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Bernhard Riemann

50 quotes

“Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”
Bernhard Riemann (1826–1866) first seen 1854 wikiquote, CC BY-SA 4.0 source
Recorded in: Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) p. 55.
“I have... set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“[M]ultiply extended magnitude is capable of different measure-relations, and consequently... space is only a particular case of a triply extended magnitude. ...[A] necessary consequence..: the propositions of geometry cannot be derived from general notions of magnitude, but... properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“[T]here may be several systems of matters of fact which suffice to determine the measure-relations of space—the most important... being that which Euclid has laid down as a foundation.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“These matters of fact are—like all matters of fact—not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is... great, and inquire about the justice of their extension beyond... observation, on the side... of the infinitely great and of the infinitely small.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“In... the development of the notion of a multiply extended magnitude... I am not practised in... undertakings... philosophical... where the difficulty lies more in the notions... than in the construction; and besides... short hints... by... Gauss in his second memoir on Biquadratic Residues..., and in his Jubilee-book, and some philosophical researches of Herbart, I could make use of no previous labours.”
Bernhard Riemann (1826–1866) first seen 1827 wikiquote, CC BY-SA 4.0 source
Recorded in: Ref: Carl Friedrich Gauss, 1) Theoria residuorum biquadraticorum, commentatio secunda (1832) presented to the Göttingen Royal Society, April 1831. English Tr: Wikisource. 2) Disquisitiones generales circa superficies cu…
“Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it "mind-.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“All thinking is, accordingly, formation of new mind masses.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Mind-masses entering the soul appear to us as ideas, the quality of the... [ideas] depending on the inner state of the... [soul].”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Forming mind-masses amalgamate, combine or compound themselves in definite degree, partly with each other, partly with older mind-masses. The manner and strength of these combinations depend on conditions... imperfectly recognised by Herbart... They depend chiefly upon the inner relationship of the mind-masses.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The soul is a compact of mind-masses combined in a most intimate and manifold manner. It grows constantly by accession of mind-masses, and upon these depend its development.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Mind-masses... are imperishable, their combinations are indissoluble; only the relative strength of these combinations is altered by the incoming of new mind masses.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Every entering mind-mass excites all related mind-masses and this excitation is the more powerful the more insignificant the diversity of their inner states (quality).”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“If... part of the related mind-masses hang together among themselves, then these are not only immediately excited but also mediately and consequently in proportion more powerfully than the rest.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The simplest and most common manifestation of the activity of older mind-masses is Reproduction, which consists in the striving of the active mind-mass to engender one similar to itself ...and accordingly strives to produce that form of motion of the matter by which it was formed.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“We now apply these laws of mental processes, to which the explanation of our own inner consciousness leads, to explain the order and adaptation observed on the earth, i.e., to explain Being and historical development.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The substratum of mental activity must be sought only in ponderable matter.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“There remains only the assumption that the ponderable masses within the rigid earth-crust are supporters of the soul-life of the earth.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“It is absurd to assume that upon the rigid earth-crust the organic originated from the inorganic. In order to explain the nascence of the lowest organisms on the earth-crust, one must assume an already existing organising principle or a thought-process, under conditions that would render organic combinations impossible. We must accordingly assume that these conditions are valid only for the life-process in the actual state of the earth's surface, and only so far as we can explain them may we estimate the possibility of life-processes under other conditions.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The adaptation observed in men, animals and plants... one part of this adaptation is explained from a thought-process in the interior of these bodies... another part, however, the adaptation of the organism, by a thought-process in a greater whole.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“[W]e must seek the cause of the adaptations observed in organisms in a unitary thought process in the earth...”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“There is within the limits of our experience no reason to seek the causes of these adaptations in a greater whole. All organisms are designed only for life upon the earth. The state of the earth's crust accordingly contains all (external) reasons of its arrangement. ...They are peculiar (individual). According to all that experience teaches we must assume that they are not repeated on other heavenly bodies.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“[T]here constantly appear new and more complete organisms. We must accordingly seek the cause in a thought process simultaneously progressing to higher order.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“From the standpoint of exact , of the explanation of nature from causes, the assumption of an earth-soul is... an hypothesis for the explanation of Being and of the historical development of the organic world.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“When the body of the lower soul dies," says Fechner, "the higher soul takes it up out of its life of contemplation into its life of ." The souls of perished creatures shall, therefore, form the elements of the soul-life of the earth.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The different thought-processes seem to differ chiefly in respect to their temporal . If plants have souls, then hours and days must be for them what seconds are for us. The corresponding period for the earth-soul, at least for its outward activity, possibly embraces many thousands of years.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Thesis. Finite, Representable. Antithesis. Infinite, System of Notions lying at the limit of the representable.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“I. Thesis. Finite elements of Space and Time. Antithesis. Continuity.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“II. Thesis. Freedom, i.e., not the power absolutely to originate, but to pass judgement between two or more possibilities. Antithesis. Determinism.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“II. Thesis. In order that decision by arbitrary power may be possible in spite of completely definite laws of the action of ideas, one must assume that the psychic mechanism itself has, or at least in its development acquires, the peculiar property of inducing the necessity of these laws. Antithesis. No one can, in case of affairs, abandon the conviction that the future is co-determined by his transactions.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“III. Thesis. A God working in Time. (Government of the world). Antithesis. A timeless, personal, omniscient, al-mighty, all-benevolent God (Providence).”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“IV. Thesis. Immortality. Antithesis. A thing in and by itself endowed with transcendental freedom, radical evil, intelligible character and lying at the basis of our temporal appearance.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“IV. Thesis. Freedom is very well compatible with sound lawfulness of the course of nature. But the concept of a timeless God is then untenable. But the restriction which omnipotence and omniscience suffer through freedom of the creature in the sense above determined, must be eliminated by the assumption of a temporally acting God, of a ruler of the hearts and destinies of men; the concept of Providence must be supplemented and in part replaced by the notion of government of the world. [No Antithesis indicated.]”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“There are no degrees of being, a gradual difference of only states or relations being thinkable. Accordingly, if an agent strives to preserve or to restore itself, then it must be a state or relation.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The word hypothesis has now a somewhat different significance from that given it by Newton. We are now accustomed to understand by hypothesis all thoughts connected with the phenomena. Newton was far from the crude thought that explanation of phenomena could be attained by abstraction.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
Recorded in: Die partiellen Differentialgleichungen der mathematischen Physik (1882) as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914) p. 239
“[[Geometry|[G]eometry]] assumes, as... given.., the notion of space and... first principles of the constructions of space. She gives definitions... merely nominal, while... true determinations appear in... s. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible. From Euclid to Legendre... this darkness was cleared up neither by mathematicians nor... philosophers... The reason... is... that the general notion of multiply extended magnitudes (...space-magnitudes... included) remained... unworked.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Measure-relations can only be studied in abstract notions of quantity, and their dependence on one another can only be represented by formulæ. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geometric representation; and thus it becomes possible to express geometrically the calculated results. In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formulæ, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superficies curvas.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“For Space, when the position of points is expressed by rectilinear co-ordinates, ds = \sqrt{ \sum (dx)^2 }; Space is therefore included in this simplest case. The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression. ...I restrict myself... to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression. ...Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum (dx)^2 }, are... only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses... I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Let us imagine that from any given point the system of shortest lines going out from it is constructed; the position of an arbitrary point may then be determined by the initial direction of the geodesic in which it lies, and by its distance measured along that line from the origin. It can therefore be expressed in terms of the ratios dx0 of the quantities dx in this geodesic, and of the length s of this line. ...the square of the line-element is \sum (dx)^2 for infinitesimal values of the x, but the term of next order in it is equal to a homogeneous function of the second order... an infinitesimal, therefore, of the fourth order; so that we obtain a finite quantity on dividing this by the square of the infinitesimal triangle, whose vertices are (0,0,0,...), (x1, x2, x3,...), (dx1, dx2, dx3,...). This quantity retains the same value so long as... the two geodesics from 0 to x and from 0 to dx remain in the same surface-element; it depends therefore only on place and direction. It is obviously zero when the manifold represented is flat, i.e., when the squared line-element is reducible to \sum (dx)^2, and may therefore be regarded as the measure of the deviation of the manifoldness from flatness at the given point in the given surface-direction. Multiplied by -¾ it becomes equal to the quantity which Privy Councillor Gauss has called the total curvature of a surface. ...The measure-relations of a manifoldness in which the line-element is the square root of a quadric differential may be expressed in a manner wholly independent of the choice of independent variables. A method entirely similar may for this purpose be applied also to the manifoldness in which the line-element has a less simple expression, e.g., the fourth root of a quartic differential. In this case the line-element, generally speaking, is no longer reducible to the form of the square root of a sum of squares, and therefore the deviation from flatness in the squared line-element is an infinitesimal of the second order, while in those manifoldnesses it was of the fourth order. This property of the last-named continua may thus be called flatness of the smallest parts. The most important property of these continua for our present purpose, for whose sake alone they are here investigated, is that the relations of the twofold ones may be geometrically represented by surfaces, and of the morefold ones may be reduced to those of the surfaces included in them...”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“An immediate consequence of these principles of explanation is that the souls of organic beings, i.e., the compacts of mind-masses, arisen during life, continue to exist after death. (Their isolated persistence is not sufficient). But in order to explain the orderly development of organic nature in which the earlier collected experiences obviously serve as basis for the later creations, it is necessary to assume that these mind-masses enter into a greater compact of mind-masses, the Earth-Soul, and that these serve a higher soul-life according to the same laws as the mind-masses engendered in our nerve-processes observe in their service of our own soul-life.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Zend-, a truly life giving word creating new life in knowledge as in faith! ...What was considered as the seat of mindless unconscious forces appears now as the workshop of the highest spiritual activity ...As Fechner in his Nanna sought to show that plants have souls, so the point of departure of his contemplations in the Zend-Avesta is the doctrine that the stars have souls. The method he employs is not that of the abstraction of general laws by induction and the application and testing of these in the explanation of nature, it is analogy. He compares the earth with our own organism, which we know to be endowed with a soul. He searches out not merely in a one-sided way the similarities, but does equal justice to the dissimilarities, too, and so arrives at the conclusion that all the... [similarities] show the earth to be a being with a soul, and that all the... [dissimilarities] indicate that it is a being with a soul far higher than our own.”
Bernhard Riemann (1826–1866) first seen 1848 wikiquote, CC BY-SA 4.0 source
Recorded in: Ref: Gustav Fechner, 1) Nanna oder über das Seelenleben der Pflanzen (1848). 2nd ed., 1899. 3rd ed., 1903. 4th ed., 1908. 2) Zend-Avesta oder über die Dinge des Himmels und des Jenseits (1851). 3 volumes. 3rd ed., 1906
“The method applied by Newton to the grounding of the Infinitesimal Calculus, and which since the beginning of this century has been recognised by the best mathematicians as the only one that furnishes sure results, is the method of limits. The method consists in this, viz., instead of considering a continuous transition from one value of a quantity to another, from one position to another, or, speaking generally, from one determination of a concept to another, one considers in the first place a transition through a finite number of intervals and then allows the number of these intervals to increase so that the distances of two successive points of division all decrease infinitely.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The concept-systems of antithesis are concepts that are indeed thoroughly determined by negative predicates but are not positively representable. Just because a precise and complete representation of these concept-systems is impossible, they are inaccessible to direct investigation and elaboration by our reflection. They may, however, be regarded as lying at the limit of the representable, i.e., we can form a concept-system lying within the representable, which passes over into the given system by simple change of magnitude ratio. By abstracting from the ratios of the quantities, the concept-system remains unchanged in case of transition to the limit. At the limit itself, however, some of the correlative concepts of the system lose their susceptibility of being represented, and those, indeed, that mediate the relation between other concepts.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“is the attempt to comprehend nature by precise concepts. According to the concepts by which we comprehend nature not only are observations completed at every instant but also future observations are pre-determined as necessary, or, in so far as the concept-system is not quite adequate therefor, they are predetermined as probable; these concepts determine what is "possible" (accordingly also what is "necessary," or the opposite of which is impossible), and the degree of the possibility (the "probability") of every separate event that is possible according to them, can be mathematically determined, if the event is sufficiently precise. If what is necessary or probable according to these concepts occurs, then the latter are thereby confirmed and upon this confirmation by experience rests our confidence in them. If, however, something happens which according to them is not expected and which is therefore according to them impossible or improbable, then arises the problem so to complete them, or if necessary, to transform them, that according to the completed or ameliorated concept-system, what is observed ceases to be impossible or improbable. The completion or amelioration of the concept-system forms the "explanation" of the unexpected observation. By this process our comprehension of nature becomes gradually always more complete and assured, but at the same time recedes even farther behind the surface of phenomena.”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“The concept-systems on which... [natural] sciences are now based arose by gradual transformation of older concept-systems, and... new modes of explanation may... always [be] traced back to contradictions or improbabilities... in the older modes of explanation. [I]t has been proved by Herbart that... concepts serving to comprehend the world, whose origin we are unable to trace.., may all be derived from this source, in so far as they are more than mere forms of combinations of simple sensuous ideas, and... accordingly... need not be derived (as according to the Kantian categories) from a peculiar constitution of the inner soul, anterior to all experience. This proof of their origin in the comprehension of what is given by sense-perception is therefore important... because only thereby their significance can be established in a manner adequate for .”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source
“Kant has rightly observed that by the resolution of the concept of a thing we can find neither that it exists nor that it is the cause of something else, and accordingly that the concepts of being and causality are not analytical but can be derived only from experience. When however he later feels himself obliged to assume that the notion of causality originates in a pre-experiential property of the cognising subject and therefore stamps it a mere rule of time-series, by which, in experience, with each observation as cause any other could be connected as effect, then is the child thrown out with the bath. (Indeed, we must derive the relations of causality from experience; but we must not fail to correct and to complete our conception of these facts of experience by reflection.)”
Bernhard Riemann (1826–1866) said by 1866 at the latest wikiquote, CC BY-SA 4.0 source