Mathematicians 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."

"With Klein, even politics has been introduced into the question: he asserts that “It would seem as if a strong naive space intuition were an attribute of the Teutonic race, while the critical, purely logical sense is more developed in the Latin and Hebrew races.” That such an assertion is not in agreement with facts will appear clearly when we come to examples. It is hardly doubtful that, in stating it, Klein implicitly considers intuition, with its mysterious character, as being superior to the prosaic way of logic and is evidently happy to claim that superiority for his countrymen. We have heard recently of that special kind of ethnography with Nazism: we see that there was already something of this kind in 1893."

"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."

"The proof that π is a transcental number will forerver mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone."

"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."

"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."

"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 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..."

"Dedekind's language in introducing irrational numbers leaves a little to be desired. He introduces the irrational &alpha; as corresponding to the cut and defined by the cut. But he is not too clear of where &alpha; comes from. He should say that... &alpha; is no more than the cut. ...Heinrich Weber told Dedekind this, and in a letter of 1888 Dedekind replied that... &alpha; 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."

"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."

"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."

"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."

"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 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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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 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."

"Some of the groundbreaking work in the treatment of n-dimensional geometry—was carried out by Hermann Günther Grassmann. ...Grassmann was responsible for the creation of an abstract science of "spaces," inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as Ausdehnungslehre... Grassmann's suggestion that BA = -AB violates one of the sacrosanct laws of arithmetic... Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions."

"Grassmann's first publication of his new system was made in 1844 in a book entitled "Die Lineale Ausdehnungslehre Ein Neuer Zweig der Mathematik." His novel and fruitful ideas were however presented in a somewhat abstruse and unusual form, with the result, as the author himself states in the preface to the second edition issued in 1878, that scarcely any notice was taken of the book by Mathematicians. He was finally convinced that it would be necessary to treat the subject in an entirely different manner in order to gain the attention of the mathematical world. Accordingly in 1862 he published "Die Ausdehnungslehre vollständig und in strenger Form bearbeitet," in which the treatment is algebraic... Since that time his great work has been more fully appreciated, but not even yet, in the opinion of the writer, at its real value."

"One may say without great exaggeration that Grassmann invented linear algebra and, with none at all, that he showed how properly to apply it to geometry. ...He ...anticipated in its most important aspects Peano's treatment of the natural numbers, published 28 years later. ...A feature of Grassmann's work, far in advance of the times, is the tendency towards the use of the implicit definition. ...The definition of a linear space (or vector space) came into mathematics, in the sense of becoming widely known, around 1920, when Hermann Weyl and others published formal definitions. ...Grassmann did not put down a formal definition—again, the language was not available—but there is no doubt that he had the concept."

"It was natural that Grassmann chose to introduce his system, not by means of a paper, but rather by means of a long and complicated book. ...such ideas as Grassmann's form of the scaler (dot) and vector (cross) products... have counterparts in modern vector analysis."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of descriptive geometry by Monge in 1795. The quaternions of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"The exchange theorem... is sometimes called the Steinitz exchange theorem after Ernst Steinitz... The result was first proved Hermann Günther Graßmann..."

"A work on tidal theory... led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagrange's work."

"It is clear... that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution."

"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept."

"I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results."

"The concept of centroid as sum led me to examine Möbius' Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and... the only point of contact which my new system of analysis had with the one that was already known."

"As the great generality of Grassmann's processes—all results being obtained for n-dimensional space—has been one of the main hindrances to the general cultivation of his system, it has been thought best to restrict the discussion to space of two and three dimensions."

"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative. Zero can never be a unit."

"From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all."

"Geometry can in no way be viewed... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry."

"The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line. This then was the first step which led to a new branch of mathematics... I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea."

"While I was pursuing the concept of geometrical product, as this idea was established by my father... I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third... and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC.] From this harmony I came to see a whole new area of analysis was opening up which could lead to important results."

"As I was reading the extract from your paper in the geometric sum and difference... I was struck by the marvelous similarity between your results and those discoveries which I made even as early as 1832... I conceived the first idea of the geometric sum and difference of two or more lines and also of the geometric product of two or three lines in that year (1832). This idea is in all ways identical to that presented in your paper. But since I was for a long time occupied with entirely different pursuits, I could not develop this idea. It was only in 1839 that I was led back to that idea and pursued this geometrical analysis up to the point where it ought to be applicable to all mechanics. It was possible for me to apply this method of analysis to the theory of tides, and in this I was astounded by the simplicity of the calculations resulting from this method."

"The wonderful and comprehensive system of Multiple Algebra invented by Hermann Grassmann, and called by him the Ausdehnungslehre or Theory of Extension, though long neglected by the mathematicians even of Germany, is at the present time coming to be more and more appreciated and studied. In order that this system, with its intrinsic naturalness, and adaptability to all the purposes of Geometry and Mechanics, should be generally introduced to the knowledge of the coming generation of English-speaking mathematicians, it is very necessary that a text-book should be provided, suitable for use in colleges and universities, through which students may become acquainted with the principles of the subject and its applications."

"[A] simpler symbol for the last function, or rather for its reciprocal... will... be... more convenient... Let...so that T is now the unknown function of the temperature... T1, T2, &c. shall represent... values of this function, corresponding to... t1, t2, &c."

"[T]he second fundamental theorem in the mechanical theory of heat... appropriately... called the theorem of the equivalence of transformations..: If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generation of the quantity of heat Q of the temperature t from work, has the equivalence-value'and the passage of the quantity of heat Q from the temperature t1 to the temperature t2, has the equivalence-value'wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected."

"We proceed now to the consideration of non-reversible cyclical processes. ...[W]e obtain the following theorem, which applies generally to all cyclical processes, those that are reversible forming the limit:—The algebraical sum of all [non-reversible] transformations occurring in a cyclical process can only be positive."

"If to the last expression we give the form...the passage of the quantity of heat Q, from the temperature t1 to the temperature t2, has the same equivalence value as... the transformation of the quantity Q from heat at the temperature t1 into work, and from work into heat at the temperature t2."