6.2 Goettingen's Tragedy as Choice of History

6.2  Goettingen's Tragedy as Choice of History

 

But surely no man knows or ever will know
The truth about the gods and all I speak of.
For even if he happens to tell the perfect truth,
He does not know it,
But appearance is fashioned over everything.

XENOPHANES

Let us read the words of Born more carefully:

My theory is in the full accordance with an atomistic instinct of so big quantity of experimenters, that the interesting attempt of Levi-Civita [7] to describe the movement of an electricity as the certain liquid moving freely under the action of its own field and not bound by any kinematic conditions, is hardly worth approval. [3]

These words can be crucial for the reconstruction of ideas and discoveries Hermann Minkowski's mind was filled with during last months of his life. Only in such coded a form could Born afford himself to disagree with the clear position of Minkowski he knew about and, thus, has given us the indirect evidence of it.

Born does not even mention the second of two Henri Poincaré's works, with the general name «About the theory of electron» [9.II]. This can be the evidence of negative Minkowski's reaction on the suggested there decision of a problem of electron, its rescue from disintegration due to the forces of scalar (surface) pressure, balancing electric repulsive forces. Such an attitude could be natural had Minkowski already stepped on the road conformable to the ideas of Levi-Civita. At that time the procedure of Poincaré was a step back and was hardly worth approval, because electron should not have really be rescued from disintegration by this means, cementing artificially its static configuration of charge. Born also thought the decision of Poincaré to be artificial and unnecessary, though on other reasons. Thus the ultimately agreeing negative appraisal of this direction of the theory progress, made it unnecessary for Born to mention the work of Poincaré, as the one not having perspective.

The work of Levi-Civita is completely another thing [7]. If Minkowski:

• highly estimated this direction of the further geometrization of the field theory and of the possibility of substantiation Minkowski's world with the help of phase-configuration analogue lying in its basis,

• was inclined to the stationary dynamic structure of electron, as at the most symmetric realization of a possibility of existence of rather stable balance between the field set of conservative hyperbolic currents of Levi-Civita and their own (resulting) field (of electron) cooperating with each other,

• shared with colleagues and Born his considerations and plans on these directions, coordinated by him in the general program,

• left after his death materials and bare notes on these subjects, obtained by Born at the suggestion of Hilbert's for studying and estimating of their importance, of possibility and expediency of their «due» edition,

then it already didn't seem possible to Born to just hold it back. He was forced, while protecting his own position (and/or carrying out certain directive of Hilbert), to look painfully for counter-evidences against such an «extremely far-fetched» nature of atomism of electron and the deviation from still forming concepts of the Special Theory of Relativity about Space and Time in the minds of contemporaries. Had not Born known of his teacher's high appraisal of Levi-Civita's ideas, had not Minkowski made him doubt about correctness and infallibility of the way Born had chosen, even the mention of Levi-Civita's work would not have followed.

Born did not manage to find something more convincing then an atomistic instinct of so wide a range of experimenters, the instinct guarding the immunity of static pattern of electron and rejecting strongly a certain liquid of Levi-Civita (and Minkowski).

Judge yourself, – whether the scientists respected by everybody, burdened with an unreasonable cargo of the responsibility, could consider so played imaginations of the colleagues – mathematicians interfering the territory of physicists and proposing seriously to build the theory of «firm» electron on the basis of a certain liquid, not even bound by any kinematic conditions, seriously and being interested enough? What other reaction could be expected from men of science who had diligently adopted the points of view of Euclid, Newton and Kant on the fundamental character of space and time, had made these conceptions an instinct, supported it with well-composed axiomatic diagrams on basis of the postulates that couldn't be doubted by mortals. And the deeper and sharper were these conceptions of a concrete scientist, the more patterns and mutual internal and external connections were opening to his intellectual look, consequently, the more of intellectual work and spiritual efforts were put by him in the development of this area, in the fostered part of his scientific «I». And so the less hope remained on his voluntary and sincere readiness to sacrifice this part of his «I» in exchange on a certain illusive prospect of gaining instead something greater and till that time tested sufficiently by nobody. In other words, – the events had real chances to develop in accordance with the thesis of Planck.

Phase and configuration spaces of analytical mechanics seemed very convenient and useful to the solving of the certain circle of problems, but were considered as especially auxiliary mathematical structures, depleted of their physical substratum and any actual physical content. The situation here very much reminded an early stage of perception of negative and complex numbers even by the largest thinkers of their days, insisting persistently on the terms «impossible», «irrational» and «imaginary», allowing their use only by virtue of a various sort of «practical» needs.

One thing is to agree to the forced use of Minkowski's world under pressure of the experimental facts which nobody managed to describe correctly and clearly in the context of conceptions about absolute space and time. And absolutely another one, is to abandon at once still not fully accepted Minkowski's world and to start construction of an «obviously fictitious» phase-configuration world for the needs of a certain liquid, as if the one solving all problems of electron at once.

The small historical information from the book of Constance Reid [12] about the tragic events previous to the appearance of this and other works of Born of that period, connected somehow or other with the reaction of Goettingen to the ideas and plans of Minkowski, that stayed unpublished…

(In contrast to Hilbert,) Minkowski was at a point of great creativity during the summer of 1908. In September he presented some of his new electrodynamical results at the annual meeting of the Society of German Scientists and Physicians, which was held in Cologne. The title he chose for this talk was «Space and time»…

He had often told his students in Goettingen, «Einstein's presentation of his deep theory is mathematically awkward – I can say that because he got his mathematical education in Zurich from me»…

Now came, in Minkowski's talk at Cologne, what has been called «the great moment of geometrization». In a period of a few moments Minkowski introduced into relativity theory his own beautifully simple mathematical idea of Space-Time by means of which the different descriptions of a phenomenon can be represented mathematically in a very simple manner…

Among the members of the audience was Max Born, whose interest in relativity had been re-aroused by the recent work of Einstein. Minkowski wanted Born to return to Goettingen as his collaborator. he needed someone with the knowledge of optics which Born had. But first he wanted his former pupil to become more familiar with his own ideas in the field. He sent Born back to Breslau with his latest electrodynamical work.

In Minkowski's work the young man found laid out «the whole arsenal of relativity mathematics… as it has been used every day since then by every theoretical physicist». Not until the beginning of December did he consider himself prepared to return to Goettingen.

«There followed several weeks when I saw Minkowski every day and talked with him. It was a happy time, full of scientific excitement but also rich in experience of a personal sort, the beginning of a true friendship so far as the difference of age and experience permits this word to be used».

Minkowski had been gone from Goettingen during the Christmas holidays, but he returned on Wednesday, January 6. The next day being Thursday, the four mathematics professor made their weekly hike, promptly at three o'clock, to the Kehrhotel on the Heinberg. In spite of the wintry hills and leafless trees, it was a pleasant excursion. The cold air rang with loud cheerful voices and laughter. Minkowski recounted «with special liveliness» his latest results in his electrodynamical work… [12,ch.XIV,p.112-113]

Exclusiveness of the growing role of Minkowski was defined not only by his personal qualities and a rare combination of depth of penetration both into the problems of pure mathematics and in the bases of electrodynamics, with its natural physical geometry, but also by his unique status of a close friend and «the private teacher of physics» of David Hilbert. The latter circumstance could increase a hundred times the weight of each of his words and instincts, to provide the unprecedented influence on a choice of perspective directions of physical researches, on the formation of instincts of the youth coming from all over the world to the training in Goettingen, and leaving it with the vision of prospect acquired there.

It could have been, – had Minkowski lived and worked several more years, had he had the time to arrange his ideas properly and to acquaint with them the scientific community. It could have been, - had Hilbert shown the genuine interest in physical program of Minkowski as he did it in relation to the nonlinear diagrams of Gustav Mie and Albert Einstein subsequently.

Almost all the necessary conditions were created for realization of a new unprecedented intellectual sheaf (group of people), which in the nearest years could become the powerful locomotive of knowledge of a geometrical nature of a field and its sources and be written forever into the annals of the history of a triumph of the creative spirit. But the Heaven itself opposed such a prospect…

On Friday Minkowski delivered a regular lecture. After that he conducted a doctoral examination.

Then, on Sunday afternoon, following dinner, he was suddenly stricken with a violent attack of appendicitis. That night the decision was made to undertake the difficult operation to remove the ruptured organ.

Through Monday Minkowski's condition worsened. He was conscious and quite clear about the hopelessness of his situation. On the hospital bed he studied the proof-sheets of some of his latest work and considered whether it would be possible to turn the still unfinished part of the work to good account.

Hilbert later recalled: «he spoke his regrets upon his fate, since he still could have accomplished much; but he decided that it would be good to correct the proof-sheets so that his latest electrodynamical works could be more easily read and better understood». he said that perhaps after his death the opposition to his new ideas could be more easily overcome…

At noon on Tuesday, January 12, 1909, Minkowski asked to see his family and Hilbert again. Hilbert left home as soon as he received the message; but by the time he reached the hospital, Minkowski was dead. Not having yet attained his forty-fifth year, he had been taken «in the full possession of his vital energy, in the middle of his most joyful work, at the height of his scientific creativity»…

«The doctors themselves stood around his eyes with tears in their eyes»…

On Thursday afternoon there was no mathematical walk. Instead, the mathematics professors provided Minkowski's body with a final escort. Again, Hilbert noted, it was exactly three o'clock.

«The strong mathematicians were like men confounded», another student wrote to his parents after the funeral. «To all appearances Klein himself found it difficult to speak calmly. Hilbert and Runge seemed disfigured, their eyes were so red wit tears». [12,ch.XIV,p.114-115]

In the summer of 1912 died the other Giant – Henri Poincaré. He was 56 years old; for 33 years he had been incredibly productive in almost all branches of mathematics [and of fundamental physics]. The year before, however, he had asked the editor of a mathematical journal to accept an unfinished paper on a problem which he considered of the highest importance:

«At my age, I may not be able to solve it, and the results obtained, susceptible of putting researches on a new unexpected path, seem to me too full of promise, in spite of the deceptions they have caused me, that I should resign myself from sacrificing them.»

It was a poignant reminder to his contemporaries that time was short. They found themselves now filled with a certain fear of death, the special characteristic of which was expressed by Vito Volterra, the leading Italian mathematician of the time, in an address on Poincaré's work:

«Among the various ways of conceiving man's affection for life, there is one in which that desire has a majestic aspect. It is quite different from the way one usually regards the feeling of the fear of death. There come moments when the mind of a scientists engenders new ideas. He sees their fruitfulness and utility, but he knows that they are still so vague that he must go through a long process of analysis to develop them before the public will be able to understand and appreciate them at their just value. If he believes then that death may suddenly annihilate this whole world of great thoughts, and that perhaps ages may go by before another discovers them, we can understand that a sudden desire to live must seize him, and the joy of his work must be confounded with the fear of having it stop forever.» [12,ch.XVI,p.133]

«Minkowski's approach to number theory [objects of physics] was geometrical, it being his aim to express algebraic conjectures about the rational numbers [physical values]. in terms of geometric figures, an approach which frequently made the proofs [fundamental physical structures] more obvious. He was deeply approached in a book on this new subject, and his letters to Hilbert were filled with his concern about the presentation of his material. All must be «klipp und klar» [short and clear]. before it went to the publisher. Although he called Poincaré «the greatest mathematician in the world», he told Hilbert, «I could not bring myself to publish things in the form in which Poincaré publishes them».». [12,ch.VI,p.40-41]

Such an exacting and very appropriate for inhabitants of Heaven Minkowski's attitude to the quality of his publications, stated in 1892 during the work in Bonn and not changed in the future, turned to be the fatal circumstance in the business of acquaintance of contemporaries and descendants with his discoveries in electrodynamics…

At Hilbert's suggestion, Max Born was entrusted by Mrs. Minkowski with the editing of her husband's physics papers. One of these Born had to reconstruct from the barest notes. He also carried on his teacher's work with a paper of his own in which he presented a new and rigorous method for calculating the electromagnetic self-energy of the electron. [12,ch.XV,p.123]

The sudden death of Hermann Minkowski seems to be the decisive event which for a long time kept claims of the atomistic instinct outside the area of its natural applicability, postponing the escape from its influence on conceptions about the nature of sources of the field which are included in the right parts of the equations of a field where the field instinct of Faraday–Maxwell should govern. The planned process of the further geometrization was stopped, and the very idea was rejected and forgotten as foolish and not worth an approval. The ideas of Levi-Civita had hung in the air, not having found the support of physicists.

The great moment of geometrization after the death of Minkowski was reduced (in what Born also took part) to the realization of very effective toolkit providing the guaranteed Lorentz-invariance of all physical values and undoubtedly useful and convenient for development and strengthening of Einstein's program. Such a doctrine opened to physicists a wide field of action on replacement of all without an exception values to their Lorentz-invariant analogues or their components. However, from the mathematical point of view this program, being obviously sufficient for gaining the Lorentz-invariant final values and the equations, being the subject to comparison with experiment, is not, strictly speaking, absolutely necessary for some category of values which appear at the intermediate stages of calculations and drop out of final theoretical predictions. The best known value of this sort is the 4-potential of an electromagnetic field. Values of a similar sort always appear in pairs and are the parts of the 4-scalar expression. It gives the possibility of replacement the group of Lorentz to another one, preserving Lorentz-invariance of this scalar as the whole. Actually, physicists faced a similar situation working with gauge invariance and with the spinor equation of Dirac. At the same time, excessively rigid requirements imposed on a nature of appropriate values and a choice of determining group, remained not overcome. And only in the second half of the XX century theorists started talking about redundancy in a set of relativistic and quantum postulates, imposed on the «complete» physical theory.

On very strange, at the first sight, an offer of Hilbert all unpublished physical works, notes and other materials on this subject, left after Minkowski's death were given to Born for studying and estimation of their importance, and later, the possibility and expediency of their edition.

Taking the decision, Hilbert couldn't know that:

• Minkowski was developing a very unexpected and revolutionary approach to electrodynamics and the bases of physical geometry, uniting these two areas of researches in the unified, strictly connected structure;

• Minkowski, remaining loyal to his geometrical approach, aligned geometrical structures under the field equations, successfully combining it with algebraic methods of the theory of groups and their invariants;

• Minkowski based on works of Italian mathematicians and their comprehensive review on «absolute differential calculation», written in 1901 by Ricci and his bright student Levi-Civita [10]. [11]

• Born did not have such an extensive mathematical luggage in combination with clear-cut vision of interrelations, that was absolutely necessary for the qualified estimation of ideas and plans of Minkowski;

• for perusal, decoding and understanding of bare notes, remarks, sketches, bookmarks for memory and so forth, made by Minkowski during the preliminary work for their subsequent use, for restoration in all this abundance the contents coded by Minkowski, it was necessary to be, at least, the second Minkowski;

• without the qualified help of the mathematicians, well acquainted with the methods of Minkowski's work, the chances on the successful performance of the mission assigned to Born were obviously infinitesimal;

• everything, that Born would write on this subjects, would be taken by the scientific world as the copy from the ideas and plans of Minkowski.

For comparison let us look at a reference pattern of Maxwell's attitude to the works of his predecessors and colleagues, of performance of his human and professional duty…

The destiny of Kavendish's works on electricity is as surprising, as the destiny of their author… Scientific activity of Kavendish contrasted sharply with prevailing then an ideal of the scientist-gentleman devoting to intriguing experiments only the leisure-times: he was completely absorbed in his researches. But his contemporaries and colleagues from the London Royal society knew nothing about the biggest part of the intense researches of Kavendish. It became known only after more than fifty years after the death of a scientist. The circumstances of the publication of works of Kavendish on electricity were such, that the idea as if the history in the best way tried to correct its own mistake involuntarily arises…

In 1870 one of descendants of «the great eremite» duke of Devonshire, being the patron of the Cambridge University, offered its management to allocate means for the foundation of specialized physics lab and an establishment of a position of the professor of experimental physics. J. C. Maxwell whom we still shall meet in a later chapter became the first «Kavendish» professor. Maxwell started energetically the construction of laboratory and at its grand opening in 1874 duke of Devonshire gave to the first director a pack of manuscripts of Kavendish who gave the name to the lab, asking to analyze them and to estimate the importance of their contests. It seemed, that for the physicist of such a category as Maxwell, it was the boring, minor work which was only taking away his time from the study of his favorite theoretical physics. However shortly after the first acquaintance with manuscripts of Kavendish Maxwell was carried away by this new, unusual to him activity. The manuscripts appeared to be the true treasury in which the results of numerous, varied and witty experiments on electricity were stored. Maxwell wrote to one of the colleagues:

«…In his manuscripts Kavendish showed his acquaintance with the laws of parallel and consecutive combination of conductors… He carried out very extensive researches in the field of conductivity of salt solutions in tubes which can be assimilated to the wires of different metals. It seems, that he deserved even the greater honour, as he outstripped Ohm before the direct currents were open. His measurements of capacity will make us sweat in Kavendish's lab., before we reach the point where he has stopped.»

The work on research of manuscripts of Kavendish took Maxwell much more time, than he had assumed. It was not stopped even despite of sharp health worsening of the scientist. Already being fatally ill Maxwell continued to prepare the edition of works of Kavendish on electricity. This edition was issued in 1879, shortly before the death of Maxwell.

The creator of the theory of an electromagnetic field approached creatively the publication of the works of his compatriot. Not only did he put the hand-written materials in a good order, he also supplied them with their comprehensive comments, many of which can be considered as small independent scientific researches. Maxwell prefaced the text of Kavendish's works with a sketch of life and activity of the scientist in which he especially in details illustrated researches on electricity. Basing on the analysis of manuscripts of Kavendish, as well as on the testimonies of his contemporaries Maxwell made almost an artistic description of the laboratory where the scientist-eremite carried out his electric experiments. [13]

Simultaneously with preparation for the edition of Henry Kavendish's manuscripts and cares about the accomplishment of the future «cradle of geniuses» – Kavendish's laboratory, Maxwell worked on the second edition of his «Bible of Electricity» – «A Treatise on Electricity and Magnetism», the first two-volume edition of which was published in 1873 and was rapidly spread. Illness and death stopped the work. Maxwell managed to prepare only nine chapters of thirty. The comparison with the first edition shows, that Maxwell planned the considerable remaking of the book. All the nine chapters were written anew. The second and the third editions were published under the edition of Maxwell's students – V. Nive and J. J. Thomson. [14]

I hasten to warn you against a false estimation of a role of Maxwell being very busy while promoting his work of preparation of the second edition of «Treatise». In fact, all these cares not so distracted, as actually positively softened the oppressing him feeling of incompleteness of the work done, associated with an absence of adequate answers to key questions he faced about the nature of the field and its sources (currents): – I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric. [21] The art of escape of these fundamental questions was still not known to Maxwell…

Maxwell, while only starting the construction of the field theory, already in his first work of a theoretic-field cycle «About Faraday's power lines», says that he has an intention only to show, how by direct application of ideas and method of Faraday the mutual relations of various categories of the effects opened by him can be found out in the best way. [13] Both in this, and in all subsequent works on the theory of an electromagnetic field, including «Treatise», he repeatedly gives due to the ideas of Faraday. The reader can even get the idea, that Maxwell only expresses the physical ideas of Faraday with the help of mathematical form: – Both in this, and in subsequent works Maxwell aimed to transfer the physical researches of Faraday to the language of mathematical formulas. [22,Ch.VII]

To these reference pattern presented to us by Genius of Maxwell, the example of the attitude of another Giant – Henri Poincaré to the ideas of Lorentz, strongly pronounced in 1905 in articles «About Dynamics of Electron» [9. I; 9. II], issued in Russian by Anatoly Alekseyevich Logunov with a lot of deep comments, can be added [15].

Poincaré [9.I]: – …The results obtained by me, are in accordance with all most important points Lorentz received; I tried only to alter and complement them to some extent. The idea of Lorentz is that the equations of an electromagnetic field will not change as a result of some transformation (which I shall name after Lorentz) of the next form: [Lorentz transformations].

Logunov [15]: – Poincaré writes: «The idea of Lorentz», but Lorentz never wrote so before Poincaré. Poincaré stated here his fundamental idea, but attributed it completely to Lorentz. He, probably, as nobody else, always extremely highly appreciated and marked everyone who had given an impetus to his ideas and the pleasure of creativity. Personal priority reasons were absolutely alien to him.

Logunov [15]: – It is necessary to emphasize, that, having established the group character of set of all pure spatial transformations together with transformations of Lorentz, leaving the equations of electrodynamics invariant, Poincaré, thus, discovered an existence in physics the fundamentally new type of symmetry, associated with the group of linear space-time transformations, which he named the Lorentz group.

Complemented by transformations of translations of spatial coordinates and the time Lorenz group forms the maximal group of the space-time transformations, leaving all the equations of movement of particles and physical fields invariant and carries the name of the group of Poincaré given to it subsequently by E. Vigner. Richard Feynman wrote about this: «It was Poincaré who offered to investigate, what was possible to do with equations, without changing their form. The idea to pay attention to properties of symmetry of physical laws belongs namely to him».

Why did Hilbert not involve in this work Arnold Sommerfeld who was well prepared for it in all respects, including ethical and moral components? Had Hilbert shown the typical to him persistence and skill to convince, the inclusion of Sommerfeld in this work would have become a matter of time. Finally, why did Hilbert not find himself personally obliged to take an active part in this work?

• Probably, it was not without the feeling of jealousy to successes of Minkowski from the side of Hilbert, which aroused during the school days in Koenigsberg, when the precocious Hermann Minkowski, although two years younger, was passing David by. That spring, «by virtue of his splendid memory and rapid comprehension» (as Hilbert later reported), Minkowski completed the eight year course at the Altstadt Gymnasium in five and a half years and went on to the local university. [12,ch.I,p.7]

Then, in the spring of 1883, came the announcement that this boy, still only 18 years old, had been awarded jointly with the well-known English mathematician Henry Smith the Grand Prix des Sciences Mathématiques. The impression which the news made in Koenigsberg can be gauged by the fact that Judge Hilbert admonished David that presuming on acquaintance with «such a famous man» would be «impertinence»… From Paris, Camille Jordan wrote to Minkowski: «Work, I pray you, to become a great mathematician». Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski. He was shortly to comment of another shy young mathematician that «with skillful treatment I am sure he would open up», and now he apparently applied such skillful treatment to Minkowski. [12,ch.II,p.12]

• Perhaps, it could be partly explained by the circumstance, that Hilbert those years was occupied with his own works on integrated equations and his place in mathematics to such an extent that he hadn't taken trouble to understand the physical works of Minkowski. He treated them as the worthless for a mathematician attempts to drag confused considerations of physicists into the inner sanctum. Besides, doing so Minkowski not only spent his own time and forces on «unworthy work», but also distracted colleagues and Hilbert himself from the great service to mathematics, instead of being his true assistant.

• It is even possible, that Minkowski's death and the following arrangements were taken by Hilbert as a certain form of betrayal from Minkowski in relation to his mission. The similar reaction of Hilbert was marked later when there was a misfortune with his son Franz. Here are the words of Richard Courant, who together with Hilbert accompanied Franz to clinic: – «From now on, – Hilbert said quietly, – I must consider myself as not having a son». «It was very sad the way he said it, but very determined». [12,Ch.XVII,p.139]

And, there appeared the most important for physics, that lost Minkowski, result of the work done by Born. Not Hermann Minkowski to whom the destiny had not given a possibility even to dispose of his revolutionary ideas and hand notes in the clear to the associates way, but Max Born published his own work on hyperbolic kinematics [3]. But, – he tries to make it look as if the uniqueness of hyperbolic movement causes the kinematics of not deformable (rigid) firm bodies on the basis of postulates of relativity, that is obviously false. Finally, this work of Born threw a shadow on the deep idea of Minkowski about the fundamental importance of the uniqueness of hyperbolic movement in the world of Minkowski and, in a sense, discredited it, having connected it in the memories of descendants with logically vicious concept of firm body of Born.

 The translation from Russian was made by Masha and Natasha Zazerska
Last modifications: November 29 2002
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The Literature Quoted:
3. Born M. Ann. d. Phys., 1909, Bd 30, S. 1
7. Levi-Civita T. Sui campi elettromagnetici puri, bei C. Ferrari, Venezia 1908; Sulle azione meccaniche etc.; Prendiconti d. Pr. Acad. dei Lincei 18, 5a.
9. Poincaré H.,
I   Sur la dynamique de l'électron. – C. R. Acad. Sci., Paris, 1905, v. 140, p. 1504
II   Sur la dynamique de l'électron. – Rend. Pal., 1906, v. 21, p. 129
10. Ricci G., Levi-Civita T. Math. Ann. 1901, v. 54, p. 125
11. Pais A. The Science and the Life of Albert EINSTEIN. Oxford University Press, 1982
12. Reid C. HILBERT (With an appreciation of Hilbert's mathematical work by Hermann Weyl), Springer – Verlag, 1970
22. Kline M. Mathematics and the Search for Knowledge, Oxford University Press, New York, 1985 (ISBN 0-19-503533-X)
 
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