Top 15 Bernhard Riemann Quotes

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.

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.)

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.

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.

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.

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.

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.]

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.

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

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.

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

I. Thesis. Finite elements of Space and Time. Antithesis. Continuity.

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

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.

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