Bernhard Riemann

Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting and revolutionary contributions to analysis, number theory, and differential geometry.

Quotes

 * 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.
 * Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) p. 55.


 * 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 (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 (1914) p. 239

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

 * Tr. William Kingdon Clifford, Nature, Vol. VIII. Nos. 183, 184


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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...

Gesammelte Mathematische Werke (1876)

 * Tr. C. J. Keyser, "On the Psychology of Metaphysics. Being the Philosophical Fragments of Bernhard Riemann." The Monist (1900) Vol. X


 * 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.


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


 * 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.


 * 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.


 * 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.


 * 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 (quality).


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


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


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


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


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


 * 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.


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


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


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


 * 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.
 * Antimonies


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


 * 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.
 * Antimonies


 * 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.]


 * 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.


 * General Relation of the Concept System of Thesis and Antithesis

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 concept-systems of antithesis are concepts that are indeed thoroughly determined by negative predicates but are not positively representable.
 * General Relation of the Concept System of Thesis and Antithesis

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.
 * Natural science is the attempt to comprehend nature by precise concepts.
 * Theory of Knowledge


 * 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.
 * Causality


 * 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.)
 * Causality

Newton was far from the crude thought that explanation of phenomena could be attained by abstraction.
 * 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.
 * Causality

Quotes about Bernhard Riemann

 * Riemann has shown that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
 * William Kingdon Clifford, On the Space-Theory of Matter (read Feb 21, 1870) Proceedings of the Cambridge Philosophical Society Vol. II 1864-76 (1876) pp. 157-158
 * Riemann was the first; I read his inaugural dissertation and his major work on Abelian functions. Starting out thus was a stroke of luck for which I have always been grateful. These are not hard to read, as long as one realizes that every word is loaded with meaning: there is perhaps no other mathematician whose writing matches Riemann's for density.
 * André Weil, The Apprenticeship of a Mathematician (1992), p. 39


 * In... 1859 Bernhard Riemann... presented a paper to the [Berlin] Academy... "On the Number of Prime Numbers Less Than a Given Quanitity." ...Riemann tackled the problem with the most sophisticated mathematics of his time... inventing for his purposes a mathematical object of great power and subtlety. ...[H]e made a guess about that object, and then remarked:"One would of course like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective..."That ... guess lay almost unnoticed for decades. Then... gradually seized... imaginations... until it attained the status of an overwhelming obsession. ...The Riemann Hypothesis... remained an obsession all through the twentieth century and remains one today, having resisted every attempt at proof and disproof. [It is] now the great white whale of mathematical research.
 * John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)


 * Riemann's style is extremely difficult. His tragically brief life was too occupied with mathematical creativity for him to devote himself to elegant exposition or to... polished presentation... Riemann's best ideas have been incorporated in later, more readable works. Nonetheless... No secondary source can duplicate Riemann's insight. Riemann was so far ahead of his time that it was 30 years before anyone could begin really to grasp his ideas... [T]he results which Siegel found in private papers were a major contribution to the field... in 1932, seventy years after Riemann discovered them. Any simplification, paraphrasing, or reworking... runs the risk of missing an important idea, of obscuring a point of view which was a source of Riemann's insight, or of introducing new technicalities or side issues which are not of real concern. There is no mathematician... whom I would trust to revise his work.
 * Harold M. Edwards, Riemann's Zeta Function (1974) Preface.


 * The profound purely scientific significance of Riemann's work in pure mathematics and mathematical physics has been long since recognised, and time is more and more disclosing the great philosophical import of portions of that same work, as for example, of the famous Habilitationschrift on "The Hypotheses that Form the Foundations of Geometry." Riemann was indeed distinctly a philosophical mathematician. Grundlage (the foundations of things), more than anything else, fascinated his marvellous genius, and his greatest work was exploration among the roots of knowledge... Some of his profoundest ideas have certainly not been duly exploited. In their German dress, they are to many people practically inaccessible. ... Like Riemann's purely philosophical ideas, which were influenced by Fechner, his psychology will, at least in its terminology, be found to be at variance in many points with the views and tendencies of the time. They have, however, a quite independent significance as throwing light upon Riemann's own intellectual development, and thus, apart from whatever intrinsic merit they may possess, form a valuable page in the history of the development of thought.
 * Edward C. Hegeler, "On the Psychology of Metaphysics Being the Philosophical Fragments of Bernhard Riemann," The Monist (1900) Vol. X, pp. 198-199


 * Einstein independently discovered Riemann's original program, to give a purely geometric explanation to the concept of "force." …To Riemann, the bending and warping of space causes the appearance of a force. Thus forces do not really exist; what is actually happening is that space itself is being bent out of shape. The problem with Riemann's approach... was that he had no idea specifically how gravity or electricity and magnetism caused the warping of space. ...Here Einstein succeeded where Riemann failed.
 * Michio Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (1995)


 * At the University of Göttingen, the shy and gifted Riemann studied mathematics under Gauss and physics under W. Weber, Gauss's collaborator in the invention of the telegraph. Like Gauss, Riemann was deeply interested in physical science, and from that source drew the inspiration for his mathematical investigations.
 * Morris Kline, Mathematics and the Physical World (1959) Ch. 26: Non-Euclidean Geometries.

...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry.
 * In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
 * Morris Kline, Mathematics and the Physical World (1959) Ch. 26: Non-Euclidean Geometries.


 * Bernhard Riemann (1826–1866) was known as “the mathematician from Göttingen.” (Of course so were Gauss and Hilbert.) He provided one answer to the question, Where do functions live?, in a time in which the more general question was, Where are people to live and grow and thrive? Here, think of the First Industrial Revolution of water and steam power, the factory, and the rise of urbanization; the novels of Charles Dickens or Émile Zola; the reconstruction of Paris under Napoleon III and Haussmann; and the soon-to-come Second Industrial Revolution of electricity, chemistry, and the further transformation of agriculture. Consequently, in the city all sorts of things were now mixed together: classes, values, roles. Those things and statuses might be separated out into a simpler less mixed-together world. Technically, such a mixture of what is not to be mixed together is called pollution. And the mathematicians and the city planners aimed to purify and make sense where there was once pollution and disorder.

This equation is of the same form as those which express the propagation of waves and other disturbances in elastic media. The author, however, seems to avoid making explicit mention of any medium through which the propagation takes place. The mathematical investigation given by Riemann has been examined by Clausius, who does not admit the soundness of the mathematical processes, and shews that the hypothesis that potential is propogated like light does not lead either to the formula of Weber, or to the known laws of electrodynamics.
 * In a memoir presented to the Royal Society of Göttingen in 1858, but afterwards withdrawn, and only published in Poggendorff's Annalen in 1867, after the death of the author, Bernhard Riemann]deduces the phenomena of the induction of electric currents from a modified form of $$\frac{\mathrm{d^2}V}{\mathrm{d}x^2} + \frac{\mathrm{d^2}V}{\mathrm{d}y^2} + \frac{\mathrm{d^2}V}{\mathrm{d}z^2} + 4 \pi \rho= \frac{1}{a^2} \frac{\mathrm{d^2}V}{\mathrm{d}t^2}$$ where $$V$$ is the electrostatic potential and $$a$$ velocity.
 * James Clerk Maxwell, A Treatise on Electricity and Magnetism (1881) Vol. 2, p. 446.


 * Riemann bloomed relatively late and did not live to see his fortieth birthday, but he revolutionized virtually everything he touched.


 * Riemann's insight followed his discovery of a mathematical looking-glass through which he could gaze at the primes. Alice's world was turned upside down when she stepped through her looking-glass. In contrast, in the strange mathematical world beyond Riemann's glass, the chaos of the primes seemed to be transformed into an ordered pattern as strong as any mathematician could hope for. He conjectured that this order would be maintained however far one stared into the never-ending world beyond the glass. His prediction of an inner harmony on the far side of the mirror would explain why outwardly the primes look so chaotic. The metamorphosis provided by Riemann's mirror, where chaos turns to order, is one which most mathematicians find almost miraculous. The challenge that Riemann left the mathematical world was to prove that the order he thought he could discern was really there.

The first of the two problems here indicated by Riemann consists in setting up the differential equation, based upon physical facts and hypotheses. The second is the integration of this differential equation and its application to each separate concrete case, this is the task of mathematics.
 * "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 (can be compared with experience)." [Riemann.]
 * Heinrich Weber, Die partiellen Differentialgleichungen der mathematischen Physik (1882) Bd. 1, Vorrede as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914) p. 239


 * After Riemann had made known his discoveries, mathematicians busied themselves with working out his system of geometrical ideas formally; chief among these were Christoffel, Ricci, and Levi-Civita. Riemann... clearly left the real development of his ideas in the hands of some subsequent scientist whose genius as a physicist could rise to equal flights with his own as a mathematician. After a lapse of seventy years this mission has been fulfilled by Einstein.
 * Hermann Weyl, Space—Time—Matter (1922) p. 102


 * It was long accepted as a fact that a metrical character could be described by means of a quadratic differential form, but the fact was not clearly understood. Riemann many years ago pointed out that the metrical groundform might, with equal right, essentially, be a homogeneous function of the fourth order in the differentials, or even a function built up in some other way. and that it need not even depend rationally on the differentials. But we dare not stop even at that point.
 * Hermann Weyl, Space—Time—Matter (1922) p. 138.

The Concepts and Theories of Modern Physics (1881)

 * John Bernhard Stallo, Ch XIV. Metageometrical Space in the Light of Modern Analysis Riemann's Essay


 * The essay of Bernhard Riemann, "On the Hypotheses which lie at the Base of Geometry," owes its great celebrity to the fact that he was a mathematical analyst of the first order, one of the favorite pupils of Gauss, under the inspiration of whose teachings, if not at his suggestion, the essay was written—by whom, in fact, it was presented, in 1854, shortly before his (Gauss's) death to the philosophical faculty of Goettingen, and by whom its cardinal propositions were expressly indorsed as an exposition of his own speculative opinions. Every intelligent reader of this essay will agree... that its intrinsic merit is not at all commensurate with the attention with which it was received and the interest with which it is still generally considered. Not only are its statements, both of the problem and of the proposed methods of solution, crude and confused, but they bear the impress throughout of Riemann's very imperfect acquaintance with the nature of logical processes and even with the import of logical terms. It is apparent... that its author was an utter stranger to the discussions respecting the nature of space which have been so vigorously carried on by the best thinkers of our time ever since the days of Kant, and that he was so little familiar with the history of logic as to be without the faintest suspicion of the manifold ambiguity of such terms as "concept" and "quantity," and of the necessity of their exact definition preliminary to an inquiry respecting the very foundations of human knowledge.

We have here five distinct propositions, which... may be stated in distinct form as follows: 1. That the nature of space is to be deduced from its concept. 2. That the concept of space can be formed and determined only by its subsumption under a higher concept. 3. That our space is a "triply extended Multiple or Aggregate," the higher concept under which its concept is to be subsumed being that of an "n-fold extended Multiple" or a "multiply extended Aggregate" (eine n-fach ausgedehnte Mannigfaltigkeit), and that—translating Riemann's phraseology into its plain logical import—the (logical) extension of this higher concept determines the number of the possible kinds of space. 4. That the conceptual possibility of space is coextensive with its empirical possibility, though not with its empirical reality. 5. That continuous quantities are coördinate with discrete quantities, i.e., are species of the same genus, both being in their nature multiples or aggregates.
 * Riemann himself modestly apologizes for the philosophical shortcomings of his essay on the ground of his inexperience in philosophical matters. But the crudeness of his speculations affords a very striking illustration... of the well-known fact that exclusive devotion to the labors of the mathematical analyst has a tendency to develop certain special powers of the intellect at the expense of its general grasp and strength. Although Sir William Hamilton, no doubt, overstated the case against the mathematicians, I believe that his suggestions are not wholly unworthy of attention, and that there is force in the words of D'Alembert (referred to by Sir William Hamilton)...


 * To imagine that conclusions respecting the nature of space and the origin of its concept can be drawn from the mere fact that space is a function of three variables, and may thus in a manner be classified with similar functions, is a mockery of all reasoning from which an old scholastic would have turned with the scornful reminder that coördination and subsumption, for the purpose of effectually aiding in the formation of a particular concept, must not only be under a genus, but under a genus proximum.


 * Riemann's third proposition that space is an "n-fold extended multiple" or a "multiply extended aggregate"... The term "Mannigfaltigkeit," as here employed, is a standing puzzle to the readers of Riemann's essay. ...Riemann adopted the term from Gauss, who was probably the originator of its employment for the designation of "space in general" (as distinguished from "flat space," in the metageometrical sense). Gauss, in turn, took the expression no doubt from Herbart... whose philosophy is to a great extent a sort of reproduction of the old Eleatic quandaries about "The One and the Many." Herbart, in fine, had obtained it from Kant, whose disciple he was, or believed himself to be, and whose phrase  Mannigfaltigkeiten der Empfindung  is variously found... in... his followers. ...space is not a "multiple" or "aggregate" at all, but... its very essence is continuity. This... follows from its conceptual nature as well as from its relativity. ...space itself is not in any intelligible sense a quantity.


 * Riemann's fourth proposition is founded on a confusion between conceptual possibility and real or empirical possibility. Conceptual possibility is determined solely by the consistency or inconsistency of the elements of the concept to be formed—it is tested simply by the logical law of non-contradiction; while empirical possibility depends upon the consistency... with the various conditions of sensible reality or... laws of nature. ...Upon this distinction depend the utility and scope of the artifice not unfre quently resorted to in certain analytical investigations of supposing the existence of a fourth spatial dimension for the purpose of reducing certain functions to a symmetrical form and this distinction too is the basis of an observation made by Boole... "Space is presented to us, in perception, as possessing the three dimensions of length, breadth, and depth. But in a large class of problems relating to the properties of curved surfaces, the rotation of solid bodies around axes, the vibration of elastic media, etc., this limitation appears in the analytical investigation to be of an arbitrary character, and, if attention were paid to the processes of solution alone, no reason could be discovered why space should not exist in four, or in any greater number of, dimensions. The intellectual procedure in the imaginary world thus suggested can be apprehended by the clearest light of analogy." Upon the same ground... Hermann Grassmann, who is sometimes referred to as one of the founders of transcendental geometry, has developed the theory of extension in its general application to an indefinite number of dimensions, although he certainly did not cherish the delusion (as seems to be supposed by Victor Schlegel) that this could be the source of inferences respecting the number of actual or empirically possible dimensions of space. On this subject we have Grassmann's own explicit declaration: "It is clear," he says, "that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a problem which is at once seen to be impossible of solution."

There are no "discrete quantities" except those which are dealt with in special (common) and general arithmetic, that is to say numbers. ...a number is not a quantity at all, nor a measure of quantity, but simply an intellectual vehicle of quantities—a purely subjective instrumentality for their comparison and admeasurement. ...quantities have been first divided into extensive quantities (space) and intensive quantities (forces, colors, sounds, and all subjective affections), and the extensive quantities have then been subdivided into continuous and discrete. Now, the fact is that all objects of apprehension, including all data of sense, are in themselves, i.e., within the act of apprehension, essentially continuous. They become discrete only by being subjected, arbitrarily or necessarily, to several acts of apprehension, and by thus being severed into parts, or coördinated with other objects similarly apprehended into wholes. To say that a datum of sensation or of subjective feeling is in itself discrete is to assert that it is absolute, and to deny that quantity is essentially relative. And to maintain (with those who speak of positive, negative, fractional, irrational, imaginary, complex, linear, or directional numbers) that number may be continuous is to ignore the plainest and most unmistakable fact in all our intellectual operations, and to misinterpret all the teachings of the history of mathematics. ...It is not to be expected... that mathematicians will cease, at this late day, to speak of arithmetical or algebraic symbols as "quantities;" but there may be a little hope... that they might return to the old phrase "geometrical (and other) magnitudes." The mischief lies, not so much in the use of a particular word, as in the employment of the same word for the denotation of objects differing from each other toto genera.
 * Riemann's fifth proposition... This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze. ...


 * The foregoing discussion has brought us to the point where the reader is in a condition.. to realize the great fundamental absurdity of Riemann's endeavor to draw inferences respecting the nature of space and the extension of its concepts from algebraic representations of "multiplicities." An algebraic multiple and a spatial magnitude are totally disparate. That no conclusions about forms of extension or spatial magnitudes are derivable from the forms of algebraic functions is evident upon the most elementary considerations.


 * If Riemann's argument were fundamentally valid, it could be presented in very succinct and simple form. It would be nothing more than a suggestion that, because algebraic quantities of the first, second, and third degrees denote geometrical magnitudes of one, two, and three dimensions respectively, there must be geometrical magnitudes of four, five, six, etc., dimensions corresponding to algebraic quantities of the fourth, fifth, sixth, etc., degree. ...the analytical argument in favor of the existence, or possibility, of transcendental space is another flagrant instance of the reification of concepts.

The Evolution of Scientific Thought from Newton to Einstein (1927)

 * A. D'Abro, book @archive.org


 * Einstein's theory, more especially the second part (the general theory), is intimately connected with the discoveries of the non-Euclidean geometricians, Riemann in particular. Indeed, had it not been for Riemann's work, and for the considerable extension it has conferred upon our understanding of the problem of space, Einstein's general theory could never have arisen.
 * Forward


 * The essence of Riemann's discoveries consists in having shown that there exist a vast number of possible types of spaces, all of them perfectly self-consistent. When, therefore, it comes to deciding which one of these possible spaces real space will turn out to be, we cannot prejudge... Experiment and observation alone can yield us a clue. To a first approximation, experiment and observation prove space to be Euclidean, and this accounts for our natural belief... merely by force of habit. But experiment is necessarily inaccurate, and we cannot foretell whether our opinions will not have to be modified when our experiments are conducted with greater accuracy. Riemann's views thus place the problem of space on an empirical basis excluding all a priori assertions on the subject. ...the relativity theory is very intimately connected with this empirical philosophy; for... Einstein is compelled to appeal to a varying non-Euclideanism of four-dimensional space-time in order to account with extreme simplicity for gravitation. ...had the extension of the universe been restricted on a priori grounds... to three dimensional Euclidean space, Einstein's theory would have been rejected on first principles. ...as soon as we recognise that the fundamental continuum of the universe and its geometry cannot be posited a priori... a vast number of possibilities are thrown open. Among these the four-dimensional space-time of relativity, with its varying degrees of non-Euclideanism, finds a ready place.
 * Forward


 * Prior to Riemann's discoveries it was thought that the absence of a boundary would necessitate the infiniteness of space. To-day we know that this belief is unjustified, for space can be finite and yet unbounded.

Riemann's exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann's works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclidean space, thus inverting in a certain sense Riemann's ideas. But in Clifford's day, this belief was mathematically untenable. Furthermore, the physical exploration of space seemed to yield unvarying Euclideanism. ...it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann's ideas to victory. (In all fairness to Einstein... he does not appear to have been influenced directly by Riemann.) Nor were Clifford's hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion, and electricity. ... Einstein had been led to recognize that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was fundamental was... one of Space-Time... possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann's ideas to space-time instead of to space... He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity... We had called their effects gravitational effects... never suspecting that they were the result of those very local variations in the geometry for which our search had been in vain....the theory of relativity is the theory of the space-time metrical field.
 * With the new views advocated by Riemann... the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure of geometry for space from place to place. ...

But in the general theory the entire situation changes when Einstein accounts for gravitation, hence for a varying lay of the metrical field, in terms of a varying non-Euclidean structure of space-time around matter. We are then compelled to recognise not only that the metrical field regulates the behaviour of material bodies and clocks, as was also the case in the special theory, but, furthermore, that a reciprocal action takes place and that matter and energy in turn must affect the lay of the metrical field. But we are still a long way from Riemann's view that the field is not alone affected but brought into existence by matter; and it is only when we consider the cosmological part of Einstein's theory that this idea of Riemann's may possibly be vindicated. And here we come to a parting of the ways with de Sitter and Eddington on one side, Einstein and Thirring on the other, and Weyl somewhere in between the two extremes.
 * Let us revert to the metrical field, as defining the space-time structure. Although Riemann had attributed the existence of the structure, or metrical field, of space to the binding forces of matter, there is not the slightest indication in Einstein's special theory that any such view is going to be developed later on; in fact, it does not appear that Einstein was influenced in the slightest degree by Riemann's ideas. ...in the special theory, the problem of determining whence the structure, or field, arises, what it is, what causes it, is not even discussed in a tentative manner. Space-time, with its flat structure, is assumed to be given or posited by the Creator.