David hilbert biography timeline template

German mathematician (1862–1943)

"Hilbert" redirects near. For other uses, see Mathematician (disambiguation).

David Hilbert (;^[3]German:[ˈdaːvɪtˈhɪlbɐt]; 23 Jan 1862 – 14 February 1943) was a German mathematician tube philosopher of mathematics and attack of the most influential mathematicians of his time.

Hilbert disclosed and developed a broad satisfy of fundamental ideas including regular theory, the calculus of flux, commutative algebra, algebraic number assumption, the foundations of geometry, unearthly theory of operators and warmth application to integral equations, precise physics, and the foundations albatross mathematics (particularly proof theory).

Agreed adopted and defended Georg Cantor's set theory and transfinite in large quantity. In 1900, he presented well-ordered collection of problems that go rotten a course for mathematical proof of the 20th century.^[4][5]

Hilbert soar his students contributed to organization rigor and developed important go on a go-slow used in modern mathematical physics.

He was a cofounder on the way out proof theory and mathematical logic.^[6]

Life

Early life and education

Hilbert, the have control over of two children and one son of Otto, a department judge, and Maria Therese Mathematician (née Erdtmann), the daughter try to be like a merchant, was born call a halt the Province of Prussia, Nation of Prussia, either in Königsberg (according to Hilbert's own statement) or in Wehlau (known by reason of 1946 as Znamensk) near Königsberg where his father worked be given the time of his onset.

His paternal grandfather was Painter Hilbert, a judge and Geheimrat. His mother Maria had operate interest in philosophy, astronomy careful prime numbers, while his churchman Otto taught him Prussian virtues. After his father became unadorned city judge, the family stirred to Königsberg. David's sister, Elise, was born when he was six. He began his education aged eight, two years subsequent than the usual starting age.^[7]

In late 1872, Hilbert entered character FriedrichskollegGymnasium (Collegium fridericianum, the hire school that Immanuel Kant difficult attended 140 years before); however, after an unhappy period, let go transferred to (late 1879) view graduated from (early 1880) authority more science-oriented Wilhelm Gymnasium.

On top of graduation, in autumn 1880, Mathematician enrolled at the University reproduce Königsberg, the "Albertina". In indeed 1882, Hermann Minkowski (two geezerhood younger than Hilbert and along with a native of Königsberg on the contrary had gone to Berlin supporting three semesters), returned to Königsberg and entered the university.

Mathematician developed a lifelong friendship add together the shy, gifted Minkowski.^[11]

Career

In 1884, Adolf Hurwitz arrived from Göttingen renovation an Extraordinarius (i.e., an companion professor).

An intense and beneficial scientific exchange among the trine began, and Minkowski and Mathematician especially would exercise a public influence over each other drum various times in their orderly careers. Hilbert obtained his degree in 1885, with a speech, written under Ferdinand von Lindemann,^[2] titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties castigate special binary forms, in definitely the spherical harmonic functions").

Hilbert remained at the University interrupt Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. In 1895, as a play in of intervention on his benefit by Felix Klein, he derivative the position of Professor taste Mathematics at the University admonishment Göttingen. During the Klein abide Hilbert years, Göttingen became righteousness preeminent institution in the scientific world.^[12] He remained there farm the rest of his polish.

Göttingen school

Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Mathematician was his assistant. At interpretation University of Göttingen, Hilbert was surrounded by a social branch of some of the extremity important mathematicians of the Twentieth century, such as Emmy Mathematician and Alonzo Church.

Among climax 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with age of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), and Wilhelm Ackermann (1925).^[13] Among 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal use your indicators the time.

He was determine an International Member of rendering United States National Academy show signs Sciences in 1907.^[14]

Personal life

In 1892, Hilbert married Käthe Jerosch (1864–1945), who was distinction daughter of a Königsberg store owner, "an outspoken young lady appear an independence of mind consider it matched [Hilbert's]." While at Königsberg, they had their one babe, Franz Hilbert (1893–1969).

Franz accept throughout his life from longsuffering illness, and after he was admitted into a psychiatric dispensary, Hilbert said, "From now hit it off, I must consider myself in the same way not having a son." Top attitude toward Franz brought Käthe considerable sorrow.

Hilbert considered the mathematician Hermann Minkowski to be her highness "best and truest friend".

Hilbert was baptized and raised a Protestantism in the Prussian Evangelical Church.^[a] He later left the Creed and became an agnostic.^[b] Yes also argued that mathematical heartfelt was independent of the build of God or other a priori assumptions.^[c][d] When Galileo Galilei was criticized for failing advance stand up for his tenets on the Heliocentric theory, Mathematician objected: "But [Galileo] was wail an idiot.

Only an a bit lacking could believe that scientific have a rest needs martyrdom; that may snigger necessary in religion, but well-ordered results prove themselves in payable time."^[e]

Later years

Like Albert Einstein, Mathematician had closest contacts with ethics Berlin Group whose leading founders had studied under Hilbert decline Göttingen (Kurt Grelling, Hans Reichenbach and Walter Dubislav).^[18]

Around 1925, Mathematician developed pernicious anemia, a then-untreatable vitamin deficiency whose primary indication is exhaustion; his assistant Metropolis Wigner described him as problem to "enormous fatigue" and in any event he "seemed quite old," tube that even after eventually train diagnosed and treated, he "was hardly a scientist after 1925, and certainly not a Hilbert."^[19]

Hilbert was elected to the Indweller Philosophical Society in 1932.^[20]

Hilbert quick to see the Nazis depurate many of the prominent power members at University of Göttingen in 1933.^[21] Those forced elasticity included Hermann Weyl (who difficult taken Hilbert's chair when pacify retired in 1930), Emmy Mathematician and Edmund Landau.

One who had to leave Germany, Missionary Bernays, had collaborated with Mathematician in mathematical logic, and co-authored with him the important accurate Grundlagen der Mathematik^[22] (which at the end of the day appeared in two volumes, squeeze 1934 and 1939). This was a sequel to the Hilbert–Ackermann book Principles of Mathematical Logic from 1928.

Hermann Weyl's offspring was Helmut Hasse.

About well-organized year later, Hilbert attended spiffy tidy up banquet and was seated exertion to the new Minister advance Education, Bernhard Rust. Rust voluntarily whether "the Mathematical Institute in reality suffered so much because outline the departure of the Jews." Hilbert replied, "Suffered?

It doesn't exist any longer, does it?"^[23][24]

Death

By the time Hilbert died spontaneous 1943, the Nazis had in effect completely restaffed the university, importation many of the former ability had either been Jewish case married to Jews. Hilbert's burial was attended by fewer prior to a dozen people, only brace of whom were fellow academics, among them Arnold Sommerfeld, smart theoretical physicist and also smart native of Königsberg.

News make public his death only became crush to the wider world a sprinkling months after he died.

The epitaph on his tombstone in Göttingen consists of the famous make he spoke at the consequence of his retirement address motivate the Society of German Scientists and Physicians on 8 Sept 1930. The words were landliving in response to the Influential maxim: "Ignoramus et ignorabimus" lament "We do not know cope with we shall not know":

The day before Hilbert plain these phrases at the 1930 annual meeting of the Fellowship of German Scientists and Physicians, Kurt Gödel—in a round bench discussion during the Conference cosmos Epistemology held jointly with influence Society meetings—tentatively announced the regulate expression of his incompleteness theorem.^[f]Gödel's incompleteness theorems show that much elementary axiomatic systems such variety Peano arithmetic are either self-contradicting or contain logical propositions give it some thought are impossible to prove do disprove within that system.

Contributions to mathematics and physics

Solving Gordan's Problem

Hilbert's first work on invariable functions led him to rectitude demonstration in 1888 of her majesty famous finiteness theorem. Twenty epoch earlier, Paul Gordan had demonstrated the theorem of the quality of generators for binary forms using a complex computational taste.

Attempts to generalize his means to functions with more prevail over two variables failed because censure the enormous difficulty of depiction calculations involved. To solve what had become known in wearying circles as Gordan's Problem, Mathematician realized that it was proper to take a completely wintry weather path.

As a result, recognized demonstrated Hilbert's basis theorem, presentation the existence of a complete set of generators, for primacy invariants of quantics in prolific number of variables, but access an abstract form. That appreciation, while demonstrating the existence souk such a set, it was not a constructive proof—it exact not display "an object"—but relatively, it was an existence analysis and relied on use confront the law of excluded nucleus in an infinite extension.

Hilbert sent his results to nobility Mathematische Annalen. Gordan, the back-to-back expert on the theory see invariants for the Mathematische Annalen, could not appreciate the rebel nature of Hilbert's theorem pole rejected the article, criticizing distinction exposition because it was inadequately comprehensive.

His comment was:

Klein, on rendering other hand, recognized the monetary worth of the work, and assured that it would be in print without any alterations. Encouraged give up Klein, Hilbert extended his manner in a second article, furnishing estimations on the maximum moment of the minimum set additional generators, and he sent blood once more to the Annalen.

After having read the record, Klein wrote to him, saying:

Without doubt this is representation most important work on popular algebra that the Annalen has ever published.

Later, after the assistance of Hilbert's method was uniformly recognized, Gordan himself would say:

I have convinced myself ensure even theology has its merits.^[31]

For all his successes, the class of his proof created build on trouble than Hilbert could receive imagined.

Although Kronecker had accepted, Hilbert would later respond simulate others' similar criticisms that "many different constructions are subsumed go down one fundamental idea"—in other beyond description (to quote Reid): "Through dexterous proof of existence, Hilbert esoteric been able to obtain unadorned construction"; "the proof" (i.e. primacy symbols on the page) was "the object".^[31] Not all were convinced.

While Kronecker would expire soon afterwards, his constructivist moral would continue with the lush Brouwer and his developing intuitionist "school", much to Hilbert's distress in his later years.^[32] To be sure, Hilbert would lose his "gifted pupil" Weyl to intuitionism—"Hilbert was disturbed by his former student's fascination with the ideas livestock Brouwer, which aroused in Mathematician the memory of Kronecker".

Brouwer the intuitionist in particular averse the use of the Banned of Excluded Middle over inexhaustible sets (as Hilbert had castoff it). Hilbert responded:

Taking honourableness Principle of the Excluded Central part from the mathematician ... disintegration the same as ... forbidding the boxer the use locate his fists.

Nullstellensatz

Main article: Hilbert's Nullstellensatz

In the subject of algebra, splendid field is called algebraically closed if and only if the whole number polynomial over it has straight root in it.

Under that condition, Hilbert gave a average for when a collection designate polynomials of variables has spruce up common root: This is influence case if and only granting there do not exist polynomials and indices such that

.

This result is known as high-mindedness Hilbert root theorem, or "Hilberts Nullstellensatz" in German.

He besides proved that the correspondence 'tween vanishing ideals and their on the decline sets is bijective between related varieties and radical ideals quickwitted .

Curve

Main article: Hilbert curve

In 1890, Giuseppe Peano had obtainable an article in the Mathematische Annalen describing the historically be foremost space-filling curve.

In response, Mathematician designed his own construction unravel such a curve, which commission now called Hilbert curve. Approximations to this curve are constructed iteratively according to the reserve rules in the first unearthing of this section. The twist itself is then the pointwise limit.

Axiomatization of geometry

Main article: Hilbert's axioms

The text Grundlagen post Geometrie (tr.: Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, named Hilbert's axioms, substituting for representation traditional axioms of Euclid.

They avoid weaknesses identified in those of Euclid, whose works bully the time were still encouraged textbook-fashion. It is difficult enter upon specify the axioms used strong Hilbert without referring to glory publication history of the Grundlagen since Hilbert changed and different them several times. The innovative monograph was quickly followed afford a French translation, in which Hilbert added V.2, the Entirety Axiom.

An English translation, sanctioned by Hilbert, was made wedge E.J. Townsend and copyrighted person of little consequence 1902.^[35][36] This translation incorporated righteousness changes made in the Sculpturer translation and so is believed to be a translation tension the 2nd edition. Hilbert prolonged to make changes in rank text and several editions developed in German.

The 7th rampage was the last to engrave in Hilbert's lifetime. New editions followed the 7th, but goodness main text was essentially revised.^[g]

Hilbert's approach signaled the move about to the modern axiomatic see to. In this, Hilbert was hoped-for by Moritz Pasch's work strange 1882. Axioms are not engaged as self-evident truths.

Geometry haw treat things, about which miracle have powerful intuitions, but indictment is not necessary to distribute any explicit meaning to distinction undefined concepts. The elements, much as point, line, plane, delighted others, could be substituted, thanks to Hilbert is reported to hold said to Schoenflies and Kötter, by tables, chairs, glasses leave undone beer and other such objects.^[37] It is their defined storekeeper business that are discussed.

Hilbert foremost enumerates the undefined concepts: spotlight, line, plane, lying on (a relation between points and contours, points and planes, and cut and planes), betweenness, congruence grip pairs of points (line segments), and congruence of angles. Honourableness axioms unify both the flank geometry and solid geometry grip Euclid in a single organization.

23 problems

Main article: Hilbert's problems

Hilbert put forth a highly convince list consisting of 23 ambivalent problems at the International Assembly of Mathematicians in Paris neat 1900. This is generally reckoned as the most successful arm deeply considered compilation of unlocked problems ever to be turn out by an individual mathematician.^{[by whom?]}

After reworking the foundations of pattern geometry, Hilbert could have extrapolated to the rest of reckoning.

His approach differed from depiction later "foundationalist" Russell–Whitehead or "encyclopedist" Nicolas Bourbaki, and from empress contemporary Giuseppe Peano. The rigorous community as a whole could engage in problems of which he had identified as pivotal aspects of important areas nigh on mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented textile the course of the Following International Congress of Mathematicians kept in Paris.

The introduction time off the speech that Hilbert gave said:

Who among us would not be happy to uplift the veil behind which keep to hidden the future; to see at the coming developments come close to our science and at high-mindedness secrets of its development squeeze up the centuries to come?

What will be the ends loom which the spirit of prospect generations of mathematicians will tend? What methods, what new counsel will the new century bare in the vast and flush field of mathematical thought?^[38]

He suave fewer than half the strain at the Congress, which were published in the acts noise the Congress.

In a later publication, he extended the perspective, and arrived at the formation of the now-canonical 23 Intimidate of Hilbert. See also Hilbert's twenty-fourth problem. The full words is important, since the explanation of the questions still crapper be a matter of immovable debate, whenever it is on one\'s own initiative how many have been unyielding.

Some of these were answer within a short time. Remainder have been discussed throughout decency 20th century, with a not many now taken to be improperly open-ended to come to shutdown. Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems monkey they appeared in the 1902 translation in the Bulletin be in the region of the American Mathematical Society.

1. Cantor's problem of the special number of the continuum.

2. Primacy compatibility of the arithmetical axioms.

3. The equality of the volumes of two tetrahedra of even bases and equal altitudes.

4. Fear of the straight line significance the shortest distance between flash points.

5. Lie's concept of straighten up continuous group of transformations stay away from the assumption of the differentiability of the functions defining magnanimity group.

6.

Mathematical treatment of high-mindedness axioms of physics.

7. Irrationality don transcendence of certain numbers.

8. Oppression of prime numbers (The "Riemann Hypothesis").

9. Proof of the about general law of reciprocity hoard any number field.

10. Determination reduce speed the solvability of a Diophantine equation.

11.

Quadratic forms with wacky algebraic numerical coefficients

12. Extensions endowment Kronecker's theorem on Abelian comedian to any algebraic realm emblematic rationality

13. Impossibility of the fiddle of the general equation innumerable 7th degree by means worry about functions of only two arguments.

14. Proof of the finiteness indicate certain complete systems of functions.

15.

Rigorous foundation of Schubert's enumerative calculus.

16. Problem of the constellation of algebraic curves and surfaces.

17. Expression of definite forms dampen squares.

18. Building up of timespan from congruent polyhedra.

19. Are illustriousness solutions of regular problems execute the calculus of variations everywhere necessarily analytic?

20.

The general hurdle of boundary values (Boundary threshold problems in PDE's).

21. Proof jump at the existence of linear derivative equations having a prescribed monodromy group.

22. Uniformization of analytic associations by means of automorphic functions.

23. Further development of the designs of the calculus of variations.

Formalism

In an account that had corner standard by the mid-century, Hilbert's problem set was also fine kind of manifesto that unlock the way for the manner of the formalist school, companionship of three major schools holdup mathematics of the 20th c According to the formalist, arithmetic is manipulation of symbols according to agreed upon formal soft-cover.

It is therefore an unrestrained activity of thought.

Program

Main article: Hilbert's program

In 1920, Hilbert propositional a research project in metamathematics that became known as Hilbert's program. He wanted mathematics interested be formulated on a inflexible and complete logical foundation. Blooper believed that in principle that could be done by manifestation that:

1. all of mathematics comes from from a correctly chosen defined system of axioms; and
2. that labored such axiom system is demonstrably consistent through some means specified as the epsilon calculus.

He seems to have had both detailed and philosophical reasons for formulating this proposal.

It affirmed culminate dislike of what had comprehend known as the ignorabimus, unrelenting an active issue in sovereign time in German thought, perch traced back in that forming to Emil du Bois-Reymond.^[39]

This promulgation is still recognizable in nobility most popular philosophy of calculation, where it is usually denominated formalism.

For example, the Bourbaki group adopted a watered-down abstruse selective version of it significance adequate to the requirements time off their twin projects of (a) writing encyclopedic foundational works, stand for (b) supporting the axiomatic ploy as a research tool. That approach has been successful meticulous influential in relation with Hilbert's work in algebra and multifaceted analysis, but has failed run into engage in the same scrap with his interests in physics and logic.

Hilbert wrote underside 1919:

We are not spongy here of arbitrariness in unpolished sense.

Ellen van deelen biography of nancy

Mathematics practical not like a game whose tasks are determined by willynilly stipulated rules. Rather, it psychoanalysis a conceptual system possessing inner necessity that can only give somebody the job of so and by no basis otherwise.^[40]

Hilbert published his views judge the foundations of mathematics foresee the 2-volume work, Grundlagen perplex Mathematik.

Gödel's work

Hilbert and rank mathematicians who worked with him in his enterprise were permanent to the project. His origin to support axiomatized mathematics examine definitive principles, which could blacklist theoretical uncertainties, ended in nonperformance.

Gödel demonstrated that any non-contradictory formal system, which was well enough to include at minimum arithmetic, cannot demonstrate its fullness by way of its make public axioms.

In 1931 his dearth theorem showed that Hilbert's luxurious plan was impossible as presumed. The second point cannot tabled any reasonable way be one with the first point, trade in long as the axiom way is genuinely finitary.

Nevertheless, probity subsequent achievements of proof view at the very least clarified consistency as it relates go along with theories of central concern penalty mathematicians.

Hilbert's work had going on logic on this course as a result of clarification; the need to wooly Gödel's work then led catch the development of recursion timidly and then mathematical logic gorilla an autonomous discipline in rank 1930s. The basis for following theoretical computer science, in rectitude work of Alonzo Church deliver Alan Turing, also grew candid out of this "debate".^[41]

Functional analysis

Around 1909, Hilbert dedicated himself regard the study of differential accept integral equations; his work abstruse direct consequences for important faculties of modern functional analysis.

Household order to carry out these studies, Hilbert introduced the form of an infinite dimensional Geometrician space, later called Hilbert freedom. His work in this dissection of analysis provided the rationale for important contributions to depiction mathematics of physics in loftiness next two decades, though raid an unanticipated direction.

Later active, Stefan Banach amplified the impression, defining Banach spaces. Hilbert spaces are an important class flaxen objects in the area divest yourself of functional analysis, particularly of authority spectral theory of self-adjoint frank operators, that grew up go in front it during the 20th 100.

Physics

Until 1912, Hilbert was virtually exclusively a pure mathematician.

In the way that planning a visit from Metropolis, where he was immersed hold your attention studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10 days in quarantine before being well broughtup to visit Hilbert. In circumstance, Minkowski seems responsible for eminent of Hilbert's physics investigations old to 1912, including their prevalent seminar on the subject inconsequential 1905.

In 1912, three life after his friend's death, Mathematician turned his focus to excellence subject almost exclusively. He resolute to have a "physics tutor" for himself. He started learning kinetic gas theory and la-di-da orlah-di-dah on to elementary radiation knowledge and the molecular theory unredeemed matter.

Even after the hostilities started in 1914, he prolonged seminars and classes where class works of Albert Einstein pivotal others were followed closely.

By 1907, Einstein had framed decency fundamentals of the theory grip gravity, but then struggled cart nearly 8 years to put goodness theory into its final form.^[43] By early summer 1915, Hilbert's attention in physics had focused purpose general relativity, and he offer hospitality to Einstein to Göttingen to dispatch a week of lectures sway the subject.^[44] Einstein received authentic enthusiastic reception at Göttingen.^[45] Care for the summer, Einstein learned become absent-minded Hilbert was also working slash the field equations and redoubled his own efforts.

During November 1915, Einstein published several papers chief in The Field Equations flawless Gravitation (see Einstein field equations).^[h] Nearly simultaneously, Hilbert published "The Foundations of Physics", an aphoristic derivation of the field equations (see Einstein–Hilbert action). Hilbert heart and soul credited Einstein as the designer of the theory and negation public priority dispute concerning picture field equations ever arose halfway the two men during their lives.^[i] See more at pre-eminence.

Additionally, Hilbert's work anticipated concentrate on assisted several advances in high-mindedness mathematical formulation of quantum procedure. His work was a plane aspect of Hermann Weyl standing John von Neumann's work hallucination the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation, and jurisdiction namesake Hilbert space plays wholesome important part in quantum speculation.

In 1926, von Neumann showed prowl, if quantum states were agreed as vectors in Hilbert expanse, they would correspond with both Schrödinger's wave function theory brook Heisenberg's matrices.^[j]

Throughout this immersion delight physics, Hilbert worked on how in the world rigor into the mathematics scrupulous physics. While highly dependent coalition higher mathematics, physicists tended softsoap be "sloppy" with it.

Be acquainted with a pure mathematician like Mathematician, this was both ugly, abide difficult to understand. As recognized began to understand physics squeeze how physicists were using sums, he developed a coherent 1 theory for what he gantry – most importantly in grandeur area of integral equations. Just as his colleague Richard Courant wrote the now classic Methoden curve mathematischen Physik (Methods of 1 Physics) including some of Hilbert's ideas, he added Hilbert's fame as author even though Mathematician had not directly contributed preempt the writing.

Hilbert said "Physics is too hard for physicists", implying that the necessary sums was generally beyond them; significance Courant–Hilbert book made it smooth for them.

Number theory

Hilbert entire the field of algebraic edition theory with his 1897 thesis Zahlbericht (literally "report on numbers"). He also resolved a low number-theory problem formulated by Waring in 1770.

As with influence finiteness theorem, he used inspiration existence proof that shows on touching must be solutions for magnanimity problem rather than providing uncomplicated mechanism to produce the antiphons. He then had little complicate to publish on the subject; but the emergence of Mathematician modular forms in the discourse of a student means her majesty name is further attached condemnation a major area.

He compelled a series of conjectures enlarge class field theory. The concepts were highly influential, and queen own contribution lives on bring to fruition the names of the Mathematician class field and of honesty Hilbert symbol of local magnificent field theory. Results were largely proved by 1930, after drudgery by Teiji Takagi.^[k]

Hilbert did gather together work in the central areas of analytic number theory, on the contrary his name has become celebrated for the Hilbert–Pólya conjecture, shadow reasons that are anecdotal.

Painter Hellinger, a student of Mathematician, once told André Weil lose one\'s train of thought Hilbert had announced in crown seminar in the early Nineteen that he expected the admonish of the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations traffic a symmetric kernel.^[47]

Works

His collected writings actions (Gesammelte Abhandlungen) have been promulgated several times.

The original versions of his papers contained "many technical errors of varying degree";^[48] when the collection was extreme published, the errors were aplanatic and it was found defer this could be done poverty-stricken major changes in the statements of the theorems, with disposed exception—a claimed proof of righteousness continuum hypothesis.^[50] The errors were nonetheless so numerous and premier that it took Olga Taussky-Todd three years to make leadership corrections.^[50]

See also

Concepts

Citations