 to 1903, he was joined at McGill by the young chemist Frederick Soddy (Nobel Prize in Chemistry, 1921) for whom he set the problem of identifying the thorium emanations. Once he had eliminated all the normal chemical reactions, Soddy suggested that it must be one of the inert gases, which they named thoron (later found to be an isotope of radon). They also found another type of thorium they called Thorium X, and kept on finding traces of helium. They also worked with samples of "Uranium X" from William Crookes and radium from Marie Curie

In 1903, they published their "Law of Radioactive Change," to account for all their experiments. Until then, atoms were assumed to be the indestructible basis of all matter and although Curie had suggested that radioactivity was an atomic phenomenon, the idea of the atoms of radioactive substances breaking up was a radically new idea. Rutherford and Soddy demonstrated that radioactivity involved the spontaneous disintegration of atoms into other, as yet, unidentified matter. The Nobel Prize in Chemistry 1908 was awarded to Ernest Rutherford "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances". 

In 1903, Rutherford considered a type of radiation discovered (but not named) by French chemist Paul Villard in 1900, as an emission from radium, and realised that this observation must represent something different from his own alpha and beta rays, due to its very much greater penetrating power. Rutherford therefore gave this third type of radiation the name of gamma ray. All three of Rutherford's terms are in standard use today – other types of radioactive decay have since been discovered, but Rutherford's three types are among the most common. 

In Manchester, he continued to work with alpha radiation. In conjunction with Hans Geiger, he developed zinc sulfide scintillation screens and ionisation chambers to count alphas. By dividing the total charge they produced by the number counted, Rutherford decided that the charge on the alpha was two. In late 1907, Ernest Rutherford and Thomas Royds allowed alphas to penetrate a very thin window into an evacuated tube. As they sparked the tube into discharge, the spectrum obtained from it changed, as the alphas accumulated in the tube. Eventually, the clear spectrum of helium gas appeared, proving that alphas were at least ionised helium atoms, and probably helium nuclei.

Gold foil experiment

Top: Expected results: alpha particles passing through the plum pudding model of the atom undisturbed. Bottom: Observed results: a small portion of the particles were deflected, indicating a small, concentrated charge. Note that the image is not to scale; in reality the nucleus is vastly smaller than the electron shell.
 
Rutherford performed his most famous work after receiving the Nobel prize in 1908. Along with Hans Geiger and Ernest Marsden in 1909, he carried out the Geiger–Marsden experiment, which demonstrated the nuclear nature of atoms by deflecting alpha particles passing through a thin gold foil. Rutherford was inspired to ask Geiger and Marsden in this experiment to look for alpha particles with very high deflection angles, of a type not expected from any theory of matter at that time. Such deflections, though rare, were found, and proved to be a smooth but high-order function of the deflection angle. It was Rutherford's interpretation of this data that led him to formulate the Rutherford model of the atom in 1911 – that a very small charged nucleus, containing much of the atom's mass, was orbited by low-mass electrons

In 1919–1920, Rutherford found that nitrogen and other light elements ejected a proton (Rutherford said "a hydrogen atom" rather than "a proton") when hit with α (alpha) particles. This result showed Rutherford that hydrogen nuclei were a part of nitrogen nuclei (and by inference, probably other nuclei as well). Such a construction had been suspected for many years on the basis of atomic weights which were whole numbers of that of hydrogen. Hydrogen was known to be the lightest element, and its nuclei presumably the lightest nuclei. Now, because of all these considerations, Rutherford decided that a hydrogen nucleus was possibly a fundamental building block of all nuclei, and also possibly a new fundamental particle as well, since nothing was known from the nucleus that was lighter. Thus, confirming and extending the work of Wilhelm Wien who in 1898 discovered the proton in streams of ionized gas, Rutherford postulated the hydrogen nucleus to be a new particle in 1920, which he dubbed the proton

In 1921, while working with Niels Bohr (who postulated that electrons moved in specific orbits), Rutherford theorized about the existence of neutrons, (which he had christened in his 1920 Bakerian Lecture), which could somehow compensate for the repelling effect of the positive charges of protons by causing an attractive nuclear force and thus keep the nuclei from flying apart from the repulsion between protons. The only alternative to neutrons was the existence of "nuclear electrons" which would counteract some of the proton charges in the nucleus, since by then it was known that nuclei had about twice the mass that could be accounted for if they were simply assembled from hydrogen nuclei (protons). But how these nuclear electrons could be trapped in the nucleus, was a mystery. 

Rutherford's theory of neutrons was proved in 1932 by his associate James Chadwick, who recognized neutrons immediately when they were produced by other scientists and later himself, in bombarding beryllium with alpha particles. In 1935, Chadwick was awarded the Nobel Prize in Physics for this discovery.

Legacy

A plaque commemorating Rutherford's presence at the University of Manchester

Nuclear physics

Rutherford's research, and work done under him as laboratory director, established the nuclear structure of the atom and the essential nature of radioactive decay as a nuclear process. Patrick Blackett, a research fellow working under Rutherford, using natural alpha particles, demonstrated induced nuclear transmutation. Rutherford's team later, using protons from an accelerator, demonstrated artificially-induced nuclear reactions and transmutation. He is known as the father of nuclear physics. Rutherford died too early to see Leó Szilárd's idea of controlled nuclear chain reactions come into being. However, a speech of Rutherford's about his artificially-induced transmutation in lithium, printed in 12 September 1933 London paper The Times, was reported by Szilárd to have been his inspiration for thinking of the possibility of a controlled energy-producing nuclear chain reaction. Szilard had this idea while walking in London, on the same day. 

Rutherford's speech touched on the 1932 work of his students John Cockcroft and Ernest Walton in "splitting" lithium into alpha particles by bombardment with protons from a particle accelerator they had constructed. Rutherford realized that the energy released from the split lithium atoms was enormous, but he also realized that the energy needed for the accelerator, and its essential inefficiency in splitting atoms in this fashion, made the project an impossibility as a practical source of energy (accelerator-induced fission of light elements remains too inefficient to be used in this way, even today). Rutherford's speech in part, read:
We might in these processes obtain very much more energy than the proton supplied, but on the average we could not expect to obtain energy in this way. It was a very poor and inefficient way of producing energy, and anyone who looked for a source of power in the transformation of the atoms was talking moonshine. But the subject was scientifically interesting because it gave insight into the atoms.

Items named in honour of Rutherford's life and work

A statue of a young Ernest Rutherford at his memorial in Brightwater, New Zealand.
Scientific discoveries
Institutions
Awards
Buildings
Major streets
Other

Incidences of cancer at Rutherford's former laboratory

The Coupland Building at Manchester University, at which Rutherford conducted many of his experiments, has been the subject of a cancer cluster investigation. There has been a statistically high incidence of pancreatic cancer, brain cancer, and motor neuron disease occurring in and around Rutherford's former laboratories and, since 1984, a total of six workers have been stricken with these ailments. In 2009, an independent commission concluded that the very slightly elevated levels of various radiation related to Rutherford's experiments decades earlier are not the likely cause of such cancers and ruled the illnesses a coincidence.

Rutherford Nitrogen-to-Oxygen Transmutation Myth

A long-standing myth existed, at least as early as 1948, running at least to 2017, that Rutherford was the first scientist to observe and report an artificial transmutation of a stable element into another element; nitrogen into oxygen. It was thought by many people to be one of Rutherford's greatest accomplishments. The New Zealand government even commemorated a stamp in honor of its belief that that the nitrogen-to-oxygen discovery belonged to Rutherford. In fact, Rutherford did detect the ejected proton in 1919 and interpreted it as evidence for disintegration of the nitrogen nucleus (to lighter nuclei). However, Blackett identified the true reaction in 1925 and showed that the actual product is oxygen. Rutherford therefore recognized "that the nucleus may increase rather than diminish in mass as the result of collisions in which the proton is expelled."

Beginning in 2017, many scientific institutions corrected their versions of this history to indicate that the discovery credit for the reaction belongs to Patrick Blackett who identified the oxygen product. These institutions included the U.S. Department of Energy Office of History and Heritage Resources, the American Institute of Physics Center for History of Physics, Imperial College London, Faculty of Natural Sciences, and the University of Cambridge, Department of Physics.

Rutherford later claimed to have discovered the occurrence of a transmutation reaction but not to have identified the oxygen product. The earliest known instance of this attribution is an article written by Rutherford in the McGill News, September 1932 and reprinted in the Journal of the Royal Astronomical Society of Canada in 1933. In this article, Rutherford, writing about himself in the third person, claimed the discovery as his own "Rutherford, in 1919, bombarded nitrogen gas with swift alpha particles from radioactive substances and found that high-speed hydrogen nuclei, or free protons as we now term them, were liberated. … This was the first time that definite evidence had been obtained that an atom could be transmuted by artificial methods…"

Rutherford made the same claim the following year. On October 11, 1933, he gave a lecture, broadcast by the British Broadcasting Corporation, called “The Transmutation of the Atom.” A transcript of the lecture was published in The Scientific Monthly in January 1934. Rutherford wrote "The first successful experiments in transmutation are comparatively recent, dating back to the year 1919. … I made in 1919 some experiments to test whether any evidence of transformation could be obtained when alpha particles were used to bombard matter. … This was the first time that definite evidence was obtained that an atom could be transformed by artificial methods." 

Rutherford never explicitly claimed that the transmutation of nitrogen resulted in oxygen in his experiments. In these 1933 and 1934 claims, he simply claimed that he had transmuted an atom. After Blackett performed the first experiments to obtain the first definite evidence that the nitrogen atom was transmuted into oxygen in 1925, and with the help of Rutherford's 1933-1934 claims that he was the first to obtain "definite evidence" for transmuting an atom, the scientific community, with very few exceptions, assumed for the next 84 years that the nitrogen-to-oxygen discovery belonged to Rutherford.

David Hilbert

From Wikipedia, the free encyclopedia

David Hilbert
Hilbert.jpg
David Hilbert (1912)
Born23 January 1862
Died14 February 1943 (aged 81)
NationalityGerman
Alma materUniversity of Königsberg (PhD)
Known forHilbert's basis theorem
Hilbert's axioms
Hilbert's problems
Hilbert's program
Einstein–Hilbert action
Hilbert space
Epsilon calculus
Spouse(s)Käthe Jerosch
ChildrenFranz (b. 1893)
AwardsLobachevsky Prize (1903)
Bolyai Prize (1910)
ForMemRS
Scientific career
FieldsMathematics, Physics and Philosophy
InstitutionsUniversity of Königsberg
Göttingen University
ThesisOn Invariant Properties of Special Binary Forms, Especially of Spherical Functions (1885)
Doctoral advisorFerdinand von Lindemann
Doctoral studentsWilhelm Ackermann
Heinrich Behmann
Otto Blumenthal
Anne Bosworth
Werner Boy
Richard Courant
Haskell Curry
Max Dehn
Rudolf Fueter
Paul Funk
Kurt Grelling
Alfréd Haar
Erich Hecke
Earle Hedrick
Ernst Hellinger
Wallie Hurwitz
Margarete Kahn
Oliver Kellogg
Hellmuth Kneser
Robert König
Emanuel Lasker
Klara Löbenstein
Charles Max Mason
Erhard Schmidt
Kurt Schütte
Andreas Speiser
Hugo Steinhaus
Gabriel Sudan
Teiji Takagi
Hermann Weyl
Ernst Zermelo
Edward Kasner
Other notable studentsJohn von Neumann
InfluencesImmanuel Kant

David Hilbert was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, 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 foundations of mathematics (particularly proof theory).

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 developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Life

Early life and education

Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, 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.

In late 1872, Hilbert entered the Friedrichskolleg Gymnasium (Collegium fridericianum, 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 the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.

Career

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 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 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.

The Mathematical Institute in Göttingen. Its new building, constructed with funds from the Rockefeller Foundation, was opened by Hilbert and Courant in 1930.

Göttingen school

Portrait of David Hilbert in the 1900s, artist Anna Gorban, the cover image of the theme issue ‘Hilbert's sixth problem, Phil. Trans. R. Soc. A 2018, 376 (2118).
 
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 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 was editor of the Mathematische Annalen, the leading mathematical journal of the time.
"Good, he did not have enough imagination to become a mathematician".
— Hilbert's response upon hearing that one of his students had dropped out to study poetry.

Later years

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 a scientist after 1925, and certainly not a Hilbert."

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 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 the departure of the Jews". Hilbert replied, "Suffered? It doesn't exist any longer, does it!"

Death

Hilbert's tomb:
Wir müssen wissen
Wir werden wissen

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 a native of Königsberg. News of his death only became known to the wider world six 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, we shall not know":
Wir müssen wissen.
Wir werden wissen.
In English:
We must know.
We will know.
The day before Hilbert pronounced these phrases at 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.

Personal life

Käthe Hilbert with Constantin Carathéodory, before 1932
 
In 1892, Hilbert married Käthe Jerosch (1864–1945), "the daughter of a Königsberg merchant, an outspoken young lady with an independence of mind that matched his own". While at Königsberg they had their one child, Franz Hilbert (1893–1969). 

Hilbert's son Franz suffered throughout his life from an undiagnosed mental illness. His inferior intellect was a terrible disappointment to his father and this misfortune was a matter of distress to the mathematicians and students at Göttingen.

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 on left the Church and became an agnostic. He also argued that mathematical truth was independent of the existence of God or other a priori assumptions.

Hilbert solves 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 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. In order 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, 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 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 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 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 have imagined 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" — 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 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 was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker".[33] 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.

Axiomatization of geometry

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 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 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 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 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.

The 23 problems

Hilbert put forth a most influential list 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 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. The introduction of 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 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 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:
  1. all of mathematics follows from a correctly chosen finite system of axioms; and
  2. 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.
Hilbert wrote in 1919:
We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.
Hilbert published his views on the foundations of mathematics in the 2-volume work Grundlagen der Mathematik.

Gödel'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, ended in failure. 

Gödel 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 Gödel'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 the work of 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 spaces are an important class of objects 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 on general relativity, and he invited Einstein to Göttingen to deliver a week of lectures on the subject. Einstein received an enthusiastic reception at Göttingen. 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". Nearly simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–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.

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 Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory. In 1926 von Neumann showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger'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 mathematics, physicists 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 physics and how 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 Methoden der mathematischen Physik (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 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 lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal.

Works

His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained "many technical errors of varying degree"; 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 exception—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.

Cooperative

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