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Bernhard Riemann Zitate
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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

“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.”

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Bernhard Riemann

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

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“The substratum of mental activity must be sought only in ponderable matter.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“I. Thesis. Finite elements of Space and Time. Antithesis.”

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Bernhard Riemann

Continuity.
Antimonies
Gesammelte Mathematische Werke (1876)

“Natural science 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.”

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Bernhard Riemann

Theory of Knowledge
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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-mass."”

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Bernhard Riemann

All thinking is, accordingly, formation of new mind masses.
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

Causality
Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

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. No one can, in case of affairs, abandon the conviction that the future is co-determined by his transactions.”

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Bernhard Riemann

Antimonies
Gesammelte Mathematische Werke (1876)

“The different thought-processes seem to differ chiefly in respect to their temporal rhythm. 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.”

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Bernhard Riemann

Gesammelte Mathematische Werke (1876)

“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.”

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Bernhard Riemann

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.
Causality
Gesammelte Mathematische Werke (1876)

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