mathematicians-from-germany

"The domain, over which the language of analysis extends its sway, is, indeed, relatively limited, but within this domain it so infinitely excels ordinary language that its attempt to follow the former must be given up after a few steps. The mathematician, who knows how to think in this marvelously condensed language, is as different from the mechanical computer as heaven from earth."

"The analysis which is based upon the conception of function discloses to the astronomer and physicist not merely the formulae for the computation of whatever desired distances, times, velocities, physical constants; it moreover gives him insight into the laws of the processes of motion, teaches him to predict future occurrences from past experiences and supplies him with means to a scientific knowledge of nature, i.e. it enables him to trace back whole groups of various, sometimes extremely heterogeneous, phenomena to a minimum of simple fundamental laws."

"Dirichlet was not satisfied to study Gauss' "Disquisitiones arithmeticae" once or several times, but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never placed on the shelf but had an abiding place on the table at which he worked....Dirichlet was the first one, who not only fully understood this work, but made it also accessible to others."

"Out of the interaction of form and content in mathematics grows an acquaintance with methods which enable the student to produce independently within certain though moderate limits, and to extend his knowledge through his own reflection. The deepening of the consciousness of the intellectual powers connected with this kind of activity, and the gradual awakening of the feeling of intellectual self-reliance may well be considered as the most beautiful and highest result of mathematical training."

"Just as the musician is able to form an acoustic image of a composition which he has never heard played by merely looking at its score, so the equation of a curve, which he has never seen, furnishes the mathematician with a complete picture of its course. Yea, even more: as the score frequently reveals to the musician niceties which would escape his ear because of the complication and rapid change of the auditory impressions, so the insight which the mathematician gains from the equation of a curve is much deeper than that which is brought about by a mere inspection of the curve."

"It is true that mathematics, owing to the fact that its whole content is built up by means of purely logical deduction from a small number of universally comprehended principles, has not unfittingly been designated as the science of the self-evident [Selbstverständlichen]. Experience however, shows that for the majority of the cultured, even of scientists, mathematics remains the science of the incomprehensible [Unverständlichen]."

"The true mathematician is always a good deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have intellectually created an ideal world, which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world, except he who knows it."

"A peculiar beauty reigns in the realm of mathematics, a beauty which resembles not so much the beauty of art as the beauty of nature and which affects the reflective mind, which has acquired an appreciation of it, very much like the latter."

"If in Germany the goddess Justitia had not the unfortunate habit of depositing the ministerial portfolios only in the cradles of her own progeny, who knows how many a German mathematician might not also have made an excellent minister."

"It may be asserted without exaggeration that the domain of mathematical knowledge is the only one of which our otherwise omniscient journalism has not yet possessed itself."

"I have come to the conclusion that the exertion, without which a knowledge of mathematics cannot be acquired, is not materially increased by logical rigor in the method of instruction."

"Mathematical knowledge, therefore, appears to us of value not only in so far as it serves as means to other ends, but for its own sake as well, and we behold, both in its systematic external and internal development, the most complete and purest logical mind-activity, the embodiment of the highest intellect-esthetics."

"I was amazed at how this outstanding connoisseur of Indian knowledge in the field of exact sciences showed himself to be so captive to Cantor’s authority."

"Georg Hamel was born in 1877 in Düren, Germany, and died in 1954 in Landshut, Germany. In 1897, Hamel went to the University of Berlin, where he was taught by Hermann Schwarz and Max Planck, to name two. Subsequently, he went to Göttingen University, where he studied with Felix Klein and David Hilbert. He was awarded a doctorate under the supervision of Hilbert in 1901. The subject of his dissertation was Hilbert’s fourth problem. In 1905, he went to Brno. It was during the period of his work in Brno that his 1905 paper on Hamel bases was written."

"The University of Berlin had been greatly influenced by the successful French research institutes, such as the Ecole Polytechnique, that had been founded by Napoleon. It had, after all, been founded during the French occupation. One of the key mathematical ambassadors was a brilliant mathematician by the name of Peter Gustav Lejeune Dirichlet. Although he was born in Germany in 1805, Dirichlet's family was of French origin. A return to his roots took him to Paris in 1822, where he spent five years soaking up the intellectual activity that was bubbling out of the academies. Wilhelm von Humboldt's brother Alexander, an amateur scientist, met Dirichlet on his travels and was so impressed that he secured him a position back in Germany. Dirichlet was something of a rebel. Perhaps the atmosphere on the streets in Paris had given him a taste for challenging authority. In Berlin, he was quite happy to ignore some of the antiquated traditions demanded by the rather stuffy university authorities, and often flouted their demands to demonstrate his command of the Latin language."

"Gordan, eine in sich geschlossene Individualität, war kräftig und einheitlich im Leben und in der Arbeit. Kein Neuerer in der Wissenschaft: er griff nur an, was seiner Art gemäß war; aber was er angriff, führte er nnermüdlich durch bis zu Ende. Aus dem Stoffe selbst heraus neue kombinatorische Methoden zu schaffen und seine Instrumente kräftig zu handhaben, das war sein mächtiges Können: er war Algorithmiker."

"A queer fellow, impulsive and one-sided. A great walker and talker - he liked that kind of walk to which frequent stops at a beer-garden or a cafe belong. Either with friends, and then accompanying his discussions with violent gesticulations, completely irrespective of his surroundings; or alone, and then murmuring to himself and pondering over mathematical problems; or if in an idler mood, carrying out long numerical calculations by heart. There always remained something of the eternal "Bursche" of the 1848 type about him – an air of dressing gown, beer and tobacco, relieved however by a keen sense of humor and a strong dash of wit. When he had to listen to others, in classrooms or at meetings, he was always half asleep."

"Gordan - anfänglich diesen begrifflichen Deduktionen gegenüber mehr ablehnend: „das ist keine Mathematik, das ist Theologie!" - ist dann zweimal (53), (69) dem diesem Beweise zugrunde liegenden Hilbertschen Endlichkeitssatze nähergetreten, indem er die gegebenen Formen F nach verschiedenen Kriterien in eine Reihe anordnete, die das Bilden eines endlichen Moduls aus ihnen deutlich machte; das erstemal in komplizierterer Weise speziell für die Invariantenformen, das zweitemal allgemein und einfach."

"In seiner eigenen Wissenschaft war es weniger ein Vertiefen in fremde Arbeiten -- denn solche las er sehr wenig -, als ein Überblick über die inneren Zusammenhänge und ein instinktives Gefühl für die Wege und Ziele der mathematischen Bestrebungen, was ihn schon aus kleinen Andeutungen Wertvolles von Minderem scheiden lieb. Aber den auf die Grundlagen gehenden Begriffsentwicklungen ist Gordan nie gerecht geworden: auch in seinen Vorlesungen hat er alle Grunddefinitionen begrifflicher Art, selbst die der Grenze, vollständig gemieden. Sein Vorlesungsprogramm hat sich nur auf die Vorlesungen allgemeiner Art, gelegentlich auch auf binäre Formentheorie, erstreckt; die Übungen waren mit Vorliebe der Algebra entnommen. Über Jacobisches, so über Funktionaldeterminanten, trug er gern vor, nie über Funktionentheoretisches, höhere Geometrie oder Mechanik; auch ließ er keine Seminarvorträge halten. Die Vorlesungen wirkten wesentlich durch die Lebhaftigkeit der Ausdrucksweise und durch eine zum Selbststudium anregende Kraft, eher als durch Systematik und Strenge."

"If now a far-reaching theory has grown . . . I attribute this result primarily to Professor Gordan. I am not here referring to his trenchant and profound labours, which shall be fully reported upon hereafter. In this place I must report what cannot be expressed in quotations or references, namely, that Professor Gordan has spurred me on when I flagged in my labours, and that he has helped me . . . over many difficulties which I should never have overcome alone."

"Der Beweis des Hilbertschen Satzes und anderer Sätze ist sehr abstrakt, aber an sich ganz einfach und darum logisch zwingend. Eben darum leitet diese Arbeit von Hilbert eine neue Epoche der algebraischen Geometrie ein. Ebenso einfach ist dann auch die Anwendung auf die Invariantentheorie, die ich hier noch weniger zergliedern kann. Die ganze Frage der Endlichkeit der Invarianten, welche Gordan seinerzeit nur mit umfangreichen Rechnungen für binäre Formen hatte erledigen können (vgl. oben S. 308), wird hier mit einem Schlage für Formen mit beliebig vielen Veränderlichen gelöst. Ihrer Eigenart entsprechend wurde diese Arbeit zunächst mit sehr verschiedener Stimmung aufgenommen. Mich hat sie damals bestimmt, Hilbert bei nächster Gelegenheit nach Göttingen zu ziehen. Gordan war anfangs ablehnend: „Das ist nicht Mathematik, das ist Theologie.“ Später sagte er dann wohl: „Ich habe mich überzeugt, daß auch die Theologie ihre Vorzüge hat.“ In der Tat hat er den Beweis des Hilbertschen Grundtheorems selbst später sehr vereinfacht (Münchener Naturforscherversammlung 1899)."

"His strength rested on the invention and calculative execution of formal processes. There exist papers of his where twenty pages of formulas are not interrupted by a single text word; it is told that in all his papers he himself wrote the formulas only, the text being added by his friends."

"Ich habe mich davon überzeugt, daß die Theologie auch nützlich sein kann."

"Hilberts Doktor-Vater Ferdinand Lindemann nannte diesen Existenzbeweis „unheimlich“ und Paul Gordan meinte (zitiert nach Otto Blumenthal ‚ Lebensgeschichte‘ in Hilberts ‚Gesammelten Abhandlungen‘, Band 3 (Berlin 1935), S. 388–429, dort S. 394): „Das ist keine Mathematik; das ist Theologie.“ Etwas später milderte Gordan seinen Ausspruch etwas ab und meinte: „Ich habe mich davon überzeugt, daß die Theologie auch nützlich sein kann“. Aber es gab auch viele Mathematiker, die sich nicht überzeugen ließen. Oskar Becker (1889–1964) beispielsweise reagierte sehr heftig und bezeichnete den Hilbert’schen Beweis des Basis-Satzes als „Schleichweg einer Schein-Konstruktion“ (O. Becker in: ‚Mathematische Existenz‘, 1927, op. cit., S. 471)."

"The proof of this theorem of Hilbert's and of others is very abstract, but in itself quite simple and hence logically compelling. And for just this reason this work of Hilbert's ushered in a new epoch of algebraic geometry. But application to invariant theory is just as simple, but I can analyze it here even less. The whole question of the finiteness of the invariants, which Gordan had been able to solve for binary forms only by means of comprehensive calculations (see p. 290), is here solved, with one stroke, for forms with arbitrarily many variables. Because of its uniqueness, this work was first received with very diverse reactions. I had then resolved to draw Hilbert to Goettingen at the earliest opportunity. Gordan at first declined, saying, "It is not mathematics, it is theology" But later he said: "I have convinced myself that even theology has its merits". In fact, Gordan himself later on much simplified Hilbert's basic theorem (Muenchener Naturforscherversammlung 1899)."

"38 Jahre von 1874 an hat Gordan in Erlangen verbracht. Sie sind für ihn gleichmäßig verlaufen: täglich Vorlesungen, Arbeit, und die unentbehrlichen Spaziergänge entweder mit Mitarbeitern … in drastisch lebhaften Zwiegesprächen, unbekümmert um alle Umgebung, oder allein in tiefem Nachdenken und seine Gedanken im Kopfe so fertig verarbeitend, dass er seine Rechnungen zuhause fast ohne Striche [Streichungen] ausführen konnte."

"Das ist keine Mathematik; das ist Theologie."

"... in 1949, a paper appeared by a German mathematician Hans Maass, which raised some rather interesting problems. You see, earlier the automorphic functions and forms — one had essentially thought of functions that were often called holomorphic, analytical. And Maass started studying functions that were not of that nature, but were instead solutions of a certain eigenvalue problem, which had a certain type of behavior with respect to the discrete group which corresponds to the modular forms. Maass also worked essentially just on the modular group and its subgroups, not on general groups."

"The investigations of Siegel on discrete groups of motions of the with a fundamental region of finite volume ... make possible a simple characterization of groups conjugate to the modular group by a minimal condition."

"Let F be a field of characteristic 0, and let F be a finite dimensional vector space over F. Let E denote the algebra of all endomorphisms of V, and let L be any Lie subalgebra of E. Among the algebraic Lie algebras contained in E and containing L, there is one that is contained in all of them, and this is called the algebraic hull of L in E. Here, an algebraic Lie algebra is defined as the Lie algebra of an algebraic group. It is an easy consequence of the definitions that if A and B are algebraic groups of automorphisms of V such that A⊂B then the Lie algebra of A is contained in the Lie algebra of B. Hence the existence of the algebraic hull of L is an immediate consequence of the following basic result: let G be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain L."

"A Lie algebra is said to be algebraic if it is isomorphic with the Lie algebra of an affine algebraic group. In view of the fact that entirely unrelated affine algebraic groups (typically, vector groups and toroidal groups) may have isomorphic Lie algebras, this notion of algebraic Lie algebra calls for some clarification. The most relevant result in this direction is due to M. Goto. It says that a finite- dimensional Lie algebra L over a field of characteristic 0 is algebraic if and only if the image of L under the adjoint representation is the Lie algebra of an algebraic subgroup of the group of automorphisms of L ..."

"... the theory of valuations may be viewed as a branch of topological algebra. In fact, historically speaking, it represents the first invasion of topology, more precisely, of early metric topology, into the domains of algebra. The introduction of metric methods into algebra has been so fruitful that today many of the deeper algebraic theories carry their mark. In this regard, one should distinguish between the classical use in algebra of the natural metric of the real or complex number fields, such as in proving the "fundamental theorem of algebra," and the much more recent use of the far less evident metrics which are derived from arithmetic notions of divisibility and which constitute the principal notion of valuation theory. Such a metric occurs for the first time in Hensel's construction of the p-adic numbers ..."

"... Lie algebras have a significance reaching beyond the domain of algebra, because they play such an important role in the theory of Lie groups. Thus, classical Lie algebra theory is strongly dominated by the fact that the finite-dimensional analytic representations of a simply connected analytic group are identifiable with the finite-dimensional representations of its Lie algebra. In the theory of infinite-dimensional representations, the connection with Lie algebra representations is somewhat tenuous, but it is nevertheless at the core of the major advances made in that theory during the last 30 years."

"Since an algebraic function w(z) is defined implicitly by an equation of the form f(z,w) = 0, where f is a polynomial, it is understandable that the study of such functions should be possible by algebraic methods. Such methods also have the advantage that the theory can be developed in the most general setting, viz. over an arbitrary field, and not only over the field of complex numbers (the classical case)."

"... H. Hasse perceived the connection between complex multiplication and the Riemann hypothesis for congruence zeta functions, which was later proved by A. Weil in a fully general form. This observation led M. Deuring to establish a purely algebraic treatment of complex multiplication of elliptic curves. He could, moreover, along the same line of ideas, determine the zeta functions of elliptic curves with complex multiplication. The definition of zeta function of an algebraic curve defined over an algebraic number field is originally due to Hasse; and Weil is the first contributor to this subject."

"It is difficult to compare a differential geometer with a function theorist, or those working on ordinary and partial differential equations with numerical analysts. Christoffel not only contributed to all these fields, but his interests extended to orthogonal polynomials and continued fractions, and the applications of his work to the foundations of tensor analysis, to geodetical science, to the theory of shock waves, to the dispersion of light. Nevertheless, it is widely recognised, at least in the German speaking countries of Europe, that Riemann was the best mathematician of the 19th century, behind Gauss and ahead of Weierstrass. In our opinion Christoffel's teacher Dirichlet, belongs to the next most important group of mathematicians which includes (in chronological order of birth) Jacobi, Kummer, Kronecker, Dedekind, Cantor and Klein. Christoffel himself should be placed in a second group following these. This second group, which may partly overlap with the former, would include such illustrious names as Möbius, von Staudt, Plücker, Heine, Du Bois-Reymond, Carl Neumann, Lipschitz, Fuchs, Schwarz, Hurwitz and Minkowski."

"In 1966 Siegel oversaw the publication of his collected works in three volumes. He spent the rest of his life editing (and writing) a fourth volume. As the story goes, he burned everything else, fearing that a historian—as he himself had done with Riemann—would get into his papers."

"... when Carl Ludwig Siegel announced that he would hold class on a University holiday, his students left the room empty on the appointed day, hiding nearby to see what he did. “Sure enough, Siegel got up front in the empty room, started in with the beautiful lecture as though he had a full room,” said Merrill Flood *35. After he had continued for a while, “we sheepishly trooped in, and listened to his lecture.”"

"Ein Bourgeois, wer noch Algebra treibt! Es lebe die unbeschrankte Individualitat der transzendenten Zahlen! ["It's a bourgeois, who still does algebra! Long live the unrestricted individuality of transcendental numbers!"]"

"One of the many importants ideas introduced by Minkowski into the study of convex bodies was that of gauge function. Roughly, the gauge function is the equation of a convex body. Minkowski showed that the gauge function could be defined in a purely geometric way and that it must have certain properties analogous to those possessed by the distance of a point from the origin. He also showed that conversely given any function possessing these properties, there exists a convex body with the given function as its gauge function."

"... Siegel kept working well into his eighties, after he had returned to Göttingen."

"I am afraid that mathematics will perish by the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up."

"Let θ be an algebraic integer and assume that all conjugates of θ, except θ itself, have an absolute value less than 1. Then –θ also has this property; on the other hand, θ is real. Without loss of generality, we may therefore suppose θ ≥ 0. Since the norm of θ is a rational integer, we have θ ≥ 1, except for the trivial case θ = 0. Recently, R. Salem ... discovered the interesting theorem that the set S of all θ is closed and that θ = 1 is an isolated point of S. Consequently there exists a smallest θ = θ1 > 1. We shall prove that θ1 is the positive zero of x3 – x – 1 and that also θ1 is isolated in S. Moreover we shall prove that the next number of S, namely the smallest θ = θ2 > θ1, is the positive zero of x4 – x3 – 1 and that θ2 is again an isolated point of S. Since θ1 = 1.324..., θ2 = 1.380..., both numbers are less than 2½; therefore our statements are contained in the following: . Let θ be an algebraic integer whose conjugates lie in the interior of the unit circle; if ±θ ≠ 0, 1, θ1, θ2, then θ2 > 2."

"Ours, according to Leibnitz, is the best of all possible worlds, and the laws of nature can therefore be described in terms of extremal principles. Thus, arising from corresponding variational problems, the differential equations of mechanics have invariance properties relative to certain groups of coordinate transformations."

"The theory of functions of several variables turns out to be essentially more difficult than the theory of one variable because of the existence of points of indeterminacy. In the case n > 1, a mere glance at the poles already indicates a behavior which is completely different from that in the case n = 1. The reason is that, in case n > 1, the poles are not isolated and, in general, there does not exist a Laurent expansion. In a neighborhood of a nonregular point we are forced to view meromorphic functions as quotients of power series."

"As regards quartic surfaces, Rohn has investigated an enormous number of special cases; but a complete enumeration he has not reached. Among the special surfaces of the fourth order the Kummer surface with 16 conical points is one of the most important. The models constructed by Plücker in connection with his theory of complexes of lines all represent special cases of the Kummer surface."

"It has been the final aim of Lie from the beginning to make progress in the theory of differential equations ..."

"Next to the elementary transcental functions the elliptic functions are usually regarded as the most important. There is, however, another class for which at least equal importance must be claimed on account of their numerous applications in astronomy and mathematical physics; these are the hypergeometric functions, so called owing to their connecton with Gauss's hypergeometric series."

"The theory of binary forms and the projective geometry of systems of points on a conic are one and the same, i.e., to every proposition concerning binary forms corresponds a proposition concerning such systems of points, and vice versa. ... Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its pointa are one and the same."

"Es ist eine Mannigfaltigkeit und in derselben eine Transformationsgruppe gegeben; man soll die Mannigfaltigkeit angehören Gebilde hinsichtlich solcher Eigenschaften untersuchen; die durch die Transformationen der Gruppe nicht geändert werden. (Given a manifold with its associated transformation group, one should investigate those structures of the manifold that have properties which are invariant under the transformation group.)"