Hilbert in my ViewA project under construction

Franz RotheDepartment of Mathematics

University of North Carolina at Charlotte

Charlotte, NC 28223

frothe@uncc.edu

December 28, 2012

Allhilbert\allhilbert.tex

Contents

I Biography 2

1 Hilbert’s life 31.1 Life at Konigsberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Wife and child . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Life at Gottingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The Gottingen school . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Later years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Work 72.1 Hilbert’s Basis Theorem and Zahlbericht . . . . . . . . . . . . . . . . . 72.2 Foundations of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 The Paris Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 ”Foundation of Physics” . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Priority remains unclear . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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2.7 Reactions to Hilbert’s work in General Relativity . . . . . . . . . . . . 122.8 International congress in Bologna 1928 . . . . . . . . . . . . . . . . . . 142.9 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

II Some Reflections 16

1 Der Annalenstreit 171.1 Mathematische Annalen . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Hintergrund des Annalenstreites . . . . . . . . . . . . . . . . . . . . . 171.3 Der Annalenstreit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Literatur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 The Einstein-Born letters 202.1 Born’s letter from 1928 . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Comment by Max Born himself . . . . . . . . . . . . . . . . . . . . . . 222.3 The Born Einstein Letters 1916-1955 . . . . . . . . . . . . . . . . . . . 24

3 Confirmation of Gravitational Waves 25

Part I

Biography

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1 Hilbert’s life

Born: 23 Jan 1862 in Konigsberg, Prussia (now Kaliningrad, Russia)Died: 14 Feb 1943 in Gottingen, Germany

1.1 Life at Konigsberg

Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, wasborn in the Province of Prussia —either in Konigsberg (according to Hilbert’s ownstatement) or in Wehlau (known since 1946 as Znamensk) near Konigsberg where hisfather worked at the time of his birth.

In the fall of 1872, he entered the Friedrichskolleg Gymnasium (Collegium frideri-cianum, the same school that Immanuel Kant had attended 140 years before), but afteran unhappy time he transferred in fall 1879 to the more science-oriented Wilhelm Gym-nasium. After graduating from the gymnasium in spring 1880, he entered the Universityof Konigsberg, the ”Albertina”.

One of Hilbert’s friends there was Hermann Minkowski —two years younger thanHilbert and also a native of Konigsberg. Minkowski was so talented he had graduatedearly from his gymnasium and had begun his studies in Berlin for three semesters. In thespring of 1882, Hermann Minkowski returned to Konigsberg and entered the university.”Hilbert knew his luck when he saw it. In spite of his father’s disapproval, he soonbecame friends with the shy, gifted Minkowski.”

In 1884, Adolf Hurwitz arrived from Gottingen as an extraordinarius, i.e. an asso-ciate professor. An intense and fruitful scientific exchange between the three began andespecially Minkowski and Hilbert would exercise a reciprocal influence on each other atvarious times in their scientific careers. Hilbert obtained his doctorate in 1885, witha dissertation Uber invariante Eigenschaften spezieller binarer Formen, insbesondereder Kugelfunktionen (”On the invariant properties of special binary forms, in particu-lar the spherical harmonic functions”), written under the supervision of Ferdinand vonLindemann.

Hilbert was a member of staff at Konigsberg from 1886 to 1895, being a Privatdozentuntil 1892, then as extraordinary professor for one year. In 1893 he was appointed a fullprofessor, and remained at the University of Konigsberg until 1895.

1.2 Wife and child

In 1892, Hilbert married Kathe Jerosch (1864–1945), ”the daughter of a Konigsbergmerchant, an outspoken young lady with an independence of mind that matched hisown”.

While at Konigsberg they had their only child Franz Hilbert (1893–1969). Franzwould suffer his entire life from an (undiagnosed) mental illness, his inferior intellect

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a terrible disappointment to his father and this tragedy a matter of distress to themathematicians and students at Gottingen.

1.3 Life at Gottingen

In 1892 Schwarz moved from Gottingen to Berlin to occupy Weierstrass’s chair, andKlein wanted to offer Hilbert the vacant Gottingen chair. However Klein failed topersuade his colleagues. So Heinrich Weber was appointed as chair. Klein was probablynot too unhappy when Weber moved to a chair at Straßburg (now Strasbourg) threeyears later. Indeed, on this occasion he was successful in his aim of appointing Hilbert.So, in 1895, as a result of intervention by Felix Klein on his behalf, Hilbert was appointedto the chair of mathematics at the university of Gottingen. He continued to teach anddo research at Gottingen for the rest of his career.

Hilbert’s eminent position in the world of mathematics meant that other institutionswould have liked to tempt him to leave Gottingen. In 1902, the University of Berlinoffered Hilbert the chair held by Fuchs. Hilbert turned down this offer from Berlin, butonly after bargain with Gottingen and persuade them to set up a new full professorshipto bring his friend Minkowski to Gottingen. Sadly, Minkowski—Hilbert’s ”best andtruest friend”—would die prematurely of a ruptured appendix in 1909.

1.4 The Gottingen school

At the University of Gottingen, Hilbert was surrounded by a social circle of some ofthe most important mathematicians of the 20th century, such as Emmy Noether andAlonzo Church. John von Neumann was his assistant.

Among his 69 Ph.D. students in Gottingen were many who later became famousmathematicians, including the famous world chess champion Lasker, Ernesto Zermelo,Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant(1910), Erich Hecke (1910), Hugo Steinhaus (1911), Wilhelm Ackermann (1925), CarlGustav Hempel.

The Gottingen atmosphere implied a constant discussion and adoption of new ideas,techniques, and problems that had originally been created or suggested by others, eitherat home or outside. Whenever they appeared to be fruitful and relevant, they wereimmediately absorbed and became current concerns of the local community and putto use for resolving open problems. Dirk Struik reports from his days as a student inGottingen the following incidence happening to Hilbert:

Once a young chap, lecturing before Hilbert’s seminar made use of a theoremthat drew Hilbert’s attention. He sat up and interrupted the speakerto ask: ”That is a really beautiful theorem, a very beautiful theorem.But who has thought it up?” The young man paused for a moment inastonishment and then replied: ”Aber Herr Geheimrat, you have thoughtit up yourself!”

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Hilbert’s response upon hearing that one of his students had dropped out to studypoetry:

”Good, he did not have enough imagination to become a mathematician”.

In 1930, the Mathematical Institute in Gottingen was opened by Hilbert and Courant inits new building which had been constructed with funds from the Rockefeller Foundation.Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leadingmathematical journal of the time.

Hilbert received many honours. In 1905 the Hungarian academy of sciences gave aspecial citation for Hilbert.

1.5 Later years

The relations between German and French mathematicians were still strained as a resultof World War I. Hilbert looked for occasions to apply some balm to the relations. Forexample in 1926, plans had been made for publication of a volume, commemoratingBernhard Riemann’s hundredth anniversary. Hilbert intended to include a paper by PaulPainleve. This important French mathematician had fiercely denounced the Germanscientific community, but Hilbert and others felt that Painleve had later been able tomove beyond these feelings. But Brouwer who was a chief editor of the mathematicalannales, too, objected. The essentially democratic setup of the journal’s operation ledto the volume being issued without any French contributor.

From this point, Hilbert made efforts to remove Brouwer from the board of thejournal. Einstein, whom Hilbert had earlier put on the board, too, clearly stated hisstrict neutrality. Einstein wrote to Brouwer and Blumenthal:

”I am sorry that I got into this mathematical wolf-pack like an innocentlamb. . . Please allow me therefore, to persist in my ’Muh-noch-Mah’ po-sition and allow me to stick to my role of astonished contemporary.”

Finally, the solution was to dissolve the old editorial board and form a new one—butwith the crucial difference. On the new cover for 1929, there would only be Herausgeberand no Mitarbeiter. In this way, it came out as just a major change in policy, ratherthan an act against one of the editors, —Brouwer.

To the international congress held in Bologna, German mathematician were invited,for the first time after the war. They had not been invited to the congress in Straßburg1920 nor to Toronto in 1924. Hilbert, the leader of the German delegation, spoke onthe fundamentals of mathematics.

At a symposium on the foundations of mathematics in Konigsberg in 1930, RudolfCarnap, Heyting, and Johann von Neumann presented papers on logicism, intuitionism,and formalism, respectively. Ever since then, there has been a tendency to characterizethe crisis of foundations as a struggle among these three. The city of Konigsberg madehim an honorary citizen on the occasion of his retirement. His address ended with six

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famous words showing his enthusiasm for mathematics, and his life devoted to solvingmathematical problems:—

We must know, we shall know.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of theSociety of German Scientists and Physicians, Kurt Godel —in a roundtable discussionduring the Conference on Epistemology held jointly with the Society meetings— tenta-tively announced the first draft of his incompleteness theorem.

In 1934 and 1939 two volumes of Grundlagen der Mathematik were published whichwere intended to lead to a ’proof theory’, and leading to a direct check for the consistencyof mathematics. Godel’s paper of 1931 showed that this aim is too ambitious andimpossible.

Hilbert lived to see the Nazis purge many of the prominent faculty members atUniversity of Gottingen in 1933. By the time Hilbert died in 1943, the Nazis hadnearly completely restaffed the university, inasmuch as many of the former faculty hadeither been Jewish or married to Jews. Those forced out included Hermann Weyl,who had taken Hilbert’s chair when he retired in 1930, Emmy Noether and EdmundLandau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbertin mathematical logic, and co-authored with him the important book Grundlagen derMathematik (which eventually appeared in two volumes, in 1934 and 1939). This was asequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928.

About a year later, Hilbert attended a banquet and was seated next to the newMinister of Education, Bernhard Rust. Rust asked, ”How is mathematics in Gottingennow that it has been freed of the Jewish influence?” Hilbert replied, ”Mathematics inGottingen? There is really none any more.”

Hilbert’s funeral was attended by fewer than a dozen people, only two of whom werefellow academics, among them Arnold Sommerfeld, a theoretical physicist and also anative of Konigsberg. The epitaph on his tombstone in Gottingen contains the famouslines he had spoken at the conclusion of his retirement address to the Society of GermanScientists and Physicians in the fall of 1930:

Wir mussen wissen. Wir werden wissen.

News of his death only became known to the wider world six months after he had died.

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2 Work

2.1 Hilbert’s Basis Theorem and Zahlbericht

Hilbert’s first work was on invariant theory. In 1888, he proved his famous Basis Theo-rem. Twenty years earlier Gordan had proved the finite basis theorem for binary formsusing a highly computational approach. Attempts to generalize Gordan’s work to sys-tems with more than two variables failed since the computational difficulties were toogreat. Hilbert himself tried at first to follow Gordan’s approach, but soon realized thata new line of attack was necessary. He discovered a completely new approach whichproved the finite basis theorem for any number of variables with an entirely abstractmethod. Although he proved that a finite basis existed, his methods did not constructsuch a basis.

Hilbert submitted a paper proving the finite basis theorem to the well-known journalMathematische Annalen. However, Gordan was the expert on invariant theory for Math-ematische Annalen and he found Hilbert’s revolutionary approach difficult to appreciate.He refereed the paper and sent his comments to Klein:—

The problem lies not with the form . . . but rather much deeper. Hilberthas scorned to present his thoughts following formal rules, he thinks itsuffices that no one contradict his proof . . . he is content to think thatthe importance and correctness of his propositions suffice. . . . for acomprehensive work for the Annalen this is insufficient.

However, Hilbert had learnt through his friend Hurwitz about Gordan’s letter to Kleinand Hilbert wrote himself to Klein in forceful terms:—

. . . I am not prepared to alter or delete anything, and regarding this paper,I say with all modesty, this is my last word so long as no definite andirrefutable objection against my reasoning is raised.

At the time Klein received these two letters from Hilbert and Gordan, Hilbert was anassistant lecturer while Gordan was the recognized leading world expert on invarianttheory and also a close friend of Klein’s. However Klein recognized the importanceof Hilbert’s work and assured him that it would appear in the Annalen without anychanges whatsoever, as indeed it did.

Hilbert expanded on his methods in a later paper, again submitted to the Mathe-matische Annalen and Klein, after reading the manuscript, wrote to Hilbert saying:—

I do not doubt that this is the most important work on general algebra thatthe Annalen has ever published.

In 1893 while still at Konigsberg Hilbert began a work Zahlbericht on algebraic numbertheory. The German Mathematical Society requested this major report three years afterthe Society was created in 1890. The Zahlbericht (1897) is a brilliant synthesis of the

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work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert’s own ideas.The ideas of the present day subject of ’Class field theory’ are all contained in this work.Rowe, in [18], describes this work as:—

. . . not really a ”Bericht” (report) in the conventional sense of the word,but rather a piece of original research revealing that Hilbert was no merespecialist, however gifted. . . . he not only synthesized the results ofprior investigations . . . but also fashioned new concepts that shaped thecourse of research on algebraic number theory for many years to come.

2.2 Foundations of Geometry

Hilbert’s work in geometry had the greatest influence in that area after Euclid. A sys-tematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axiomsand he analyzed their significance. He published Grundlagen der Geometrie in 1899,putting geometry in a formal axiomatic setting. It was a major influence in promotingthe axiomatic approach to mathematics—indeed one of the major characteristics of thisscience throughout the 20th century. The Foundations of Geometry have continued toappear in new editions, the 14th edition has appeared in 1999.

2.3 The Paris Problems

Hilbert’s famous speech The Problems of Mathematics was delivered to the SecondInternational Congress of Mathematicians in Paris. Several versions of the talk appearedin print, and they were all longer and more detailed than the actual talk. Hilbert’sfamous 23 Paris problems have challenged—and still today challenge—mathematiciansto solve fundamental questions. Many of the problems were solved during this century,and each time one of the problems was solved it was a major event for mathematics.

It was a speech full of optimism for mathematics in the coming century and he feltthat open problems were the sign of vitality in the subject:—

The great importance of definite problems for the progress of mathematicalscience in general . . . is undeniable. . . . as long as a branch of knowledgesupplies a surplus of such problems, it maintains its vitality. . . . everymathematician certainly shares . . . the conviction that every mathemat-ical problem is necessarily capable of strict resolution . . . we hear withinourselves the constant cry: There is the problem, seek the solution. Youcan find it through pure thought. . .

Hilbert’s problems included the continuum hypothesis, the well ordering of the reals,Goldbach’s conjecture, the transcendence of powers of algebraic numbers, the Riemannhypothesis, the extension of Dirichlet’s principle and many more. The list shows thatHilbert’s mathematical horizons were unusually. Nevertheless, of course, important

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contemporary fields of research were left out. Especially two major contemporary openproblems, Fermat’s last theorem and Poincare’s three-body problem, were mentioned inthe introduction, but not counted among the 23 problems.

Finally, there exists a tentative 24th problem that was not published as part of thefinal list of Hilbert’s twenty-three problems, but was included in David Hilbert’s originalnotes. Hilbert’s 24th problem asks for a criterion of simplicity in mathematical proofsand the development of a proof theory with the power to prove that a given proof is thesimplest possible. The 24th problem was rediscovered by the German historian RudigerThiele in 2000. Rudiger Thiele’s paper [?] gives the full text from Hilbert’s notes:

The 24th problem in my Paris lecture was to be: Criteria of simplicity, orproof of the greatest simplicity of certain proofs. Develop a theory ofthe method of proof in mathematics in general. Under a given set ofconditions there can be but one simplest proof. Quite generally, if thereare two proofs for a theorem, you must keep going until you have derivedeach from the other, or until it becomes quite evident what variant con-ditions and aids have been used in the two proofs. Given two routes, it isnot right to take either of these two or to look for a third; it is necessaryto investigate the area lying between the two routes.

Attempts at judging the simplicity of a proof are in my examination ofsyzygies and syzygies between syzygies (see Hilbert 42, lectures XXXI-IXXXIX). The use or the knowledge of a syzygy simplifies in an essentialway a proof that a certain identity is true. Because any process of ad-dition is an application of the commutative law of addition etc., andbecause this always corresponds to geometric theorems or logical con-clusions, one can count these [processes], and, for instance, in provingcertain theorems of elementary geometry (the Pythagoras theorem, the-orems on remarkable points of triangles), one can very well decide whichof the proofs is the simplest.

2.4 Mathematical Physics

Today Hilbert’s name is often best remembered through the concept of Hilbert space.Irving Kaplansky explains Hilbert’s work which led to this concept:

Hilbert’s work in integral equations in about 1909 led directly to 20th-centuryresearch in functional analysis—the branch of mathematics in which func-tions are studied collectively. This work also established the basis for hiswork on infinite-dimensional space, later called Hilbert space, a conceptthat is useful in mathematical analysis and quantum mechanics.

Making use of his results on integral equations, Hilbert contributed to the development ofmathematical physics by his important memoirs on kinetic gas theory and the theory of

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radiations. Together with Richard Courant, he wrote the two volume book on methodsof mathematical physics.

2.5 ”Foundation of Physics”

Hilbert’s work ”Foundation of Physics” from November 1915 makes him the co-discovererof the relativistic theory of gravitation. Like in the foundations of geometry, he stressesthe axiomatic point of view, from which his derivation of Einstein’s field equation ofgeneral relativity proceeds. The other most important source of Hilbert’s approach isGustav Mie’s attempt to generalize electrodynamics. In the publication of 1916, Hilbertputs Einstein first, then Mie:

The tremendous problems formulated by Einstein, as well as the penetratingmethods he devised for solving them, and the far reaching and originalconceptions by means of which Mie produced his electrodynamics, haveopened new ways to the research of the foundations of physics.

In what follows I would like to derive—in the sense of the axiomatic method—essentially from two axioms, a new system of fundamental equations ofphysics that display an ideal beauty, and which in my opinion simulta-neously contain the solutions to the problems of both Einstein and Mie.

Hilbert does not mention the contribution of Max Born, who clarified the ideas ofGustav Mie how to generalize electrodynamics, and put them in the form of a variationalprinciple for a system with infinitely many degrees of freedom—as a generalization of thewell known variational principle for the Lagrange function in Hamiltonian mechanics.

The derivation of Einstein’s field equation from the variational principle for theEinstein-Hilbert action is still the most clear and simple approach to general relativity. Itis used in the physical literature, too. Hilbert submitted his article on 20 November 1915,five days before Einstein submitted his article containing the correct field equations. Inthe printed version of his paper, Hilbert added a reference to Einstein’s conclusive paperand a concession to the latter’s priority:

The differential equations of gravitation that result are,

in my view, in accordance with those of Einstein’s recently presented, im-portant works on the general theory of relativity.

In his original talk of November 1915, Gustav Mie’s contribution to Hilbert’s approachwas even mentioned before Einstein’s. But soon Hilbert recognized Einstein’s prior-ity as the first researcher who completely understood the physics of general relativity.Nevertheless, the paper expressed in the conclusion far reaching hopes:

. . . not only our conception of space, time and motion have been modifiedfrom their foundation in the direction suggested by Einstein, but I am

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also convinced that starting from the basic equations established here, theinnermost—and so far concealed—processes occurring inside the atomwill be finally illuminated. In particular, a general reduction of all phys-ical constants to mathematical ones must be possible, and with it thepossibility must be brought closer, that the principle physics be trans-formed into a science of the kind of geometry: this is certainly the greatestglory of the axiomatic method that, as we see in this case, makes use ofthe powerful tools of analysis, namely, the variational calculus and thetheory of invariants.

Hilbert’s hopes that his ”foundations of physics” would and could achieve what his hasdone to geometry with his foundations back in 1899 turned out to be an exaggeration.Hilbert’s talk of 1924 stresses again the central role of the axiomatic analysis, but nowwith a more cautious attitude:

I am convinced that the theory I present here contains an enduring core,and provides a framework within which there is enough room for futureconstruction of physics in the sense of a field-theoretical unifying ideal.In any case, it is epistemologically interesting to see how the few simpleassumptions that I express axioms I, II, III and IV suffice to reconstructthe whole theory.

In his 1915 talk, Hilbert had expressed the opinion that electrodynamics is a phe-nomenon derived from gravitation. In the 1924 version, this connection is formulatedmore cautiously:

This is the exact mathematical expression of the connection (Zusammen-hang) between gravitation and electrodynamics that dominates the entiretheory.

In the version published in the Mathematische Annalen (1924), instead of Hilbert’sirrepressible optimism, we find a short and very cautions opening:

Whether the field-theoretical unifying ideal is indeed a definite one, or whatadditions and modifications will eventually be necessary in order to allowfor the theoretical foundation of the existence of negative and positiveelectrons, as well as the logically consistent construction of the laws thatare valid inside the atom—to answer these questions remains a task forthe future.

2.6 Priority remains unclear

Hilbert said in the introduction of his presentation November 1915:

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The far reaching deliberations and original conceptions by means of whichGustav Mie produced his electrodynamics, and the tremendous problemsformulated by Einstein, as well as the penetrating methods he devised forsolving them, have opened new ways to the research of the foundationsof physics.

In what follows I would like to derive—in the sense of the axiomatic method—from three axioms, a new system of fundamental equations of physics thatdisplay an ideal beauty, and which in my opinion simultaneously containthe solutions to the problems posed.

Many have claimed that in 1915 Hilbert discovered the correct field equations for generalrelativity before Einstein, but never claimed priority. The article [11] show Einstein’sarticle appeared on 2 December 1915 but the proofs of Hilbert’s paper (dated 6 December1915) do not contain the field equations.

As the authors of [11] write:—If Hilbert had only altered the dateline to read ”submitted on 20 November 1915,

revised on [any date after 2 December 1915, the date of Einstein’s conclusive paper],”no later priority question would have arisen.

2.7 Reactions to Hilbert’s work in General Relativity

Here are some samples of the reactions to Hilbert’s work in physics and indicationsof the further development during the time 1915-1924. Thus we can understand theshifting attitude of Hilbert seen above. On November 23, 1915, in a letter from Berlin,Max Born reported to Hilbert he had heard from his Berlin friends that Hilbert had”already cleared up gravitation”, and asks for an offprint of Hilbert’s paper. Born’sletter contains a first noteworthy comparison of Einstein’s and Hilbert’s work:

Einstein himself says that he has already solved the problem. But it seemsto me that his considerations (which I know only from conversation) area particular case of yours.

An openly critical judgement of Hilbert’s work on the foundation of physics is given byEinstein in a letter written to Hermann Weyl in November 1916. Here Einstein statedhis opinion very clearly:

Hilbert’s assumption about matter appears childish to me in the sense ofa child who does not know any of the tricks of the world outside. I amsearching in vain [in Hilbert’s approach] for a physical indication that theHamilton function for matter can be formulated from the ϕν ’s withoutdifferentiation. At all events, mixing the solid considerations originatingfrom the relativity postulate with such bold, unfounded hypotheses aboutthe structure of the electron or matter cannot be sanctioned. I gladly

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admit that the search for a suitable hypothesis, or for a Hamiltonianfunction for the structural makeup of the electron, is one of the mostimportant tasks of the theory today. The ”axiomatic method” can be oflittle use here, though.

Hermann Weyl notes in the edition of Raum-Zeit-Materie from 1923:

In his first communication [November 1915], Hilbert formulated the invariantfield equations, simultaneously with Einstein, and independently of him,but in the framework of Mie’s hypothetical theory of matter.

In 1921, the young Wolfgang Pauli wrote his Enzyklpadie Artikel, at the initiative ofArnold Sommerfeld. This is one of the most significant and influential early accountsof the history of general relativity. Felix Klein took an active role in advising Pauli onthe contents. Klein insisted to clearly state Hilbert’s effort in the physics of gravitation.We read in Pauli’s article:

At the same time as Einstein, and independently, Hilbert formulated thegenerally covariant field equations. . . . His presentation, though, wouldnot seem to be acceptable to physicists, for two reasons. First, the ex-istence of a variational principle is introduced as an axiom. Secondly,of more importance, the field equations are not derived for an arbitrarysystem of matter, but are specifically based on Mie’s theory of matter.

Max Born has written in his letter to Einstein (number 45) from April 7th, 1923:

. . . I hear that you have a new theory about the connection between gravita-tional and electromagnetic fields, which allegedly points to a relationshipbetween gravitation and the earth’s magnetic field. I am very curious.Most of what is published about relativistic problems leaves me cold. Ifind Mie’s pulpy effusions horrible.

Hilbert follows all this halfheartedly, as he is completely preoccupied withhis new basic theory of mathematics and logic. What I know of it seemsto me the greatest step forward imaginable in this field. But for the timebeing most mathematicians refuse to recognise it.

. . . Yours Max Born

Here is the comment by Max Born himself:

The rumour about Einstein’s new investigation, in which he attempted theunification of his theory of gravity with Maxwell’s theory of the electro-magnetic field, proved to be correct. At that time he began his oftenrepeated, although unavailing, attempts to develop a unified field theoryalong these lines.

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Hilbert’s efforts to find a new basis for mathematics enthralled and fasci-nated me to begin with. Later, I was no longer able to follow. I hadsome correspondence with Einstein about these problems, when they be-came the cause of dispute between Hilbert and the Dutch mathematicianBrouwer (letter number 58).

Edward Nelson from Princeton University (nelson@math.princeton.edu) has recentlymade the following comment:

In strong contrast to the other great scientific debate of the twentieth cen-tury, that between Niels Bohr and Einstein, the debate between Brouwerand Hilbert was acerbic, with uncollegial words and acts, primarily onthe part of Hilbert It strikes me as a curious historical fact that Bohrpersuaded physicists, who used to study the real world, to give up theirbelief in the objective reality of the physical world whereas mathemati-cians, who study an abstract world that we ourselves create, followedHilbert (who was a Platonist at heart) in refusing to abandon our beliefin the objective reality of mathematical entities.

In 1934 and 1939 two volumes of Grundlagen der Mathematik were published which wereintended to lead to a ’proof theory’, and leading to a direct check for the consistencyof mathematics. Godel’s paper of 1931 showed that this aim is too ambitious andimpossible.

2.8 International congress in Bologna 1928

In 1928, to the international congress held in Bologna, German mathematician wereinvited, for the first time after the war. They had not been invited to the congressin Straßburg 1920 nor to Toronto in 1924. Hilbert, the leader of the German delega-tion, spoke on the fundamentals of mathematics. The second of Hilbert’s twenty-threeproblems posed in his 1900 paper asked for a proof of consistency for the axioms of arith-metic. He had devoted a great deal of thought and effort to this topic over the years.Unlike several of his younger colleagues, he was convinced that it was possible to findan demonstration, beyond any debate, showing that mathematics formed a completeand consistent whole. Returning to the foundations of mathematics, Hilbert added twomore basic questions, bringing the number to three:

• First, is mathematics complete, in the technical sense that every statement (suchas ”every integer is the sum of four squares”) could be either proved, or disproved.

• Second, is mathematics consistent, in the sense that the statement ”2 + 2 = 5”could never be arrived at by a sequence of valid steps of proof.

• And thirdly, is mathematics decidable? By this he meant, did there exist a definitemethod which could, in principle, be applied to any assertion, and which was

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guaranteed to produce a correct decision as to whether that assertion was true.(Hodges, Turing: The Enigma, 1983, 91).

Three years later, the Czech mathematician Kurt Godel published ”Uber formal un-entscheidbare Satze der Principia mathematica und verwandter Systeme I” (Monatsheftefur Mathematik und Physik 38 [1931]: 173-98), in which he answered the first two ofHilbert’s questions in the negative: mathematics was neither complete nor consistent.Hilbert’s third question, known as the Entscheidungsproblem (decision problem), wasaddressed independently in 1936 by Church, Post, and Turing each of whom presentedproofs that mathematics was also not decidable; Turing’s proof involved the creation ofhis hypothetical ”universal computing machine” (Turing machine).

2.9 Concluding remark

Hilbert contributed to many branches of mathematics, including invariants, algebraicnumber fields, functional analysis, integral equations, mathematical physics, and thecalculus of variations. Hilbert’s mathematical abilities were nicely summed up by OttoBlumenthal, his first student:—

In the analysis of mathematical talent one has to differentiate between theability to create new concepts that generate new types of thought struc-tures and the gift for sensing deeper connections and underlying unity.In Hilbert’s case, his greatness lies in an immensely powerful insightthat penetrates into the depths of a question. All of his works containexamples from far-flung fields in which only he was able to discern aninterrelatedness and connection with the problem at hand. From these,the synthesis, his work of art, was ultimately created. Insofar as the cre-ation of new ideas is concerned, I would place Minkowski higher, and ofthe classical great ones, Gauss, Galois, and Riemann. But when it comesto penetrating insight, only a few of the very greatest were the equal ofHilbert.

Remark. Sources for this essay are the book [?] of Leo Corry,David Hilbert and the Axiomatization of Physics (1898-1918).The Born-Einstein Letters 1916-1955, with the subtitle Friendship, Politics and

Physics in Uncertain Times.The Born-Einstein Letters [?] are a superb collection of letters, never written for

publication.I have used the article of J. J. O’Connor and E. F. Robertson, July 1999 and internet

sources as

http://www.answers.com/topic/david-hilbert#ixzz1mTbbmzDH

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Part II

Some Reflections

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1 Der Annalenstreit

1.1 Mathematische Annalen1 Die Mathematischen Annalen (abgekurzt Math. Ann. oder Math. Annal.) sind einemathematische Fachzeitschrift. Die Zeitschrift wurde 1868 durch Alfred Clebsch undCarl Gottfried Neumann begrundet und galt fur viele Jahrzehnte als eine der weltweithochrangigsten Fachzeitschriften fur Mathematik. Die ersten 80 Bande 1868–1920 er-schienen im Teubner-Verlag, danach wurde die Zeitschrift vom Julius-Springer-Verlagfortgefuhrt. Bis in die 1960er Jahre erschienen die Zeitschriftenbeitrage im Wesentlichenin deutscher Sprache, heute jedoch auf Englisch.

Zu den Herausgebern zahlten international hoch angesehene Mathematiker:Alfred Clebsch (1869–1872), Carl Gottfried Neumann (1869–1876), Felix Klein (1876–

1924), Adolph Mayer (1876–1901), Walther von Dyck (1888–1921), David Hilbert (1902–1939), Otto Blumenthal (1906–1938), Albert Einstein (1920–1928), Constantin Carathodory(1925–1928), Erich Hecke (1929–1947), Bartel Leendert van der Waerden (1934–1968),Franz Rellich (1947–1955), Kurt Reidemeister (1947–1963), Richard Courant (1947–1968), Heinz Hopf (1947–1968), Gottfried Kothe (1957–1971), Heinrich Behnke (1938–1972), Max Koecher (1968–1976), Lars Grding (1970–1978), Konrad Jorgens (1972–1974), Fritz John (1968–1979), Peter Dombrowski (1970–1983), Louis Boutet de Mon-vel (1979–1983), Wulf-Dieter Geyer (1979–1983), Elmar Thoma (1974–1990), Win-fried Scharlau (1984–1990), Hans Grauert (1963–1991), Heinz Bauer (1971–1992), HansFollmer (1990–1993), Friedrich Hirzebruch (1961–1996), Reinhold Remmert (1970–1996),etc.

1.2 Hintergrund des Annalenstreites

Schwerwiegende Differenzen zwischen den Professoren der damals fuhrenden mathema-tischen Institute in Gottingen und Berlin fuhrten Ende der 1920er Jahre zu einem ern-sthaften Konflikt unter den Herausgebern der Zeitschrift, der als Annalenstreit bekanntwurde. Mehrer Ursachen kamen zusammen:

• Rivalitat zwischen den Instituten in Gottingen und Berlin;

• Fachliche Streit uber die Grundlegung der Mathematik. Der von Luitzen EgbertusJan Brouwer begrundete Intuitionismus (in Berlin) stand dem Formalismus vonDavid Hilbert oder Richard Courant in Gottingen gegenuber.

• Unterschiedlichen politischen Auffassungen der Beteiligten. Die in Berlin hauptsachlichvon Ludwig Bieberbach vertretene deutsch-nationale Gesinnung stand der poli-tisch liberalen Haltung der Gottinger Mathematiker gegenuber.

1aus Wikipedia, der freien Enzyklopadie

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• Die Frage der Teilnahme oder Boykott des ersten internationalen Mathematik-erkongress in Bologna.

Viele deutsche Wissenschafter in der Weimarer Republik waren nach der schmerzhaftenund fur viele unerklarlichen Niederlage im Ersten Weltkrieg von Nichtakzeptanz dereigenen schwachen Regierung und von einem enormen Nationalstolz gepragt. Sie sahendie Wissenschaft als geeignetes Mittel an, die ”Starke Deutschlands” aufzuzeigen.

Als 1928 erstmals nach dem Krieg wieder deutsche Mathematiker zum Interna-tionalen Mathematikerkongress in Bologna eingeladen wurden, 1920 in Straßburg und1924 in Toronto war das nicht der Fall gewesen, kam es zu einer neuen Auseinanderset-zung zwischen der nationalen Fraktion um Bieberbach und Brouwer und der liberalen,nicht streng deutschnationalen Seite um David Hilbert und seinen Gottinger Kollegen.

Bieberbach war der Meinung, man sollte dem Kongress fernbleiben, zum Einen, weiler vermutete, dass das ”Conseil International de Recherche”, welches der deutschenWissenschaft nicht gerade positiv gegenuberstand, an der Organisation des Kongressesbeteiligt war.

Außerdem stand ein Ausflug ins ”befreite Sudtirol”, das durch den Krieg an Ital-ien gefallen war, auf der Tagesordnung, fur einen national denkenden Menschen wieBieberbach eine Beleidigung hochsten Ausmaßes.

Hilbert, Courant und weitere Gottinger befurworteten die Teilnahme, die BerlinerMathematiker um Bieberbach lehnten sie ab. Ihnen schwebte von jeher eine Zusamme-narbeit zwischen Mathematikern, auch international vor. So fuhrte Hilbert mit seinenGottinger Kollegen Landau und Courant die deutsche Delegation beim Mathematik-erkongress an, wahrend aus Berlin um Bieberbach keine Mathematiker teilnahmen.

1.3 Der Annalenstreit

Dies fuhrte in der Folge auch zum so genannten Annalenstreit innerhalb der DMV. Schon1925 verhinderten die Mitherausgeber der Zeitschrift ”Mathematische Annalen” LudwigBieberbach und der Hollander L.E.J. Brouwer die vom geschaftsfuhrenden HerausgeberOtto Blumenthal beabsichtigte Aufnahme von einigen franzosischen Beitragen in einenRiemann-Gedenkband.

Nach dem Kongress von Bologna setzte Hilbert durch, dass Brouwer nicht langerals Mitherausgeber der Annalen fungieren sollte, indem ein neuer Vertrag mit demSpringerverlag geschlossen wurde. Demzufolge sollten nur noch Hilbert, Blumenthalund Hecke als Herausgeber auf der Titelseite gefuhrt werden. Bieberbach und Brouwerprotestierten bei Ferdinand Springer, allerdings erfolglos, worauf sie ihm mangelndesNationalgefuhl vorwarfen.

Die Geschehnisse innerhalb der DMV wahrend des Nationalsozialismus hangen engmit Machtkampfen in der Vereinigung wahrend der Weimarer Republik zusammen.Ruckblickend zeigt sich, dass bereits lange vor der offiziellen Machtergreifung durch denNationalsozialismus bereitwillig nationale Grunde fur die Durchsetzung eigener fachpoli-tischer Interessen herangefuhrt wurden.

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1.4 Literatur

Heinrich Behnke: Ruckblick auf die Geschichte der Mathematischen Annalen In: Math.Ann. 200 (1973), S. I-VII.

http://www.math.uni-hamburg.de/home/loewe/2007-08-I/L12.pdf

http://www5.in.tum.de/lehre/seminare/math_nszeit/SS03/vortraege/de-math/

http://www.survivor99.com/pscience/2006-6/philosophy/

Folder%20%20of%20Logic/luitzen_egbertus_jan_brouwer.htm

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2 The Einstein-Born letters

2.1 Born’s letter from 1928

Institut for Theoretical Physics of the University Gottingen, Bunsenstr. 920 February, 1928

Dear EinsteinAfter consultation with Harald Bohr, who is in Gottingen this term, I want to write

2 to you about a matter which is—strictly speaking—none of my business, but whichnevertheless has caused me alarm and uneasiness on many occasions. I am referring tothe Hilbert and Brouwer affair. Up to now I have merely followed if from a distance,and have only recently been initiated into all the details by Bohr and Courant.

In this way I learnt that you remained neutral with regard to Hilbert’s letter toBrouwer, on the grounds that one should permit people to be as foolish as they wish.I find this quite reasonable, of course, but you seem not to be quite in the picture onsome points, and so I want to write briefly and you about it. There will probably be aconference soon at Springer’s about the matter, and Bohr told me that he considered itvery important for the inner editorial staff to present a united front.

I would therefore ask you please to maintain your present neutrality, and not to takeany action against Hilbert and his friends. It would help to restore my peace of mind,as well as Bohr’s and that of many other people, if you could write a few words to meabout this.

I would like to tell you briefly why this business interests me. It only matters tome because I am worried and concerned about Hilbert. Hilbert is seriously ill, and hasprobably not very long to live. Any excitement is dangerous for him, and means losingsome of the few hours left to him in which to live and to work.

He still has, however, a powerful will to live, and considers it his duty to completehis new basis for mathematics with whatever strength left to him. His mind is clearerthan ever, and it is an act of extreme callousness on Brouwer’s part to spread therumour that Hilbert is no longer responsible. Courant and other friends of Hilbert’shave frequently said that the sick man should be protected against any excitement,and Brouwer has misrepresented this to mean that one should no longer take Hilbert’sactions and opinions seriously.

Hilbert is quite in earnest about his proposed action against Brouwer. He talked tome about it a few weeks ago, but only in quite general terms and without going into anydetail. In his opinion Brouwer is an eccentric and maladjusted person to whom he didnot wish to entrust the management of the Mathematische Annalen. I think Hilbert’sevaluation of Brouwer has been shown to be correct in view of Brouwer’s most recentactions. In my experience, Hilbert’s judgment is almost always clear and to the pointin human affairs.

2This the letter number 58 from the collection [?]

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I have followed the previous history of the whole business, including the quarrelabout the visit to the Congress in Bologna, from a distance only. But I do know thatthe visit to this Congress was a heavy burden to Hilbert anything of this kind meant atremendous exertion for him because of his illness.

Hilbert is not politically very left-wing; on the contrary—for my taste and even morefor yours,—he is rather reactionary. But when it comes to the question of the intercoursebetween scientists of different countries, he has a very sharp eye for detecting what isbest for the whole. Hilbert considered—as we all did—that Brouwer’s behavior in thisaffair, where he was even more nationalistic than the Germans themselves, was utterlyfoolish.

But the worst of it all was that the Berlin mathematicians were completely takenin by Brouwer’s nonsense. I would like to add that the Bologna business was notthe decisive factor — only the occasion for Hilbert’s decision to remove Brouwer. Ican understand this in Erhard Schmidt’s case, for he always did lean to the right inpolitics, as a result of his basic emotions. For Mises and Bieberbach, however, it is arather deplorable symptom./ I talked to Mises about it in August, during our journey toRussia, and he said right at the beginning of our discussion that the people in Gottingenwere blindly follow9ng Hilbert, and that he was probably no longer responsible.

Thus the allegation about Hilbert’s weakened mental power was made even then. Ithen immediately broke off my discussion with Mises, for I do not consider him significantenough to allow himself the liberty of passing judgment on Hilbert.

I also enclose a paper which Ferdinand Springer sent to Bohr and Courant. Thisshows that Brouwer and Bieberbach have threatened to denounce Springer as lackingin national feeling, and that they would do him harm if he remained loyal to Hilbert. Ineed not tell you what I think of such behaviour.

Forgive me for bothering you with so long a letter. My only desire is to see thatHilbert’s earnest intentions are put into effect without causing him any unnecessaryexcitement. I would have no objection to your showing this letter, or part of it, toSchmidt, if you consider it correct. As an old friend of Schmidt’s I believe that it ispossible to negotiate successfully with him even if he is of a different opinion.

I hope that you yourself are feeling much better now. I get news of you from time totime in Margot’s letters to my wife. Those two are very close friends indeed, and suiteach other.

I myself am busy completing a book on quantum mechanics, which i have beenwriting for the last year. Unfortunately I have overtaxed my strength a little in doingthis, and will probably have to go on leave for a time during January. It is really not atall easy to find the time and strength for that kind of work, on top of all the lecturesand other professional duties.

With kindest regards, also from my wife to yoursYours Max Born

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Figure 2.1: Einstein has nicknamed this dispute the ’frog-mouse battle’

2.2 Comment by Max Born himself

Harald Bohr, a bother of the physicist Niels Bohr, was a notable mathematician whofrequently visited us in Gottingen.

David Hilbert, my revered teacher and friend, was then—and still is—considered tobe the foremost mathematician of his time. At that time he was busy trying to findsounder logical foundations for mathematics, in order to eliminate the intrinsic contra-dictions found by Bertrand Russell and others in the theory of infinite sets, withoutsacrificing any previous mathematical knowledge.

This led him to consider true mathematics as a kind of logical game with symbols,for which arbitrary axioms are found. The latter, however, should be applied by a‘metamathematics’ based on evident, real conclusions. Brouwer rejected this concept ofmathematics, and suggested another, termed intuitionism. The two ways of thinkingdiffered in one essential result. Hilbert’s concept justified the so-called existence proofs,whereby the existence of a certain number or a mathematical truth is deduced from thefact that to assume the contrary would lead to a contradiction.

Brouwer, however, postulated that the existence of a mathematical structure couldonly be taken for granted if a method could be found that would actually construct it.As it happened, many of Hilbert’s greatest mathematical achievements were preciselysuch abstract proofs of existence, which for some time had not only been accepted bythe mathematical world, but had been celebrated as great feats.

It is therefore no wonder that Brouwer’s behavior greatly upset Hilbert, and that heexpressed his opposition in no uncertain terms; whereupon Brouwer replied with evengreater rudeness.

To make matters worse, a political quarrel broke on top of the scientific one. Afterthe 1914-1918 war, ‘International Unions’ had been founded for all principal branchesof science; the Germans, however, had been excluded from them. The hatred directed

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against Germany gradually diminished, and at the time this letter was written (1928)the German mathematicians were about to be admitted to the ‘International Union forMathematics’, on the occasion of a large mathematical congress at Bologna.

But a group of ‘national’ German mathematicians protested against this; they feltthat it would not be right to join the Union without further ado—after having been ex-cluded for such a long time—and that one should protest against it in Bologna. Threeimportant Berlin mathematicians were amongst the leaders of this movement: Bieber-bach, who was a good analyst; von Mises, a research worker of some significance, whowas also concerned with theoretical physics; and Erhard Schmidt, the most outstandingof the three. Schmidt and I (Max Born) had been friends ever since student days and,although politically we were poles apart, we always remained on the best terms. Butthe Dutchman Brouwer was more nationalistic than all these proved to be.

Hilbert went to Bologna, despite his grave illness, and faced his adversaries. Asfar as I can remember, he got his way and the germans joined the Union. But thewhole business had annoyed him so much that he expelled Brouwer from the manage-ment of the Mathematische Annalen. This started a new storm amongst the Germanmathematicians. But Hilbert finally got the upper hand.

The whole affair was, strictly speaking, no concern of mine. But, as I said in the let-ter, I was moved to intervene by my anxiety about the state of Hilbert’s health. Hilbertsuffered from pernicious anaemia, and would no doubt have died within a short timehad not Minot in the United States discovered the specific remedy, a liver extract, justin time. This was not yet commercially available, but the wife of the Gottingen mathe-matician Edmund Landau was a daughter of Paul Ehrlich, the founder of chemotherapyand discoverer of Salvarsan. It was due to his good offices that Hilbert was able toreceive regular supplies of the extract and so to live for many more years.

I doubt whether my letter to Einstein had any influence on the course of the greatmathematical quarrel.

As for the further development of the fundamental problems of mathematics, Brouwerhad many supporters to begin with, including some important ones such a HermannWeyl. But gradually Hilbert’s abstract interpretation was, after all, realized to be byfar the more profound. Things took a new turn when Godel discovered the existence ofmathematical theorems which can be proved to be incapable of proof. Today, mathe-matics is more abstract than ever, and exactly the same is true for theoretical physics.

The journey to Russia I mention was a kind of wandering physicists’ congress, or-ganized in Leningrad by Joffe, who has been mentioned before. It began in Leningrad,and was continued first to Moscow and then in Nizhni-Novgorod; there the participantsboarded a Volga steamer and travelled down river, stopping at all the large towns enroute to continue the congress. The whole thing was very fascinating and stimulating,but extremely fatiguing. I went as far as Saratov, and from there returned to Germanyby train.

The book about quantum mechanics I mentioned at the end of the letter was writtenin collaboration with Jordan over a period of several years.

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2.3 The Born Einstein Letters 1916-1955

Friendship, Politics and Physics in Uncertain TimesEinstein, Albert; Born, Max;Thorne, Kip S.; Buchwald, Diana Kormos; Heisenberg, WernerISBN-10: 1403944962 ISBN-13: 9781403944962

The highlight of this book by Nobel Prize-winning physicist Max Born (1882 to1970) is the letters he and Nobel Prize-winning physicist Albert Einstein (1879 to 1955)exchanged between the years 1916 and 1955. These letters (that were never meant tobe published) show the human side of these brilliant physicists.

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3 Confirmation of Gravitational Waves

”Hier eine Methode zum indirekten Nachweis von Gravitatioswellen mittels eines Pulsarsin einem Doppelsternsystem.” Genau! Ich durfte einem Vortrag von Joe Taylor (ererhielt 1984 den Nobelpreis fur Physik fur genau diesen Nachweis) im Januar 2009in HH life zuhoren. Der Nachweis funktioniert so: Wenn 2 Pulsare sich umkreisenso werden große Massen sehr schnell bewegt. Dies ist eine Voraussetzung f’ur einenerfolgversprechenden Versuch des Nachweises. Im Nenner fur die Berechnung der Starkeder Gravitationswellen steht namlich die Lichtgeschwindigkeit in 5. Potenz! Die Zahlwird also sehr klein. Wenn sich die Entfernung der Pulsare verkleinert so wird demDoppelsystem Energie entzogen, genau wie der ISS beim Flug um die Erde. Bei derRaumstation geschieht dies durch Reibung an den obersten Schichten der Atmosphre,beim Doppel-Pulsarsystem durch Abgabe von Gravitationswellen. Taylor gelang diehaargenaue Vermessung des Orbits und die Quantifizierung des Energieverlusts. Erkonnte die Entfernungsverkleinerung mit der Vorhersage durch Einstein vergleichen undkam auf die vorhergesagten Werte. Er konnte damit die qualitative und quantitativeBestatigung von Einsteins Gleichungen fur die Gravitationswellen fuhren und zeigen,dass die Physik wie wir sie kennen auch in tausenden Lichtjahren Entfernung unterextremen Bedingungen funktioniert. Das war einen Nobelpreis wert. Ich bin vollerBewunderung!

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