Academics From Germany

"Ehrenfest had always emphasized the importance of Klein's lectures to his students, and we read many of those that circulated in lithograph form. They are full of sweeping insights that reveal the interconnections between different mathematical fields: geometry, function theory, number theory, mechanics, and the internal dialectics of mathematics that manifest themselves through the concept of a group. During my stay in G6ttingen, Courant invited me to help prepare Klein's lectures on the history of nineteenth and early twentieth century mathematics for publication, which I did. These first appeared in Springer's well-known "yellow series," and they remain, with all their personal recollections, the most vivid account of the mathematics of this period."

"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."

"From outside Germany, Klein epitomized the cultured German elite. Self-assured, handsome, highly educated, and married to Hegel's granddaughter, he had all the perquisites of a German professor with a devoted cadre of students. Within Germany, however, there was a split between the school of analysis typified by the great and influential German mathematician Karl Weierstrass, and the proponents of more geometric methods associated with Riemann. Klein had identified himself, and his students, with the latter, and thereby contributed to widening the rift—for Klein's enthusiasm was the sort that divides as much as it unifies."

"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."

"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.)"

"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."

"In ordinary geometry a surface is conceived as a locus of points; in Lie's geometry it appears as the totality of all the spheres having contact with the surface."

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

"Every one knows that the fine phrase "God geometrizes" is attributed to Plato, but few know where this famous passage is found, or the exact words in which it was first expressed. Those who, like the author, have spent hours and even days in the search of the exact statements, or the exact references, of similar famous passages, will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations, with their exact references, bearing on one of the most time-honored, and even today the most active and most fruitful of all the sciences, the queen-mother of all the sciences, that is, mathematics."

"Literametrics' may become to philology, what 'Biometrics' has already become to the biological sciences."

"Dedekind's language in introducing irrational numbers leaves a little to be desired. He introduces the irrational α as corresponding to the cut and defined by the cut. But he is not too clear of where α comes from. He should say that... α is no more than the cut. ...Heinrich Weber told Dedekind this, and in a letter of 1888 Dedekind replied that... α is not the cut itself but something distinct, which corresponds to the cut and brings about the cut. Likewise, while the rational numbers generate cuts, they are not the same as the cuts. He says we have the mental power to create such concepts."

"Julius Wilhelm Richard Dedekind stands out as one of the most prominent contributors of the 19th century to the theory of algebraic numbers. He wrote various important memoirs on the binomial equation and on the theory of modular and Abelian functions, but is best known for his treatises Was sind und was sollen die Zahlen? (1888) and Stetigkeit und irrationale Zahlen (1872). In the latter work he set forth his idea of the Schnitt (cut) in relation to irrational numbers,—an idea he had in mind as early as 1858."

"The beauty of [ Eudoxus' ] theory of proportions [ expounded in Book V of Euclid's Elements ] was its adaptability to this new climate. ...The length \sqrt2 is determined by the two sets of positive rationalsL_\sqrt2 = \{r: r^2 < 2\}, \qquad U_\sqrt2 = \{r: r^2 > 2\}Dedekind... decided to let \sqrt2 be this pair of sets! In general, let any partition of the positive rationals into sets L, U such that any member of L is less than any member of U be a positive real number. This idea, now known as the Dedekind cut, is more than just a twist of Eudoxus; it gives a complete and uniform construction of all real numbers, or points on a line, using just the discrete, finally resolving the fundamental conflict in Greek mathematics."

"The tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line... by means of numbers. This assumption is... equivalent to the assertion that a perfect correspondence can be established... The great success of analytic geometry... gave this assumption an irresistible pragmatic force. It was essential to include this principle... But how? Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. ...The very vagueness of all intuition renders such a substitution... highly acceptable. ... On the one hand there was the logically consistent concept of a real number and its aggregate, the arithmetic continuum; on the other hand, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two... to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."

"The above comparison of the domain R of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Everything must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains."

"In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.e., in the following principle: "If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions." ...every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed."

"Although the real theory might have been less useful than the complex in obtaining properties of special functions, its significance for the development of mathematics as a whole has been incomparably greater. It was in the real variable that the necessity for a rigorous theory of the number system of analysis was first recognized. ...the reconstruction of the real number system by Weierstrass in the 1860's and by Dedekind and Cantor in the 1870's led in the last three decades of the nineteenth century, to a profound reconsideration of the nature of all mathematical reasoning. This in turn initiated some of the most searching examinations of all deductive reasoning since the days or Aristotle. Thus the theory of the functions of a real variable since the 1870's has increasingly acquired more than merely a local interest: its problems, solved and unsolved, are significant in fields far distant from technical mathematics."

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this."

"If a is any definite number, then all numbers of the system R fall into two classes, A1 and A2, each of which contains infinitely many individuals; the first class A1 comprises all numbers a1 that are < a, the second class A2 comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second. In every case the separation of the system R into the two classes A1, A2 is such that every number of the first class A1 is less than every number of the second class A2."

"The way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes—which itself is nowhere carefully defined—and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself."

"That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers."

"In discussing the notion of the approach of a variable magnitude to a fixed limiting value, [Dedekind] had recourse, as had Cauchy before him, to the evidence of the geometry of continuous magnitude. ...Dedekind's approach was somewhat different from that of Weierstrass, Méray, Heine, and Cantor in that, instead of considering in what manner the irrationals are to be defined so as to avoid the vicious circle of Cauchy, he asked himself... what is the nature of continuity? ...The philosophy and mathematics of Leibniz had led him to agree with Galileo that continuity was a property concerning conjunctive aggregation, rather than a unity or coincidence of parts. Leibniz had regarded a set as forming a continuum if between any two elements there was always another element of the set. ...Ernst Mach likewise regarded this property of denseness of an assemblage as constituting its continuity, but... rational numbers... possess the property of denseness and yet do not constitute a continuum. Dedekind...found the essence of... continuity, not by a vague hang-togetherness, but in the nature of the division of the line by a point. ...in any division of the points into two classes such that every point of the one is to the left of every point of the other, there is one and only one point which produces this division. This is not true of the ordered system of rational numbers."

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic."

"Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated... as characteristic of a body of numbers [Zahlkőrper] and which consists in this, that the four fundamental operations are always performable with any two individuals in R, i.e., the result is always an individual of R, the single case of division by the number zero being excepted."

"The system R forms a well-arranged domain of one dimension extending to infinity on two opposite sides. What is meant by this is sufficiently indicated by my use of expressions borrowed from geometric ideas; but just for this reason it will be necessary to bring out clearly the corresponding purely arithmetic properties in order to avoid even the appearance as if arithmetic were in need of ideas foreign to it."

"The modern theory of functions of one real variable was first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and Heine, and then carried further, principally, by Weierstrass, Schwarz, Du Bois-Reymond, Thomae, and Darboux. Hankel established the principle of the condensation of singularities; Dedekind and Cantor gave definitions for irrational numbers..."

"If a, c are two different numbers, there are infinitely many different numbers lying between a, c."

"As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis."

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858."

"What advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself."

"One cannot grasp freedom in faith without hearing simultaneously the categorical imperative: One must serve through bodily, social and political obedience the liberation of the suffering creation out of real affliction. ... Consequently, the missionary proclamation of the cross of the Resurrected One is not an opium of the people which intoxicates and incapacitates, but the ferment of new freedom. It leads to the awaking of that revolt which, in the "power of the resurrection" ... follows the categorical imperative to overthrow all conditions in which man is a being who labors and is heavily laden,"

"Christian identity can be understood only as an act of identification with the crucified Christ, to the extent to which one has accepted the proclamation that in him God has identified himself with the godless and those abandoned by God."

"Schlick ( [Wende] p.8 ) interprets Wittgenstein's position as follows: philosophy "is that activity by which the meaning of propositions is established or discovered" ; it is a question of "what the propositions actually mean. The content, soul, and spirit of science naturally consist in what is ultimately meant by its sentences; the philosophical activity of rendering significant is thus the alpha and omega of all scientific knowledge"."

"The Vienna Circle was a discussion group of philosophically interested specialists who came together in 1923 and from 1925 to 1936 met regularly once a week in an institute of Vienna University. These gatherings were conducted by Moritz Schlick, the physicist and philosopher who was appointed professor of the philosophy of inductive sciences in 1922. Over the years, members included Hans Hahn, Otto Neurath, Philipp Frank, Viktor Kraft, Herbert Feigl, Friedrich Waismann, Rudolf Carnap, Kurt Godel, Karl Menger, Bela Juhos and others. There was no conscious aim of radically revising traditional views on the task and place of philosophy, but the members were on the whole well aware that current findings of research into the foundations of logic, mathematics and the natural sciences had important philosophic consequences. Among subjects for discussion were Wittgenstein's Tractatus, the possibility of reducing all concepts of science to what is directly given in experience, the setting up of a criterion of meaningfulness for non-logical utterances, the character of the basic propositions of empirical science, and the devising of a meta-language for the syntactic analysis of scientific language systems."

"The members of the Vienna Circle (Moritz Schlick, Rudolf Carnap, , Hans Hahn, , Fritz Waismann, Kurt Godel, Otto Neurath and others) are working out a ‘Logical Empiricism’. Following Mach and Poincare, but above all Russell and Wittgenstein, all the sciences are treated uniformly. Carnap’s Logischer Aufbau der Welt (1928) shows in which direction future systematic work will move. Wittgenstein’s Tractatus Logico- Philosophicus (1921) clarified, among other things, the position of logic and mathematics; besides the statements that make additions to what is meaningful, there are the ‘tautologies’ that show us which transformations are possible within language. By its syntax the language of science excludes anything that is meaningless from the very beginning."

"The 'physical' does not mean any particular kind of reality, but a particular kind of denoting reality, namely a system of concepts in the natural sciences which is necessary for the cognition of reality. 'The physical' should not be interpreted wrongly as an attribute of one part of reality, but not of the other ; it is rather a word denoting a kind of conceptual construction, as, e.g., the markers 'geographical' or 'mathematical', which denote not any distinct properties of real things, but always merely a manner of presenting them by means of ideas."

"Philosophy is not a system of propositions, and not a science."

"Philosophy... is that activity by which the meaning of propositions is established or discovered. Philosophy elucidates propositions, science verifies them. In the latter we are concerned with the truth of statements, but in the former with what they actually mean."

"Stakeholder theory regards the firm as a nexus of stakeholders, commonly defined as groups or individuals who can affect or are affected by the achievement of the firm's goals (Freeman 1984). Depending on the ground for examining a firm's stakeholders, we can differentiate three approaches (Jones 1995):"

"Homburg, Workman, and Krohmer (1999) examine marketing’s influence within the firm and, in a survey of U.S. and German companies, find that marketing had substantial influence — at least ten years ago. They also find that marketing’s influence is related to (1) external contingency variables, such as the frequency and unpredictability of market-related changes; (2) competitive strategies; and (3) institutional determinants, such as whether the chief executive officer (CEO) has a marketing background."

"Cannon and Homburg (2001) have analyzed the customer firm costs, and concluded that lowering customer cost can be a key way to create value for customers and gain larger customer shares."

"We define CSR as a firm's voluntary consideration of stakeholder concerns both within and outside its business operations."

"Homburg and Pflesser [2000] develop a multilayer model of market-oriented organizational culture. Following the model by Schein [1984] and framing a construct consisting of the four latent variables (1) shared basic values supporting market orientation, (2) norms for market orientation, (3) artifacts of market orientation, and. (4) market-oriented behaviors, Homburg and Pflesser [2000] provide evidence that culture influences market performance and indirectly also financial performance."

"Despite the high relevance of corporate social responsibility (CSR) in current business practice and the considerable research on CSR outcomes in consumer markets, investigations of its influence on organizational business relationships are scarce. Relying on instrumental stakeholder theory the authors develop and empirically test a framework of the influence of a supplier's CSR engagement on organizational customer outcomes. Findings from an examination of 200 cross-industry supplier-customer dyads reveal positive effects of two facets of a supplier's CSR efforts on customer loyalty through distinct mechanisms. Business practice GSR fosters customers' trust, whereas philanthropic CSR strengthens customer-company identification. The authors distinguish a supplier's actual GSR engagement and customers' perception of these GSR activities. In addition, they consider central contingency factors reflecting uncertainty and dependence in business-to-business relationships that determine the effectiveness of GSR."

"The notion of corporate social responsibility (CSR) has gained momentum and is currently of strategic importance for many companies. Among Fortune 500 companies, as many as 90% have explicit CSR initiatives, more than half publish a separate annual CSR report, and most have senior executives responsible for CSR (Luo and Bhattacharya 2009; McKinsey & Company 2009)."

"Cross-functional KAM companies stand out with respect to both performance in the market and adaptiveness."

"[Key Account Management is] the extent to which an organization achieves better relationship outcomes for its Key Accounts than for its average accounts."

"Marketing Management: A Contemporary Perspective provides a fresh new perspective on marketing from some of the leading researchers in Europe. The book offers students and practitioners the comprehensive coverage they need to make the right decisions to create and implement highly successful marketing strategies. This exciting new book combines scholarly international research with relevant and contemporary examples from markets and brands across the world."

"[Strategic account management programs include] special activities... such as pricing, products, services, distribution, and information sharing’ and that they involve ‘in addition to marketing and sales, functional groups such as manufacturing, research and development, and finance."

"Norms guide market-oriented behaviors within organizations. More specifically, we focus on market orientation as the concrete object of these norms. If, for example, the members of an organization share the value of openness of internal communication, a specific norm related to that value is the openness of market-related internal communication. As another example, the norm of market-related responsibility of employees is a specification of the more general shared value of responsibility of the employees... The difference between values and norms is that norms guide behaviors in a specific context, whereas values represent general guidelines."