Bernhard Riemann: Possibility

“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.”

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)

Idea , Law , Decision

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

Bernhard Riemann

Determinism.
Antimonies
Gesammelte Mathematische Werke (1876)

Freedom

“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

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

Way , Question , Study , Parting

“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

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

Way , Imagination

“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

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

“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.”

Bernhard Riemann

Theory of Knowledge
Gesammelte Mathematische Werke (1876)

Confidence , Mathematics , Nature , Problem

“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

Gesammelte Mathematische Werke (1876)

Life

“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

Antimonies
Gesammelte Mathematische Werke (1876)

Law , Idea , Decision , Future

“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.”

Bernhard Riemann

Gesammelte Mathematische Werke (1876)

Soul