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Bernhard Riemann Zitate

Georg Friedrich Bernhard Riemann war ein deutscher Mathematiker, der trotz seines relativ kurzen Lebens auf vielen Gebieten der Analysis, Differentialgeometrie, mathematischen Physik und der analytischen Zahlentheorie bahnbrechend wirkte. Er gilt als einer der bedeutendsten Mathematiker. Wikipedia  

✵ 17. September 1826 – 20. Juli 1866
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Bernhard Riemann: Zitate auf Englisch

“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”

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Bernhard Riemann

quality
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

General Relation of the Concept System of Thesis and Antithesis
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

No one can, in case of affairs, abandon the conviction that the future is co-determined by his transactions.
Antimonies
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Causality
Gesammelte Mathematische Werke (1876)

“II. Thesis. Freedom, i. e., not the power absolutely to originate, but to pass judgement between two or more possibilities. Antithesis.”

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Bernhard Riemann

Determinism.
Antimonies
Gesammelte Mathematische Werke (1876)

“The souls of perished creatures shall… form the elements of the soul-life of the earth.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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 which are but imperfectly recognised by Herbart… They depend chiefly upon the inner relationship of the mind-masses.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein https://archive.org/details/TheEvolutionOfScientificThought (1927) p. 55.

“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.”

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Bernhard Riemann

On the Hypotheses which lie at the Bases of Geometry (1873)

“From the standpoint of exact natural science, 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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

On the Hypotheses which lie at the Bases of Geometry (1873)

“Zend-Avesta, a truly life giving word creating new life in knowledge as in faith! …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 former show the earth to be a being with a soul, and that all the latter indicate that it is a being with a soul far higher than our own.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

On the Hypotheses which lie at the Bases of Geometry (1873)

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Antimonies
Gesammelte Mathematische Werke (1876)

“III. Thesis. A God working in Time. (Government of the world). Antithesis. A timeless, personal, omniscient, al-mighty, all-benevolent God”

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Bernhard Riemann

Providence
Antimonies
Gesammelte Mathematische Werke (1876)

“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”

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Bernhard Riemann

can be compared with experience
Die partiellen Differentialgleichungen der mathematischen Physik (1882) as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book https://books.google.com/books?id=G0wtAAAAYAAJ (1914) p. 239

“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.)”

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Bernhard Riemann

Causality, p. 214
Gesammelte Mathematische Werke (1876)

“There remains only the assumption that the ponderable masses within the rigid earth-crust are supporters of the soul-life of the earth.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

On the Hypotheses which lie at the Bases of Geometry (1873)

“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…”

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Bernhard Riemann

On the Hypotheses which lie at the Bases of Geometry (1873)

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

No Antithesis indicated.
Gesammelte Mathematische Werke (1876)

“Thesis. Finite, Representable. Antithesis. Infinite, System of Notions lying at the limit of the representable.”

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Bernhard Riemann

Antimonies
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

On the Hypotheses which lie at the Bases of Geometry (1873)

“Every mind-mass strives to produce a like formed mind-mass and accordingly strives to produce that form of motion of the matter by which it was formed.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“Mind-masses entering the soul appear to us as ideas, the quality of the latter depending on the inner state of the former.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“Mind-masses, once formed, are imperishable, their combinations are indissoluble; only the relative strength of these combinations is altered by the incoming of new mind masses.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

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