Quotes by and about David Hilbert
1. A quote from Thomas Arthur Alan Broadbent.
Source: T A A Broadbent, Review: Gesammelte Abhandlungen. II by David Hilbert; Gesammelte Abhandlungen. III by David Hilbert, The Mathematical Gazette 52 (379) (1968), 61-62. (1965).
Under Hilbert, Göttingen reached its peak as one of the great mathematical centres of the world. No one in recent years has surpassed his dual capacity, for seeing and overcoming the central difficulty of some major topic, and for propounding new problems of vital importance.2. A quote from John David North.
Source: J D North, Review: Gesammelte Abhandlungen, by David Hilbert (1970); Hilbert Gedenkband, by K Reidemeister (1971), The British Journal for the History of Science 6 (2) (1972), 208-209.
It is now nearly thirty years since David Hilbert died, in Gottingen, at the age of eighty-one, the greatest mathematician of his generation. His attitudes and methods were so influential in the world at large that there can scarcely be a mathematician who is not familiar with some aspect of his work. With the exception of two books published alone (Grundlagen der Geometrie and Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen), four more published with the collaboration of others (R Courant on mathematical physics, W Ackermann on logic, S Cohn-Vossen on geometry, and P Bernays on mathematical foundations), and about twenty papers, most of which would have meant duplication, Hilbert's monumental collected works were first published in three volumes between 1932 and 1935.3. A quote from Victor Vinnikov.
Source: V Vinnikov, We shall know: Hilbert's apology, Math. Intelligencer 21 (1) (1999), 42-46.
On January 23, 1930, David Hilbert reached the mandatory retirement age of 68. Among the many honours bestowed upon him, he was made an "honorary citizen" of his native town of Königsberg. The honorary citizenship was presented to Hilbert on September 8, 1930, at the meeting of the Society of German Scientists and Physicians which was held that year in Königsberg. Hilbert's acceptance address was entitled "Natural Philosophy and Logic"4. A quote from Rüdiger Thiele.
Source: R Thiele, Hilbert and his twenty-four problems, in Mathematics and the historian's craft (CMS Books Math./Ouvrages Math. SMC, 21, Springer, New York, 2005), 243-295.
Hilbert, best known for his axiomatic foundations of mathematics and his formalist viewpoint, knew the value of important problems. As his disciple and biographer Otto Blumenthal (1876-1944) put it: "Hilbert is the man of problems. He collects and solves existing problems; he poses new ones." Indeed, it is just by the solution of concrete problems that mathematics will be developed; in the end, problem solving and theory building go hand in hand. That's why Hilbert risked offering a list of unsolved problems instead of presenting new methods or results, as was usually done at meetings. "He who seeks for methods without having a definite problem in mind seeks for the most part in vain," Hilbert told his Paris audience.5. A quote from David E Rowe.
Source: D E Rowe, From Königsberg to Göttingen: a sketch of Hilbert's early career, Math. Intelligencer 25 (2) (2003), 44-50.
Probably the most memorable moments for Hilbert, when he thought back on the years from 1886 to 1892, were the almost dally walks with Hurwitz. He recalled how together they wandered through nearly every comer of mathematics, with his friend and former teacher acting as guide. No doubt they often discussed ideas that Hilbert was working on in the context of his lecture courses. K6nigsberg was just about the last place one could find mathematics students during the late 1880s and early 1890s, and Hilbert's courses were often attended by no more than two or three auditors, sometimes even fewer! He complained about these circumstances occasionally, but never seemed to be really bothered. His goal was to become a truly universal mathematician. Indeed, his lecture courses, supplemented by nearly dally discussions with Hurwitz, served as his primary vehicle for attaining that purpose, and they spanned practically every area of higher mathematics of the day: from invariant theory, number theory, and analytic, projective, algebraic, and differential geometry, to Galois theory, potential theory, differential equations, function theory, and even hydrodynamics. During his entire nine years on the Königsberg faculty he never lectured on any subject more than once, with the exception of a one-hour course on determinants. Surely Hilbert read a great deal, but more important to him still were the opportunities to talk about mathematics. Thus, right from the start, the power of the spoken word that stimulates the mathematical imagination played a central role in his work. Without it, his phenomenal success in Göttingen simply would not have been possible.6. Letter from Hilbert to Klein, 24 July 1890.
Source: D E Rowe, Hilbert's early career: encounters with allies and rivals, Math. Intelligencer 27 (1) (2005), 72-82.
It seems to me that the mathematicians of today understand each other far too little and that they do not take an intense enough interest in one another. They also seem to know--so far as I can judge--too little of our classical authors (Klassiker); many, moreover, spend much effort working on dead ends.7. Extract from the Preface of Hilbert's 'Zahlbericht'.
Thus we see how far arithmetic, the Queen of mathematics, has conquered broad areas of algebra and function theory to become their leader. The reason that this did not happen sooner and has not yet developed more extensively seems to me to lie in this, that number theory has only in recent years become known in its maturity ... Nowadays the erratic progress characteristic of the earliest stages of development of a subject has been replaced by steady and continuous progress through the systematic construction of the theory of algebraic number fields. The conclusion, if I am not mistaken, is that above all the modern development of pure mathematics takes place under the banner of number: the definitions given by Dedekind and Kronecker of the concept of number lead to an arithmetization of function theory and serve to realize the principle that, even in function theory, a fact can be regarded as proven only when in the last instance it has been reduced to relations between rational integers8. A quote from David E Rowe.
Source: D E Rowe, Mathematics made in Germany: on the background to Hilbert's Paris lecture, Math. Intelligencer 35 (3) (2013), 9-20.
Despite his impressive achievements before 1900, Hilbert's glory years still lay ahead of him. His was a career that took many surprising turns, nearly all of which enhanced his fame, power, and influence. No single event contributed more to these than his Paris lecture with its famous list of unsolved problems. With the advantage of hindsight, this episode stands out as a virtual moment of metamorphosis in Hilbert's career, including the whole spectrum of mathematical interests that engaged him both before and afterward. Even within the close-knit circle of his friends and Mitarbeiter, none could well have imagined that the author of the 'Zahlbericht' would after the turn of the century virtually turn his back on algebra and number theory to take up entirely new problems in geometry, integral equations, the calculus of variations, and mathematical physics. At the same time, Hilbert would set forth a vision for a new axiomatic approach to mathematics and physics, a veritable modern-day Leibnizian mathesis universalis. It was the stuff from which legends are born.9. An extract from Hilbert's 1928 address to the International Congress.
Source: Guillermo P Curbera, Mathematicians of the world, unite! The International Congress of Mathematicians - a human endeavor (A K Peters Ltd., Wellesley, MA, 2009).
10. Two quotes from Hermann Weyl.
The International Congress of Mathematicians had problems over German members following World War I. In 1928 the Congress was held in Bologna. Germans were invited (following two congresses when they were not) but some chose to boycott the congress. Hilbert addressed this issue in his opening speech:
It makes me very happy that after a long, hard time all the mathematicians of the world are represented here. That is as it should be and as it must be for the prosperity of our beloved science. Let us consider that we as mathematicians stand on the highest pinnacle of the cultivation of the exact sciences. We have no other choice than to assume this highest place, because all limits, especially national ones, are contrary to the nature of mathematics. It is a complete misunderstanding of our science to construct differences according to people and races, and the reasons for which this has been done are very shabby ones. Mathematics knows no races ... For mathematics the whole cultural world is a single country.
Source: H Weyl, Hermann Obituary: David Hilbert. 1862-1943, Obituary Notices of Fellows of the Royal Society of London 4 (1944), 547-553.
10.1. Hilbert was of slight build. Above the small lower face with its goatee there rose the dome of a powerful, in later years bald, skull. He was physically agile, a tireless walker, a good skater, and a passionate gardener. Until 1925 he enjoyed good health. Then he fell ill of pernicious anaemia. Yet this illness only temporarily paralyzed his restless activity in teaching and research. He was among the first with whom the liver treatment, inaugurated by G R Minot at Harvard, proved successful; undoubtedly it saved Hilbert's life at that time.
10.2. In Hilbert simplicity and rigour go hand in hand. The growing demand for rigour, imposed by the critical reflections of the nineteenth century upon those parts of mathematics which operate in the continuum, was felt by most investigators as a heavy yoke that made their steps dragging and awkward. Full of longing and with uneasiness they looked back upon Euler's era of happy-go-lucky analysis. With Hilbert rigour figures no longer as enemy, but as promoter of simplicity. Yet the secret of Hilbert's creative force is not plumbed by any of these remarks. A further element of it, I feel, was his sensitivity in registering hints which revealed to him general relations while solving special problems. This is most magnificently exemplified by the way in which, during his theory of numbers period, he was led to the enunciation of his general theorems on class fields and the general law of reciprocity.
JOC/EFR November 2014
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