## Abstract

Riemann’s text links in a novel manner different thematic threads that combine mathematics, physics and philosophy, and Helmholtz in addition brings the physiology of the senses into the discussion. Therefore, in order to historically situate Riemann’s text, let us first sketch the history of the space problem as developed in those sciences. The starting point of geometrical research is Euclid (fl. 300 bc) . As is well known, in his Elements, from a few definitions, postulates and axioms, he developed a planar and spatial geometry in a constructive manner. This then dominated the subsequent development of geometry so strongly that often and for a long time, it was considered as being without alternatives. While the relationship of Euclidean geometry to the philosophy of Plato was unproblematic, it did not fit with Aristotelian physics. Euclidean space is homogeneous, that is, any point in it is like any other, and isotropic, that is, in all directions it looks the same. No point and no direction is in any way distinguished. Aristotle (384–322 bc) in contrast thought of the world as a collection of places. According to him, the location of an object is determined by its bounding surface. Every body has its natural place to which it tries to move. Thus, the world is heterogeneous. Because objects naturally fall downwards, the vertical direction is different from other directions. Thus, Aristotelian space is not isotropic. Contrasting Euclid and Aristotle in this manner leads us already to the fundamental question of the relationship between geometry and physics, or in a slightly different formulation, to the question about the relation between the geometric space and the objects filling it. From the point of view of physics, this also raises the question about the existence of the vacuum , the empty space devoid of any content that was required for the ancient atomic theory of Democritus (ca. 460–370 bc) and Leucippus (fifth century bc), but which was considered impossible by Parmenides and Aristotle. Euclidean space is infinite, and the question of the finiteness or infinity of physical space was also controversial in antiquity, with Aristotle again standing on the opposing side. For him, infinity could just exist as potentiality in time, but not actually in space. A new point was then introduced by the artists and art theorists of the Italian Renaissance. They wanted to represent objects no longer in their real, objective size or display persons with size corresponding to their significance, but show them as they presented themselves subjectively to the eye of the beholder. For this purpose, they had to utilize the objectively valid laws of geometrical optics, which in turn follow the rules of Euclidean geometry. In a certain sense, they replaced the physics of bodies by a physics of light rays which had to correspond to Euclidean geometry. This may also have been facilitated or inspired by the needs of cartography required for the rise of maritime trade, which was also concerned with the adequate representation of spatial relations. Anyway, linear perspective , which is said to have been discovered by the Florentine architect and artist Filippo Brunelleschi (1377–1466) and which found its first representation, in the book “Della Pittura” (1435) by the writer and scholar Leon Battista Alberti (1404–1472), is the Euclidean construction of the projection from the three-dimensional space on a two-dimensional surface. This inspired then Kepler (1571–1630) and Desargues (1591–1661) to a new treatment of conic sections. In the hands of mathematicians, this led (only) in the first half of the nineteenth century to the development of projective geometry , which then in connection with the ideas developed by Riemann, Klein and other mathematicians of the second half of the nineteenth century became a part of algebraic geometry.

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## Notes

- 1.
The concept of infinity in antiquity, however, differed from the modern one which was shaped by Cantor’s view of modern mathematics. The infinite was understood not as actually existing, but as a potentiality, in the constructive sense that, for example, a straight line can be extended forever, without reaching an end, but without having to assign to all points of this infinite straight line a prior existence. For a systematic analysis of the historical development of the concept of infinity s. J. Cohn,

*Geschichte des Unendlichkeitsproblems im abendlndischen Denken bis Kant.*Leipzig, Wilhelm Engelmann, 1896. - 2.
Samuel Y. Edgerton, The Renaissance Rediscovery of Linear Perspective, Basic Books, 1975.

- 3.
See for instance J. V. Field,

*The invention of infinity*, Oxford, New York etc., 1997. - 4.
See the detailed presentation by Kirsti Andersen, The Geometry of an Art. The History of the Mathematical Theory of Perspective from Alberti to Monge. Berlin etc., Springer, 2007.

- 5.
For a systematic presentation of the entire historical development, we refer to E. Cassirer,

*Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit*, 4 vols, Darmstadt, Wiss. Buchgesellschaft, 1974 (reprint of the 3rd edition of volumes 1,2 from 1922, of the 2nd edition of volume 3 from 1923, and of the 2nd edition of volume 4 from 1957). A concise introduction to the development of anti-Aristotelianism and mechanical philosophy until Newton is given by Daniel Garber, Physics and foundations, in: Katherine Park, Lorraine Daston (eds.),*The Cambridge History of Science*, Vol. 3, Early modern science, Cambridge, Cambridge Univ. Press 2006, pp. 21–69. - 6.
In addition to Cassirer, Vol. 1, loc.cit., see also the article on Telesio in R. Eisler,

*Philosophenlexikon,*Berlin, 1912, p. 741f. - 7.
Galileo put the focus on the empirically measurable development of physical processes instead of their derivation from final principles. He believed that the world therefore cannot be readily deduced from revealed principles, but can only be painstakingly explored empirically and needs to be measured. With these views and the underlying atomistic conceptions, Galileo ultimately demolished the scholastic natural philosophy of the Middle Ages that had been formed through the reception of Aristotle. According to scholastic philosophy, the world was a structure with an order designed for and revealed to man. (See, eg, the concise analysis of E.A. Burtt,

*The metaphysical foundations of modern science.*Mineola. Dover, 2003 (reprint of the 2nd ed., 1932)). This philosophy also provided a foundation for the Aristotelian distinction between form and substance which in turn was needed for the doctrine of the Eucharist. This was an important component of the worldview of the Catholic Church that had hardened in the Counter-Reformation. This was a basic reason for opposition of the Church against the Galilean conception of nature. This is in contrast to the beginning of the dissolution of the Aristotelian world view in the Italian natural philosophy of the sixteenth century, as outlined above, which had still encountered papal benevolence. Instead of the Aristotelian substances, which could be given different forms, and which then in turn could be converted while preserving their form in the miracle of transubstantiation, in the Galilean view, there were only formless atoms whose qualitative properties such as color were constituted only in the process of perception. Then such a miracle became impossible, or plausible, at best, as a crude sensory illusion. And of course, also the Copernican heliocentric system did not harmonize with a human-oriented plan of creation. These seem to be the deeper underlying reasons for the resistance that Galileo encountered from the leading intellectual representatives of the Catholic Church. This is different from popular expositions, where petty disputes about the literal interpretation of certain biblical passages, like that where Joshua allegedly made the sun stand still when taking Jericho, are presented as the reason for the persecution of Galileo by the church. Bible passages could well be interpreted allegorically also by the Catholic Church, if this was deemed useful or necessary for systematic reasons. Scriptures probably served rather as material for rhetorical tricks at a time when the methods advocated by Machiavelli could also be employed in intellectual discussions. Even with physical experiments it is often unclear whether they were actually carried out or their results were only claimed on the basis of intuitive plausibility as evidence for a systematic theory, see, eg, Alexandre Koyr, Galile et l’exprience de Pise: propos d’une legende, in: Annales de l’Universit de Paris, 1937. Although Pietro Redondi, Galileo eretico, Torino, Einaudi, 1983 (English translation: Galilei heretic, Princeton Univ. Press, 1987), found evidence that the church actually saw justification of the doctrine of transsubstantion, which was central for the Counter-Reformation, endangered by Galileo and reprimanded him for this, the consequences of this discovery are probably not incorporated to their full extent in the history of science discussion. - 8.
The classical treatment is Alexandre Koyré,

*From the closed world to the infinite universe*, Baltimore, Johns Hopkins Univ. Press, 1957. Cassirer, loc. cit., contains more material and in some regards penetrates more deeply into the matter. - 9.
For a modern account of Cartesian physics, see Daniel Garber,

*Descartes’ metaphysical physics*, Chicago, London, The University of Chicago Press, 1992. - 10.
Newton held the view that the Cartesian conception of matter as characterized by expansion just confuses the essential properties of space and bodies with each other. Newton considered impenetrability as an important characteristic of the physical bodies and refuted the Cartesian theory with physical arguments, Isaac Newton, Mathematical Principles of Natural Philosophy,

^{3}1726. Nevertheless, Alexandre Koyr, Newtonian Studies, Chicago, Univ. Chicago Press, 1965, argues for a decisive influence of Descartes on Newton. Even historians of science seem to have their favorite heroes. Newton’s struggle with the conceptions of Descartes can be seen perhaps most clearly from the posthumous manuscript, which was probably written before the drafting of the Principia, which is usually quoted by its opening words “De gravitatione.,, ” and which was first published with English translation in A.R. Hall and M. Boas Hall, Unpublished scientific papers of Isaac Newton, Cambridge, Cambridge Univ. Press, 1962, pp. 89–156. - 11.
Kepler considered the attraction of the earth as a kind of magnetic force, inspired by the study of magnetism by William Gilbert (1543–1603) and the latter’s discovery that the earth also behaves like a magnet, which can explain the properties of the compass. Remarkably, to this day, physics has not succeeded in capturing magnetism and gravitation in a unified theory, as we shall explain in more detail below.

- 12.
For a succinct but very clear exposition see Richard S. Westfall, The construction of modern science. Mechanisms and mechanics. John Wiley, 1971; Cambridge, Cambridge Univ. Press, 1977. A detailed analysis of the Newtonian concept of force and its historical genesis and preparation can be found in Richard S. Westfall, Force in Newton’s physics, London, MacDonald, 1971. See also Ferdinand Rosenberger, Isaac Newton und seine physikalischen Prinzipien, Leipzig, Ambrosius Barth, 1895, reprint Darmstadt, Wiss. Buchges., 1987.

- 13.
It is a noteworthy fact in the history of science that in the hands of Kepler, this idea was still fertile and pioneering for physics. In particular, this lead to his insight into the cause of the tides. While Galileo had wanted to explain the tides by the earth’s rotation and reversely believed to have thereby found a proof for the earth’s rotation and thus for the correctness of the Copernican system (although this argument contradicted his own principle of relativity ), Kepler attributed the tides to the influence of the moon, that is, to an action over a spatial distance. Had this idea not been accepted, the great system of Newton would not have been possible either. But when the general acceptance of the Newtonian theory made the questioning of this idea difficult, the further progress of physics was hindered.

- 14.
See for instance Koyré,

*Closed world,*loc. cit. Such views were still advocated by the leading scientists of the eighteenth century, Leonhard Euler, although Euler was deeply religious. - 15.
The idea of space as an expression of God’s omnipresence had already been developed by the Cambridge Platonists and, in particular, by Newton’s friend Henry More (1641–1687) and that of time as an expression of God’s eternity and constant presence by Isaac Barrow (1630–1677), the colleague of More and the teacher, colleague and friend of Newton. See the presentation in E.A. Burtt,

*The metaphysical foundations of modern science*, Mineola. Dover, 2003 (reprint of the 2nd ed., 1932). For Newton, space and time even were the sensoria of God, and this naturally led to God’s characterization as the self-perception of reality. Here, however, we cannot discuss such later developments in detail. - 16.
See the famous polemics between Leibniz and Newton follower Samuel Clarke (1675–1729), for example reproduced in G.W. Leibniz,

*Hauptschriften zur Grundlegung der Philosophie*, part I, pp. 81–182, translated by A. Buchenau, ed. by E. Cassirer, Hamburg, Meiner, 1996 (reissue of the 3rd ed., 1966). - 17.
For a systematic presentation and analysis of Leibniz’s concept of space in the context of his philosophy, I refer to Vincenzo De Risi, Geometry and Monadology. Leibniz ’ Analysis Situs and Philosophy of Space, Basel etc., Birkhauser, 2007, pp. 283–293. The structural considerations of Leibniz went far beyond the discussion of his time, but because they were not systematically published and therefore not properly understood by his contemporaries, they did not have a sustained effect.

- 18.
But even this argument of Mach did not provide the final resolution. That was supplied only in the general theory of relativity, as will be explained in more detail below.

- 19.
Newton did not allow himself in the “Principia” to pursue the question of the cause of gravitation. According to his empiricist attitude, through careful observation of the phenomena, he wanted to come inductively to laws which then in mathematical formulation and by mathematical methods allowed for empirically testable prediction of further phenomena. In this sense we should understand his famous quote “Hypotheses non fingo”. However, at other occasions, he did speculate about an ether mediating gravity and other physical forces, see E.A. Burtt, loc. cit. For instance, on p. 350 of the 1979 Dover republication of his

*Opticks*, he speculates about an ether that mediates gravitation. The standard version or interpretation of Newtonian physics where gravity acts immediately and across empty space, that is, without the help of a medium, was developed by Newton’s followers and derived in a philosophical framework by Kant. - 20.
See Daniel Garber, loc. cit.

- 21.
Kant contrasts the mathematical approach of the mechanical philosophy which tries to explain physics through the mathematical analysis of a priori concepts with his own metaphysical-dynamical approach that uses forces as basic ingredients and therefore depends on experience. See Michael Friedman,

*Kant’s construction of nature*, Cambridge, Cambridge Univ. Press,^{2}2012. - 22.
Actually, Leibniz’ views are more complex, but this is not the place to go into details. See for instance M. Guerault,

*Dynamique et métaphysique leibniziennes*, Paris, Les Belles Lettres, 1934, and many other texts dealing with Leibniz’ natural philosophy. - 23.
See for instance Daniel Garber,

*Leibniz: Body, Substance, Monad*, Oxford, Oxford Univ. Press, 2009, who, however, emphasizes the changes that Leibniz’ natural philosophy underwent during his lifetime. - 24.
See the comprehensive analysis of Kant’s natural philosophy by Michael Friedman,

*Kant’s construction of nature*, loc. cit. - 25.
Immanuel Kant,

*Kritik der reinen Vernunft,*1781, in his*Werkausgabe Bd. III/IV*, ed. W. Weischedel, Frankfurt, 1977. I shall use the translation*Critique of Pure Reason*by Paul Guyer and Allen W. Wood, Cambridge etc., Cambridge University Press, 1998. This edition, like that of Weischedel, gives the paginations for both the first (from 1781) and the second edition (from 1787) of the*Critique*; for instance, A86/B118 means p. 86 of the first and p. 118 of the second edition. - 26.
Where axioms here should not be interpreted in a modern sense, following Hilbert, as arbitrary stipulations.

- 27.
In this context, it can only contribute to confusion when Paul Franks in the

*Oxford Handbook of Continental Philosoph*y, Oxford etc., Oxford Univ. Press, 2007, pp. 243–286 (about Helmholtz especially pp. 269–276), edited by Brian Leiter and Michael Rosen, classified Helmholtz as a Neo-Kantian, because the so-called Neo-Kantians were his most important philosophical adversaries, besides people like Hering whom he referred to as a nativist. Worth mentioning in this context is G. Schiemann,*Wahrheitsgewissheitsverlust. Hermann von Helmholtz’ Mechanismus im Anbruch der Moderne.*Eine Studie zum bergang von klassischer zu moderner Naturphilosophie. Darmstadt, Wiss. Buchges., 1997. Schiemann works out especially how Helmholtz’ approach of a justification of experience in the conditions of physical measurements differs from the Kantian point of departure of the transcendental subject, and examines the systematic changes that Helmholtz’ natural philosophy ’underwent in the course of his life. See also some essays in the anthology David Cahan (Ed.), Hermann von Helmholtz and the Foundations of nineteenth-century science, etc. Berkeley, Univ. California Press, 1993. - 28.
See the references below in the reception history.

- 29.
This had already been contemplated by Leibniz, see De Risi, loc. cit. Leibniz then went on to attempt to prove the three-dimensionality of space.

- 30.
Immanuel Kant,

*Von dem ersten Grunde der Unterschiede der Gegenden im Raume*, 1768, in ibid.,*Vorkritische Schriften bis 1768, Werkausgabe Bd. II*, hrsg. v. W. Weischedel, Frankfurt, 1977, S. 991–1000; English translation in Kant,*Theorerical Philosophy, 1755–1770*, transl. and ed. D. Walford, with R. Meerbote, Cambridge, Cambridge Univ. Press, 1992. This example is taken up again in ibid.,*Prolegomena zu einer jeden knftigen Metaphysik die als Wissenschaft wird auftreten knnen,*1783, in: the same,*Schriften zur Metaphysik und Logik 1*,*Werkausgabe Bd. V*, edited by W. Weischedel, Frankfurt, 1977, pp. 111–264, §13; English translation in Kant,*Prolegomena to Any Future Metaphysics That Will Be Able to Come Forward as Science*, transl. and ed. G. Hatfield, Cambridge, Cambridge Univ. Press, rev. ed., 2004. - 31.
See for example Hermann Weyl,

*Philosophy of Mathematics and Natural Science*, Princeton, Princeton Univ. Press, 1949, 2009 (translated from the German). - 32.
For a comparison of the positions of Leibniz and Kant on this issue and a newer overview of the literature on the subject, refer to Vincenzo De Risi, Geometry and Monadology. Leibniz’s analysis situs and Philosophy of Space, etc. Basel, Birkhauser, 2007, pp. 283–293.

- 33.
Carl Friedrich Gau, Werke, Gttingen, 1870–1927, reprint Hildesheim, New York, 1973; Vol. II, p. 177.

- 34.
The fact that from mathematical axioms conclusions can be drawn that are not obvious, is a central theme of the philosophy of mathematics. The Platonic approaches view mathematics as an opportunity or a tool to discover eternal truths. Weyl,

*Philosophy*, however, emphasizes the constructive and creative nature of mathematics. - 35.
Kant,

*Critique of Pure Reason,*2nd ed., Introduction, B16 (p. 145) (emphasis in the original). - 36.
Ibid., A141/B180 (p. 273).

- 37.
Ibid., A221/B268 (p. 323).

- 38.
Euclidean space is also referred to as “flat”, and the word “curvature” should then simply express the deviation from this plane, straight shape.

- 39.
Alexandre Koyré,

*Etudes galiléennes*, Paris, Hermann, 1966, therefore tries to deny Galileo knowledge of the law of inertia, even if this law is implicitly assumed and explicitly expressed in the passages quoted by him from Galileo and his successors Cavalieri (1598–1647) and Torricelli (1608–1647) and Gassendi (1592–1655) several times. Simply, unlike Newton, he did not make this the basis of his physical theory, because he had considered a motion without the influence of other bodies as unphysical. - 40.
The Cartesian coordinates, however, are only implicit in the geometry of Descartes and were not constructed by him explicitly. But since Descartes laid the conceptual foundations, it is still justified to name these coordinates after him. See, for example Mariano Giaquinta, La forma delle cose, Roma, Edizioni di Storia e Letteratura 2010 or A. Ostermann, G. Wanner, Geometry by Its History, Berlin, Heidelberg, Springer, 2012.

- 41.
As we shall explain below, the actual logical relationship is rather the other way around: One obtains the metrical structure of Euclidean space by interpreting the magnitudes of coordinate differences on each coordinate axis of a Cartesian space as distances and by declaring different coordinate axes as perpendicular to each other. Thus, Euclidean space possesses a metric structure which as such is not yet contained in the Cartesian concept, while the Cartesian coordinate space is determined in a way not provided by the Euclidean concept. The clear separation of geometric facts and their different descriptions in different coordinate systems is then just one of the essential achievements of Riemann.

- 42.
The question of whether it is appropriate to attribute to the vacuum the geometric structure of Euclidean space leads into modern physics, which will be discussed below.

- 43.
On this, see Alexandre Koyré,

*From the closed world to the infinite universe,*Baltimore, Johns Hopkins Press, 1957. It is remarkable that cosmology today returns to the idea of a finite cosmos, amongst other reasons, to “explain” the emergence of the universe from a singular beginning, the Big Bang, and thus to regain the historical dimension in contrast to a truly infinite, but static universe. - 44.
However, more general spatial concepts were then introduced after Riemann, which give up this condition of the approximability by a Euclidean space. An example are the so-called topological spaces. Also the concept of Riemannian manifold is later developed to the extent that only so-called differentiable manifolds, but no longer more general manifolds satisfy this approximability condition. Thus, Euclidean space will finally lose its special role. More details will be presented below, when we analyze Riemann’s text.

- 45.
See Euclid,

*The Thirteen Books of Euclid’s Elements*. Translated from the Text of Heiberg with Introduction and Commentary by Sir Thomas L. Heath, 3 Vols, Reprint of the 2nd edition, Dover, 1956, 2000. - 46.
English translations in Roberto Bonola, Non-Euclidean Geometry. A Critical and Historical Study of its Developments, Dover, 1955. For more information, please refer to the bibliography.

- 47.
For details, see for instance B.R. Torretti,

*Philosophy of Geometry from Riemann to Poincaré*, Dordrecht, Boston, Lancaster,^{2}1984, 63f, 381. - 48.
See E. Scholz,

*Riemanns frhe Notizen zum Mannigfaltigkeitsbegriff und zu den Grundlagen der Geometrie*, Arch. Hist. Exact Sciences 27, 1982, 213–282. - 49.
C. F. Gau,

*Disquisitions générales cira superficies curas*. Commentationes Societatis Gottingensis, 1828, 99–146;*Werke,*Bd. 4, 217–258; English translation in Peter Dombrowski, 150 years after Gau’*“Disquisitiones generales circa superficies curvas”,*Astérisque 62, Paris, 1979. - 50.
This was first shown by Leonhard Euler (1707–1783), see Opera omnia, Leipzig, Berlin, Zurich, 1911–1976, 1st series, vol. XXVIII, pp. 1–22. Unless all the normal sections have the same curvature, these two intersection curves of extremal curvature are uniquely determined and intersect each other at right angles.

- 51.
For a modern presentation, see for instance J. Eschenburg, J. Jost,

*Differentialgeometrie und Minimalflchen*, Heidelberg, Berlin,^{3}2013. - 52.
In his private notes, Riemann cited in particular the philosopher Johann Friedrich Herbart (1776–1841) and mentions him also at the beginning of his habilitation address, see the same,

*Sämtliche Werke*in chronologischer Reihenfolge herausgegeben von Karl Kehrbach und Otto Flügel, 19 vols., Langensalza, 1882–1912, reprinted Aalen, Scientia Verlag, 1964, in particular Psychologie als Wissenschaft., 2 parts, Vol. 5, 177–402, and Vol. 6, 1–339 (originally published in 1824/25). In 1809 Herbart became Kant’s successor as the chair for philosophy in Knigsberg and in 1834 took over the chair of philosophy in Göttingen. He represents the transition from idealism to realism in German philosophy of the nineteenth century. He criticizes Kant from an empiricist and association psychology position. The individual being is for him a unit, a feature bundle that by coming together with others acquires different characteristics that in each case can represent different continua. Thus snow is white, when the eye sees it, cold, when the hand touches it. These continua can be conceived spatially. He emphasizes in particular the historical contingency and conditionality of the notion of space, which for him, according to his considerations just presented, was only an example of a continuous sequel. The relationship between the ideas of Herbart and the concepts Riemann is discussed in Benno Erdmann,*Die Axiome der Geometrie. Eine philosophische Untersuchung der Riemann-Helmholtzschen Raumtheorie,*Leipzig, Leopold Voss, 1877, pp. 29–33, Luciano Boi, Le problme de l’espace mathmatique, Berlin, Heidelberg, Springer, 1995, pp. 129–136. Erhard Scholz,*Herbart’s influence on Bernhard Riemann,*Historia Mathematica 9, 413–440, 1982, on the other hand comes to the conclusion that ultimately the influence of Herbart’s thoughts on the Riemannian manifold concept was rather minor, even if Riemann may have been guided by some general principles of Herbart, as that for each area of science a main concept needs to be worked out, or that ideas such as tone or color are not only quantitatively different, but also subject to different types of mathematical laws and consequently should be investigated by the methods of mathematics. In this context, we also refer to the presentation in Pulte,*Axiomatik und Empirie*, pp. 375–388. - 53.
See the corresponding quote from the Dean’s Office Archive in Laugwitz,

*Riemann.*p. 218.

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Jost, J. (2016). Historical Introduction. In: Jost, J. (eds) On the Hypotheses Which Lie at the Bases of Geometry. Classic Texts in the Sciences. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26042-6_2

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