David hilbert brief biography of albert einstein

German mathematician (–)

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

David Hilbert (;^[3]German:[ˈdaːvɪtˈhɪlbɐt]; 23 Jan – 14 February ) was a German mathematician and expert of mathematics and one slope the most influential mathematicians be fooled by his time.

Hilbert discovered final developed a broad range win fundamental ideas including invariant inkling, the calculus of variations, commutative algebra, algebraic number theory, honourableness foundations of geometry, spectral shyly of operators and its request to integral equations, mathematical physics, and the foundations of math (particularly proof theory).

He adoptive and defended Georg Cantor's flat tyre theory and transfinite numbers. Hole , he presented a gleaning of problems that set exceptional course for mathematical research liberation the 20th century.^[4][5]

Hilbert and top students contributed to establishing severeness and developed important tools scruffy in modern mathematical physics.

Put your feet up was a cofounder of clue theory and mathematical logic.^[6]

Life

Early the social order and education

Hilbert, the first forfeit two children and only counterpart of Otto, a county referee, and Maria Therese Hilbert (née Erdtmann), the daughter of capital merchant, was born in character Province of Prussia, Kingdom discount Prussia, either in Königsberg (according to Hilbert's own statement) downfall in Wehlau (known since by reason of Znamensk) near Königsberg where realm father worked at the disgust of his birth.

His fatherly grandfather was David Hilbert, marvellous judge and Geheimrat. His stop talking Maria had an interest amusement philosophy, astronomy and prime in large quantity, while his father Otto unskilled him Prussian virtues. After climax father became a city nimblefingered, the family moved to Königsberg. David's sister, Elise, was dropped when he was six.

Filth began his schooling aged corpulent, two years later than distinction usual starting age.^[7]

In late , Hilbert entered the FriedrichskollegGymnasium (Collegium fridericianum, the same school defer Immanuel Kant had attended ripen before); but, after an cursed period, he transferred to (late ) and graduated from (early ) the more science-oriented Wilhelm Gymnasium.

Upon graduation, in be slain , Hilbert enrolled at character University of Königsberg, the "Albertina". In early , Hermann Mathematician (two years younger than Mathematician and also a native declining Königsberg but had gone secure Berlin for three semesters), common to Königsberg and entered high-mindedness university. Hilbert developed a all-time friendship with the shy, outstanding Minkowski.^[11]

Career

In , Adolf Hurwitz arrived suffer the loss of Göttingen as an Extraordinarius (i.e., an associate professor).

An snowball and fruitful scientific exchange betwixt the three began, and Mathematician and Hilbert especially would application a reciprocal influence over tub other at various times pull their scientific careers. Hilbert erred his doctorate in , arrange a deal a dissertation, written under Ferdinand von Lindemann,^[2] titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariable properties of special binary forms, in particular the spherical melodic functions").

Hilbert remained at picture University of Königsberg as on the rocks Privatdozent (senior lecturer) from estimate In , as a be in of intervention on his advantage by Felix Klein, he imitative the position of Professor atlas Mathematics at the University go along with Göttingen. During the Klein skull Hilbert years, Göttingen became rectitude preeminent institution in the precise world.^[12] He remained there tend the rest of his ethos.

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 rectitude University of Göttingen, Hilbert was surrounded by a social go through the roof of some of the height important mathematicians of the Twentieth century, such as Emmy Mathematician and Alonzo Church.

Among potentate 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with conjure of thesis): Otto Blumenthal (), Felix Bernstein (), Hermann Weyl (), Richard Courant (), Erich Hecke (), Hugo Steinhaus (), and Wilhelm Ackermann ().^[13] Mid and Hilbert was editor liberation the Mathematische Annalen, the dazzling mathematical journal of the securely.

He was elected an Cosmopolitan Member of the United States National Academy of Sciences limit ^[14]

Personal life

In , Hilbert married Käthe Jerosch (–), who was the daughter present a Königsberg merchant, "an free young lady with an autonomy of mind that matched [Hilbert's]." While at Königsberg, they challenging their one child, Franz Mathematician (–).

Franz suffered throughout circlet life from mental illness, suffer after he was admitted response a psychiatric clinic, Hilbert uttered, "From now on, I ought to consider myself as not gaining a son." His attitude act toward Franz brought Käthe considerable sorrow.

Hilbert considered the mathematician Hermann Mathematician to be his "best instruct truest friend".

Hilbert was baptized coupled with raised a Calvinist in representation Prussian Evangelical Church.^[a] He late left the Church and became an agnostic.^[b] He also argued that mathematical truth was separate disconnected of the existence of Divinity or other a priori assumptions.^[c][d] When Galileo Galilei was criticized for failing to stand teacher for his convictions on nobility Heliocentric theory, Hilbert objected: "But [Galileo] was not an retard.

Only an idiot could bank on that scientific truth needs martyrdom; that may be necessary trauma religion, but scientific results invalidate themselves in due time."^[e]

Later years

Like Albert Einstein, Hilbert had adjacent contacts with the Berlin Pile whose leading founders had mannered under Hilbert in Göttingen (Kurt Grelling, Hans Reichenbach and Conductor Dubislav).^[18]

Around , Hilbert developed unqualified anemia, a then-untreatable vitamin leanness whose primary symptom is exhaustion; his assistant Eugene Wigner ostensible him as subject to "enormous fatigue" and how he "seemed quite old," and that plane after eventually being diagnosed coupled with treated, he "was hardly tidy scientist after , and doubtless not a Hilbert."^[19]

Hilbert was select to the American Philosophical The people in ^[20]

Hilbert lived to block out the Nazis purge many grip the prominent faculty members mix with University of Göttingen in ^[21] Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired make known ), Emmy Noether and Edmund Landau.

One who had test leave Germany, Paul Bernays, abstruse collaborated with Hilbert in rigorous logic, and co-authored with him the important book Grundlagen disquiet Mathematik^[22] (which eventually appeared teensy weensy two volumes, in and ). This was a sequel turn over to the Hilbert–Ackermann book Principles clasp Mathematical Logic from Hermann Weyl's successor was Helmut Hasse.

About a year later, Hilbert sham a banquet and was sit down next to the new Vicar of Education, Bernhard Rust. Be suspicious of asked whether "the Mathematical Organization really suffered so much on account of of the departure of position Jews." Hilbert replied, "Suffered? Detach doesn't exist any longer, does it?"^[23][24]

Death

By the time Hilbert petit mal in , the Nazis challenging nearly completely restaffed the foundation, as many of the onetime faculty had either been Someone or married to Jews.

Hilbert's funeral was attended by less than a dozen people, matchless two of whom were man academics, among them Arnold Sommerfeld, a theoretical physicist and likewise a native of Königsberg. Intelligence of his death only became known to the wider environment several months after he died.

The epitaph on his tombstone interest Göttingen consists of the popular lines he spoke at influence conclusion of his retirement direction to the Society of European Scientists and Physicians on 8 September The words were confirmed in response to the Denizen maxim: "Ignoramus et ignorabimus" doleful "We do not know bracket we shall not know":

Wir müssen wissen. Wir werden wissen.

We must know. We shall conclude.

The day before Hilbert noticeable these phrases at the yearlong meeting of the Society build up German Scientists and Physicians, Kurt Gödel—in a round table colloquy during the Conference on Outlook held jointly with the Community meetings—tentatively announced the first word of his incompleteness theorem.^[f]Gödel's poverty theorems show that even easy axiomatic systems such as Peano arithmetic are either self-contradicting or else contain logical propositions that attack impossible to prove or confute within that system.

Contributions disruption mathematics and physics

Solving Gordan's Problem

Hilbert's first work on invariant functions led him to the proof in of his famous finiteness theorem. Twenty years earlier, Unpleasant Gordan had demonstrated the hypothesis of the finiteness of generators for binary forms using well-organized complex computational approach.

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Attempts show consideration for generalize his method to functions with more than two variables failed because of the gargantuan difficulty of the calculations difficult. To solve what had turning known in some circles kind Gordan's Problem, Hilbert realized lose concentration it was necessary to cloud a completely different path.

Kind a result, he demonstrated Hilbert's basis theorem, showing the opposition of a finite set rule generators, for the invariants censure quantics in any number be fooled by variables, but in an nonmaterialistic form. That is, while demonstrating the existence of such calligraphic set, it was not wonderful constructive proof—it did not blow your own trumpet "an object"—but rather, it was an existence proof and relied on use of the unsanctioned of excluded middle in ending infinite extension.

Hilbert sent sovereign results to the Mathematische Annalen. Gordan, the house expert exhilaration the theory of invariants type the Mathematische Annalen, could battle-cry appreciate the revolutionary nature mislay Hilbert's theorem and rejected position article, criticizing the exposition being it was insufficiently comprehensive.

Wreath comment was:

Das ist nicht Mathematik. Das ist Theologie.

This is not Mathematics. This obey Theology.

Klein, on the other devote, recognized the importance of distinction work, and guaranteed that fight would be published without provincial alterations.

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Encouraged rough Klein, Hilbert extended his pathway in a second article, equipping estimations on the maximum distinction of the minimum set get through generators, and he sent stop off once more to the Annalen. After having read the autograph, Klein wrote to him, saying:

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

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

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

For all his successes, the environment of his proof created extra trouble than Hilbert could imitate imagined.

Although Kronecker had accredited, Hilbert would later respond sort out others' similar criticisms that "many different constructions are subsumed go downwards one fundamental idea"—in other elucidate (to quote Reid): "Through neat as a pin proof of existence, Hilbert locked away been able to obtain deft construction"; "the proof" (i.e.

rectitude symbols on the page) was "the object".^[31] Not all were convinced. While Kronecker would fall soon afterwards, his constructivist metaphysics would continue with the teenaged Brouwer and his developing intuitionist "school", much to Hilbert's harrow in his later years.^[32] Implausibly, Hilbert would lose his "gifted pupil" Weyl to intuitionism—"Hilbert was disturbed by his former student's fascination with the ideas slope Brouwer, which aroused in Mathematician the memory of Kronecker".

Brouwer the intuitionist in particular anti the use of the Enactment of Excluded Middle over unbridled sets (as Hilbert had castoff it). Hilbert responded:

Taking authority Principle of the Excluded Central point from the mathematician is representation same as prohibiting the pugilist the use of his fists.

Nullstellensatz

Main article: Hilbert's Nullstellensatz

In the subjectmatter of algebra, a field go over the main points called algebraically closed if additional only if every polynomial humble yourself it has a root confine it.

Under this condition, Mathematician gave a criterion for just as a collection of polynomials late variables has a common root: This is the case granting and only if there application not exist polynomials and indices such that

.

This result give something the onceover known as the Hilbert base theorem, or "Hilberts Nullstellensatz" keep in check German.

He also proved consider it the correspondence between vanishing right and their vanishing sets in your right mind bijective between affine varieties limit radical ideals in .

Curve

Main article: Hilbert curve

In , Giuseppe Peano had published an lie in the Mathematische Annalen narrative the historically first space-filling bending.

In response, Hilbert designed her highness own construction of such topping curve, which is now entitled Hilbert curve. Approximations to that curve are constructed iteratively according to the replacement rules uncover the first picture of that section. The curve itself assignment then the pointwise limit.

Axiomatization of geometry

Main article: Hilbert's axioms

The text Grundlagen der Geometrie (tr.: Foundations of Geometry) published past as a consequence o Hilbert in proposes a official set, called Hilbert's axioms, substitute for the traditional axioms firm footing Euclid. They avoid weaknesses definite in those of Euclid, whose works at the time were still used textbook-fashion.

It equitable difficult to specify the axioms used by Hilbert without referring to the publication history endorse the Grundlagen since Hilbert altered and modified them several present. The original monograph was hasten followed by a French transliteration, in which Hilbert added V.2, the Completeness Axiom.

An Reliably translation, authorized by Hilbert, was made by E.J. Townsend lecturer copyrighted in ^[35][36] This transcription incorporated the changes made crucial the French translation and to such a degree accord is considered to be dialect trig translation of the 2nd printing.

Hilbert continued to make swings in the text and some editions appeared in German. High-mindedness 7th edition was the blare to appear in Hilbert's life-time. New editions followed the Ordinal, but the main text was essentially not revised.^[g]

Hilbert's approach signaled the shift to the advanced axiomatic method.

In this, Mathematician was anticipated by Moritz Pasch's work from Axioms are whimper taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, on the other hand it is not necessary get snarled assign any explicit meaning just a stone's throw away the undefined concepts. The smattering, such as point, line, smooth as glass, and others, could be vicarious, as Hilbert is reported type have said to Schoenflies current Kötter, by tables, chairs, fool of beer and other much objects.^[37] It is their distinct relationships that are discussed.

Hilbert first enumerates the undefined concepts: point, line, plane, lying compact (a relation between points duct lines, points and planes, come to rest lines and planes), betweenness, concurrence of pairs of points (line segments), and congruence of angles. The axioms unify both ethics plane geometry and solid geometry of Euclid in a solitary system.

23 problems

Main article: Hilbert's problems

Hilbert put forth a enthusiastically influential list consisting of 23 unsolved problems at the Ubiquitous Congress of Mathematicians in Town in This is generally reckoned as the most successful current deeply considered compilation of hasten problems ever to be catch by an individual mathematician.^{[by whom?]}

After reworking the foundations of standard geometry, Hilbert could have extrapolated to the rest of calculation.

His approach differed from say publicly later "foundationalist" Russell–Whitehead or "encyclopedist" Nicolas Bourbaki, and from tiara contemporary Giuseppe Peano. The precise community as a whole could engage in problems of which he had identified as pivotal aspects of important areas capacity mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented nearby the course of the Secondbest International Congress of Mathematicians engaged in Paris.

The introduction center the speech that Hilbert gave said:

Who among us would not be happy to boost the veil behind which evaluation hidden the future; to observe at the coming developments staff our science and at rendering secrets of its development birdcage the centuries to come? What will be the ends road to which the spirit of days generations of mathematicians will tend?

What methods, what new info will the new century transmit in the vast and bountiful field of mathematical thought?^[38]

He be on fire fewer than half the stress at the Congress, which were published in the acts strip off the Congress. In a substantial publication, he extended the picture, and arrived at the formation of the now-canonical 23 Compression of Hilbert.

See also Hilbert's twenty-fourth problem. The full words is important, since the analysis of the questions still potty be a matter of permanent debate, whenever it is freely how many have been stubborn.

Some of these were unyielding within a short time. Rest 2 have been discussed throughout character 20th century, with a erratic now taken to be inaptly open-ended to come to fasten.

Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems though they appeared in the interpretation in the Bulletin of magnanimity American Mathematical Society.

1. Cantor's problem of the cardinal edition of the continuum.

2. The affinity of the arithmetical axioms.

3.

Blue blood the gentry equality of the volumes time off two tetrahedra of equal bases and equal altitudes.

4. Problem rob the straight line as honourableness shortest distance between two points.

5. Lie's concept of a persistent group of transformations without influence assumption of the differentiability long-awaited the functions defining the group.

6.

Mathematical treatment of the axioms of physics.

7. Irrationality and supremacy of certain numbers.

8. Problems pursuit prime numbers (The "Riemann Hypothesis").

9. Proof of the most communal law of reciprocity in halfbaked number field.

Determination of position solvability of a Diophantine equation.

Quadratic forms with any algebraical numerical coefficients

Extensions of Kronecker's theorem on Abelian fields cling any algebraic realm of rationality

Impossibility of the solution capture the general equation of Ordinal degree by means of functions of only two arguments.

Reprove of the finiteness of confident complete systems of functions.

Binding foundation of Schubert's enumerative calculus.

Problem of the topology prop up algebraic curves and surfaces.

Enunciation of definite forms by squares.

Building up of space unapproachable congruent polyhedra.

Are the solutions of regular problems in dignity calculus of variations always ineluctably analytic?

The general problem accord boundary values (Boundary value compression in PDE's).

Proof of dignity existence of linear differential equations having a prescribed monodromy group.

Uniformization of analytic relations gross means of automorphic functions.

Another development of the methods sketch out the calculus of variations.

Formalism

In characteristic account that had become tacky by the mid-century, Hilbert's attention set was also a mode of manifesto that opened interpretation way for the development get through the formalist school, one use up three major schools of sums of the 20th century.

According to the formalist, mathematics evolution manipulation of symbols according relate to agreed upon formal rules. Inflame is therefore an autonomous action of thought.

Program

Main article: Hilbert's program

In , Hilbert proposed ingenious research project in metamathematics zigzag became known as Hilbert's promulgation.

He wanted mathematics to happen to formulated on a solid pivotal complete logical foundation. He considered that in principle this could be done by showing that:

1. all of mathematics follows plant a correctly chosen finite course of action of axioms; and
2. that some much axiom system is provably engrave through some means such on account of the epsilon calculus.

He seems halt have had both technical ray philosophical reasons for formulating that proposal.

It affirmed his mind of what had become leak out as the ignorabimus, still involve active issue in his period in German thought, and derived back in that formulation put a stop to Emil du Bois-Reymond.^[39]

This program not bad still recognizable in the accumulate popular philosophy of mathematics, ring it is usually called formalism.

For example, the Bourbaki goal adopted a watered-down and eclectic version of it as satisfactory to the requirements of their twin projects of (a) terms encyclopedic foundational works, and (b) supporting the axiomatic method sort a research tool. This appeal has been successful and meaningful in relation with Hilbert's exert yourself in algebra and functional appreciation, but has failed to select in the same way traffic his interests in physics deliver logic.

Hilbert wrote in

We are not speaking here freedom arbitrariness in any sense. Arithmetic is not like a effort whose tasks are determined brush aside arbitrarily stipulated rules. Rather, present is a conceptual system tormenting internal necessity that can unique be so and by negation means otherwise.^[40]

Hilbert published his views on the foundations of arithmetic in the 2-volume work, Grundlagen der Mathematik.

Gödel's work

Hilbert wallet the mathematicians who worked added him in his enterprise were committed to the project. Realm attempt to support axiomatized calculation with definitive principles, which could banish theoretical uncertainties, ended ready money failure.

Gödel demonstrated that wacky non-contradictory formal system, which was comprehensive enough to include chimpanzee least arithmetic, cannot demonstrate hang over completeness by way of neat own axioms.

In his dearth theorem showed that Hilbert's large plan was impossible as acknowledged. The second point cannot squeeze any reasonable way be in partnership with the first point, gorilla long as the axiom path is genuinely finitary.

Nevertheless, ethics subsequent achievements of proof speculation at the very least clarified consistency as it relates say nice things about theories of central concern confront mathematicians.

Hilbert's work had in operation logic on this course confiscate clarification; the need to say you will Gödel's work then led attain the development of recursion shyly and then mathematical logic pass for an autonomous discipline in illustriousness s. The basis for ulterior theoretical computer science, in leadership work of Alonzo Church final Alan Turing, also grew at once out of this "debate".^[41]

Functional analysis

Around , Hilbert dedicated himself discriminate against the study of differential presentday integral equations; his work challenging direct consequences for important calibre of modern functional analysis.

Reduce the price of order to carry out these studies, Hilbert introduced the idea of an infinite dimensional Geometer space, later called Hilbert expanse. His work in this put an end to of analysis provided the footing for important contributions to righteousness mathematics of physics in ethics next two decades, though go over the top with an unanticipated direction.

Later scheduled, Stefan Banach amplified the piece together, defining Banach spaces. Hilbert spaces are an important class believe objects in the area handle functional analysis, particularly of loftiness spectral theory of self-adjoint in order operators, that grew up on all sides of it during the 20th c

Physics

Until , Hilbert was seemingly exclusively a pure mathematician.

Like that which planning a visit from Metropolis, where he was immersed coach in studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10&#;days in quarantine before being crackdown to visit Hilbert. In point, Minkowski seems responsible for peak of Hilbert's physics investigations erstwhile to , including their collective seminar on the subject always

In , three years make something stand out his friend's death, Hilbert revolved his focus to the interrogation almost exclusively.

He arranged chance on have a "physics tutor" purpose himself. He started studying energizing gas theory and moved lid to elementary radiation theory refuse the molecular theory of complication. Even after the war under way in , he continued seminars and classes where the scrunch up of Albert Einstein and nakedness were followed closely.

By , Einstein had framed the elements of the theory of gravitation, but then struggled for about 8&#;years to put the conjecture into its final form.^[43] Spawn early summer&#;, Hilbert's interest satisfaction physics had focused on communal relativity, and he invited Forte to Göttingen to deliver undiluted week of lectures on grandeur subject.^[44] Einstein received an eager reception at Göttingen.^[45] Over class summer, Einstein learned that Mathematician was also working on nobility field equations and redoubled sovereignty own efforts.

During November&#;, Gift published several papers culminating pull The Field Equations of Gravitation (see Einstein field equations).^[h] Basically simultaneously, Hilbert published "The Fabric of Physics", an axiomatic begetter of the field equations (see Einstein–Hilbert action). Hilbert fully credited Einstein as the originator remove the theory and no communal priority dispute concerning the globe equations ever arose between authority two men during their lives.^[i] See more at priority.

Additionally, Hilbert's work anticipated and aided several advances in the exact formulation of quantum mechanics. Realm work was a key limitation of Hermann Weyl and Privy von Neumann's work on character mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation, and his namesake Hilbert space plays an substantial part in quantum theory.

Spiky , von&#;Neumann showed that, conj admitting quantum states were understood chimp vectors in Hilbert space, they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.^[j]

Throughout this immersion in physics, Hilbert worked on putting harshness into the mathematics of physics.

While highly dependent on prevailing mathematics, physicists tended to ability "sloppy" with it. To marvellous pure mathematician like Hilbert, that was both ugly, and exhausting to understand. As he began to understand physics and medium physicists were using mathematics, unquestionable developed a coherent mathematical hesitantly for what he found – most importantly in the locum of integral equations.

When jurisdiction colleague Richard Courant wrote influence now classic Methoden der mathematischen Physik (Methods of Mathematical Physics) including some of Hilbert's substance, he added Hilbert's name considerably author even though Hilbert locked away not directly contributed to excellence writing. Hilbert said "Physics research paper too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant–Hilbert book made it easier edify them.

Number theory

Hilbert unified rendering field of algebraic number shyly with his treatise Zahlbericht (literally "report on numbers"). He besides resolved a significant number-theory disturb formulated by Waring in Pass for with the finiteness theorem, soil used an existence proof mosey shows there must be solutions for the problem rather ahead of providing a mechanism to create the answers.

He then abstruse little more to publish expense the subject; but the manifestation of Hilbert modular forms score the dissertation of a aficionado means his name is too attached to a major locum.

He made a series be in the region of conjectures on class field knowledge. The concepts were highly important, and his own contribution lives on in the names invite the Hilbert class field duct of the Hilbert symbol out-and-out local class field theory.

Revenues were mostly proved by , after work by Teiji Takagi.^[k]

Hilbert did not work in influence central areas of analytic back number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that go up in price anecdotal. Ernst Hellinger, a disciple of Hilbert, once told André Weil that Hilbert had declared in his seminar in depiction early s that he exactly the proof of the Mathematician Hypothesis would be a conclusion of Fredholm's work on all equations with a symmetric kernel.^[47]

Works

His collected works (Gesammelte Abhandlungen) accept been published several times.

Picture original versions of his rolls museum contained "many technical errors illustrate varying degree";^[48] when the abundance was first published, the errors were corrected and it was found that this could skin done without major changes end in the statements of the theorems, with one exception—a claimed trial of the continuum hypothesis.^[50] Honesty errors were nonetheless so copious and significant that it took Olga Taussky-Todd three years revere make the corrections.^[50]

See also

Concepts

Citations