Bernhard Riemann - Collected Quotes

“The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting.” (Bernhard Riemann, “Foundations of a general theory of functions of a variable complex magnitude”, 1851)

"But even if it is of interest to grasp the possibility of this mode of treatment of geometry, the execution of the latter would be extremely fruitless, as by this means we would not find new theorems, and what seems simple and clear in the presentation in space, would thereby get involved and difficult." (Bernhard Riemann, "Nachlass", cca. 1852-1853) [on Euclid’s parallels axiom]

 "Therefore, one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginery quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann, "Nachlass", cca. 1852-1853)

"If in [a continuous manifold] one passes from a certain [point] in a definite way to another, the [points] 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, than all the [points] 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." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"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, forward 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 foundation of geometry", 1854)

“It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible.” (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"[…] let us take a continuous function of position within the given the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness. Every system of points where the function has a constant value, forms then a continuous manifoldness of fewer dimensions than the given one. These manifoldnesses pass over continuously into one another as the function changes; we may therefore assume that out of one of them the others proceed, and speaking generally this may occur in such a way that each point passes over into a definite point of the other; the cases of exception (the study of which is important) may here be left unconsidered. […] By repeating then this operation n times, the determination of position in an n-ply extended manifoldness is reduced to n determinations of quantity […]."  (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"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 we can only determine the more or less and not the how much." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

“Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics.” (Bernhard Riemann, 1854)

"By the concepts through which we conceive nature, not only are our perceptions complemented in each moment, but also future perceptions are singled out as necessary, or, insofar as the conceptual system is not complete enough for that purpose, determined as probable […] The conceptual systems which underlie them now [the exact sciences], have been formed by gradual change of older conceptual systems, and the reasons which resulted in new modes of explanation, can be reduced to contradictions or improbabilites, which turned up in the older modes of explanations." (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

“For an understanding of Nature, questions about the infinitely large are idle questions. It is different, however, with questions about the infinitely small. Our knowledge of their causal relations depends essentially on the precision with which we succeed in tracing phenomena on the infinitesimal level.” (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

“The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.” (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’. All thinking is, accordingly, formation of new mind masses.” (Bernhard Riemann, “Gesammelte Mathematische Werke”, 1876)

"Just give me the insights. I can always come up with the proofs!" (Bernhard Riemann)