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In order to present an historical discussion on spherical geometry in relativity between the 19th–20th centuries, the current research divided into two parts–papers is presented. Part I: Reflections on geometry–physics relationship is an excursus focusing on the role played by geometry in history and its relationship with physics. Part II: Reflections on Late Relationship Geometry–Physics is an epistemological investigation on the history of the foundations of geometry in special relativity focusing in 4 S (O, x, y, z, t). In this sense, the two parts are complementary. By considering the large literature, we decided to present the main and basic accredited references, maybe useful for very young scholars, as well.
Keywords: relationship geometry–physics–mathematics, foundations, historical epistemology of science.
1. On geometry and continuum
The fundamental theory supporting the Newtonian pattern of continuum in mathematics as far as the study of physical phenomena is concerned can be attributed to Euclid. We can maintain that he connected in a symbiotic link the then well known properties of geometry and arithmetic elevating them in part to being the founding properties of the self–evident theories. The precariousness of the fifth axiom on the parallel straight line was well-known (and possibly accounts for the delay in its application). With Euclid’s theory being typically deductive, the derivation of that axiom was automatically obtained from the first four axioms. In fact:
[…] Euclid […], after examining and comparing the results of dealing with a precise topic set about to divide them harmoniously [according to an Aristotelian–axiomatic pattern] filling in the gaps, so as to create an organic whole (Loria, 1914, p. 279)1.
Euclid was, however, not able to prove the fifth axiom based on the first four. The Greek Posidonius from Rhodes (135 B.C.? 1 B.C.?), is supposed to be one of the first learned men to have shown his disappointment regarding the fifth axiom. He tried to re–arrange Euclidean geometry, attempting to prove the axiom of the straight lines in order to include it in theorems. Ptolomaeus also tried a logical deduction of the fifth axiom from the first nine (and from the first 28 of Euclid’s theorems too, although they were not based on it). Yet Proclus (fl. V B.C) condemned this method, maintaining that even if two straight lines came close to each other over the infinite, this would not imply any proof of an overlap at one point.2 The whole debate is clearly focused upon the concept of continuum and upon the choice of a particular type of infinite (actual or potential). It highlights the problem of whether to conceive the universe as being flat or to accept the theory of relativity. Only in the third decade of the 19th century, when in Nicolaj Inanovic Lobacevskij (1793–1856) and János Bolyai’s (1802–1912) works, and in the complex theories first by Carl Friedrich Gauss (1777–1855) and then Jules–Henri Poincaré (1854–1912), (Paty 1999) was it possible to claim the existence of a geometry different from that conceived by Euclid (Pisano, 2006b; Pisano & Casolaro, 2011); a geometry where the first four axioms were valid but not the fifth, a so–called non Euclidean geometry, but still perfectly coherent and logical. To sum up, the hypothesis of a geometrical space conceived differently from the Euclidean plane contributed to the pattern of the mathematical continuum, dealing with the interpretation of an ideal physics. It is interesting to recall Poincaré’s view, warning that
[…] but it would be a mistake to conclude from that that geometry is, even part, an experimental science. if it were experimental, it would only be approximative and provisory. And what a rough approximation it would be! Geometry would be only the study of the movements of solids bodies ... (Poincaré, [1905] 2007, p. 70, line 8)3.
2. On geometry and physics
When geometry is used in the study of ideal images4, a body (a virtual one) is the result of a process of idealization of reality, naturally elaborated by our mind; and the empirical performance in geometry, as Poincaré thought, represents but the occasional stimulus required to make the image materialize (Poincaré, 2003, p.117). According to Poincaré the geometrical axioms are neither synthetic previous comments nor experimental facts, but they are conventions. Therefore, it would be futile to wonder about the truth of such statements as they are already set as conventions. As well as being nonsense – according to Poincaré –wondering whether the metrics system is true and the old measurement system is false, an element which comes to be essential in the study of Poincaré’s geometry is the concept of group –meant either as a file containing information and properties valid under determined conditions or as a new way of conceiving geometry. In this sense, it is not possible to choose one type of geometry rather than another, because there is no such thing as a better geometrical group. It is well–known that in classical physics space represents a system where the natural phenomena are described and exist independently from the occurring events. In theory, this way of perceiving the physical sciences permits one to interpret the movement of a punctual–form body compared to an absolute space (Aristotle), even in the event that the description of the phenomenon is not anticipated by an experiment. In fact, classical physics assumes the mathematical sciences as a control element of reality, so that within the Cartesian reference model a (material) point comes to be associated unequivocally with a real number; for example, according to Newtonian theory any variation (in continuum5) of a physical magnitude is also a mathematical function of space and time. Concepts of space and time (Jammer. 1957, 1960. 1961) have, on the whole, been set as the foundation for subsequent theoretical physics. Evidently, due to the power of calculations and the authority of the Newtonian tradition, which gave birth to the physical–mathematical tradition, they were instituted as a theoretical pattern, influencing most of the physicists until the beginning of the 20th century: the continuum is an indisputable means of giving a mathematical interpretation to physical phenomena. It should be maintained that physics itself in 1900 often dealt with the interpretation of some physical effects (through the mathematics of continuum) perceived not always by direct observation, but by the study of built–up mathematical models, so to speak, built–up in papers in order to interpret both the micro and macro reality; also the scientific instruments used for the experiments were arranged according to those models. Let us just consider the operating principle underlying the detection of particles rather than the kinetic model of gases in the homonymous theory; or let us recall the problems encountered with the use of the standard model in modern physics. The opinion was supported by some mathematicians of the time that the mathematical issues proving so helpful in physics theory were the most reliable and relevant part of mathematics as a whole. No alternative to the close connection between physics and mathematics was considered possible, yet this was disputed.6 The context, though containing many underlying uncertainties, undoubtedly produced interesting scientific innovations, such as the birth of special relativity (1905) and general relativity (1916–19), even suggesting a total change in the physical (Newtonian paradigm) categories and in the (Kantian) philosophical categories of time and space. In this perspective, Einstein’s relativity theories have completely changed the foundations of classical Newtonian science, highlighting that, from a physical point of view, the acknowledgment of space, even in the absence of visible phenomena, is nonsense (as well as the absence of absolute time); that is why it is appropriate to imagine a space whose properties are related to the observer and to the body’s motion. Furthermore, as the speed of a body is the result of the variation of its position instant by instant, it is incorrect to consider space as being independent from time. This idea was properly developed by Einstein (Einstein, 1905, 891–921; D’Agostino, 1995, pp. 167–178) through the hypothesis of a fourth dimension “t” which, with the coordinates of space, should describe a phenomenon by quadrivector. Fundamental problems7 emerged: Hermann Minkowski (1864–1909) regarding the choice of space, more than twenty mathematical problems set by David Hilbert (1862–1943), the debate about logic in quantum mechanics, the introduction of differential geometry and the mathematics of the tensors8 in general relativity.
3. On spherical geometry and special relativity
In the first decades of the past century, space–time and mass–energy concepts became very important in distancing physics from the past but seemed to be ahead of their time. Most notably, these concepts appeared to be a real problem for the traditionalist physicist (Hirosighe, 1976). A new adventure in physics followed but this perturbed the scientific authorities of Newtonianism and Descartesian space. According to Einstein, a plane Universe (Euclidean Modelling) is an (materially) empty universe, devoid of matter since the presence of matter causes a curvature of the space. Regarding the inner phenomena of the solar system, it is well-known that general relativity brings light variations (which are rather remarkable with regard to the applicability of Newtonian theory) and space is slightly deformed. Yet for ca. 109 light year distances, the matter in the universe dramatically affects the geometrical nature of space, meaning that the Euclidean theory is not more basically applicable over huge distances. In fact, Einstein’s theory of general relativity is merely based on an elliptic model analogous to Bernhard Riemann’s (1826–1866) non–Euclidean geometry model. Half a century earlier, in his works (Smith, 1958) on k–variety space surfaces (n>k dim.), Riemann concluded that the physical space is 3–dimensioned with a constant curvature. From an exclusively geometrical point of view, the problem consists of the rigorousness of a model with regard to non–plane geometry by means of the theory of vectors. The perpendicular vectors to a plane at any point all have the same direction. Thus, in succession, performing on plane spaces, of three, four dimensions and so on, the perpendicular vectors to the space all have the same direction. Whereas, for a curved space, a direction of a perpendicular vector to the space changes at any given point, so the components of a normal vector to the space at one point (precisely at the plane tangent to the surface in that very point) result as functions of the coordinates. One can approach the non–Euclidean geometries from various points of view (Boi, Flament & Salanskis, 1992):
Elementary approach (Gauss, Bolyai and Lobacewski), developing by observations analogous to those involved within the Elements, except for some variations inside the postulates corresponding to the hypothesis of the acute angle.
Metrics–differentia approach, (Riemann) that consists of the extension to a variety of three or more dimensions of the differential geometry of the surfaces introduced by Gauss, with particular regard to the surfaces having constant curvature.
Grouping approach which, exploiting (as much as possible) the free mobility of figures in space, studies those particular transformations, the motions keeping all distances unaltered.
Projective approach, that states the metrics properties of the figures in a projecting form.
Riemann assumed that the properties singling out physical space from other kinds of space can be deduced by experiment only; consequently, he thought that the axioms of Euclidean geometry could be approximately shown and related to physical space. Moreover, his elliptic model (1854), which can be considered an extension of Gauss’ differential geometry, stated that within a space, where the curvature changes from one place to another, owing to the presence of matter, and instant by instant (owing to the matter motion) the Euclidean geometry laws can no longer be applied. Therefore, in order to determine the adequate nature of a physical space, space and matter (consequently, time) should be related to each other. Assuming an analytical approach9to the local behaviour of space, Riemann used the metrics–differential method. For the linear elements, he defined the distance between two points as follows:
have ranks ??2
A? and represent, respectively, a space 2S (analogous to the Euclidean plane) as a hyperplane of3 S , and a space 1 S (analogous to the Euclidean straight line) as a hyperplane of2S . According to Einstein’s approach, the functionsgij, that, in (2), represent the coefficients of the variables x, y, z, t, include the effects of gravitational masses in the space. Such effects (causing a curvature at any point) are the source of the orientation change of the normal vector to the surface. By introducing time as coordinate the analysis is combined with the 4–dimensional space–time à la Minkowski. The set of the points–events in (x, y, z, t) defines a continuum by 4–dimensions and represents a geometrical space4S . In fact, the space 3 S of the points (x, y, z) of the Euclidean model, assumed as subspace of4S , can be considered as a hyperplane of space–time à la Minkowsky. If we analyse the structure of the subspaces of 4S and the analytical Euclidean pattern, we can deduce:
By t=0 (or t=const.), we have the hyperplane3 S , which is the 3–dimensional Euclidean geometrical space where the laws of the classic kinematics are still valid;
Whereas, if one of the coordinates is constant, we have hyperplanes 3 S of 4S which featuring kinematics and relativistic models over 2S is analogous to the Euclidean plane.
Let us see an example. A set of the points P(x, y, 0, t) ? P(x, y, t) is a 3 S space, whose coordinates (x, y, t) mark the points–events of the plane 2S (x, y) extended to kinematics by the introduction of the time coordinate. In such a context, it should be taken into account that metrics is defined by
ds???? (3)
Such a relation, presenting a minus sign in the fourth term of the quadratic expression, can be geometrically justified, by adding to the real term (O,x,y,z) of the reference S3 a forth imaginary orthogonal axis at S3 where the time coordinate is multiplied by the imaginary unit i. According to a dimensional
4. Some final remarks
Between the 16th and 18th centuries, modern science reached a peak. Many factors, social and political
were part of this process. A crucial transformation of scientific theories characterized the development of
physics and mathematics in astronomy (Pannekoek, 1961).
Table 1
Paradigm lost?
Astronomy should not question religion or philosophy and did not aim to confront the absolute power of
the past. In this sense, geometry cannot theoretically be abstracted from the physical evolution. As a matter of fact, according to the classical conception, geometry expresses a set of properties with regard to the motion of bodies and the law of the propagation of light, sometimes obtained through the abstraction from time and forces. Thus, by the application of geometry to kinematics (that is the theory of movement in space–time) and, later on, to dynamics (by the introduction of forces) interpretative approximations closer to the physical phenomenon, shall result. This should all be further clarified thanks to the theory of general relativity. Furthermore, we notice an attempted although unaccomplished axiomatic formulation of special relativity: even by means of isotropy of the space or by the law of light propagation, one always assumes a constant to be determined or a parameter specified. Now, as a result of historical writing, it follows that the physical universe does not include Euclidean geometry only, as a privileged paradigm, but additionally includes other models.
In conclusion, the foundations of astronomy seem to be partially recovered by means of other new theories based on mechanics: should we have to wait a long time to see the foundations of astronomy replaced? Finally, since 1800 the history of astronomy has not been merely a collection of discoveries and tables, so should it, as an interpretative history, be integrated into the history of physics? (Pisano, 2012) A debate upon the role played by non–Euclidean geometries and their relationship to both the Riemann studies and modern conjectures on the expansion of the hyperbolic universe seems necessary.
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Keywords: relationship geometry–physics–mathematics, foundations, historical epistemology of science.
1. On geometry and continuum
The fundamental theory supporting the Newtonian pattern of continuum in mathematics as far as the study of physical phenomena is concerned can be attributed to Euclid. We can maintain that he connected in a symbiotic link the then well known properties of geometry and arithmetic elevating them in part to being the founding properties of the self–evident theories. The precariousness of the fifth axiom on the parallel straight line was well-known (and possibly accounts for the delay in its application). With Euclid’s theory being typically deductive, the derivation of that axiom was automatically obtained from the first four axioms. In fact:
[…] Euclid […], after examining and comparing the results of dealing with a precise topic set about to divide them harmoniously [according to an Aristotelian–axiomatic pattern] filling in the gaps, so as to create an organic whole (Loria, 1914, p. 279)1.
Euclid was, however, not able to prove the fifth axiom based on the first four. The Greek Posidonius from Rhodes (135 B.C.? 1 B.C.?), is supposed to be one of the first learned men to have shown his disappointment regarding the fifth axiom. He tried to re–arrange Euclidean geometry, attempting to prove the axiom of the straight lines in order to include it in theorems. Ptolomaeus also tried a logical deduction of the fifth axiom from the first nine (and from the first 28 of Euclid’s theorems too, although they were not based on it). Yet Proclus (fl. V B.C) condemned this method, maintaining that even if two straight lines came close to each other over the infinite, this would not imply any proof of an overlap at one point.2 The whole debate is clearly focused upon the concept of continuum and upon the choice of a particular type of infinite (actual or potential). It highlights the problem of whether to conceive the universe as being flat or to accept the theory of relativity. Only in the third decade of the 19th century, when in Nicolaj Inanovic Lobacevskij (1793–1856) and János Bolyai’s (1802–1912) works, and in the complex theories first by Carl Friedrich Gauss (1777–1855) and then Jules–Henri Poincaré (1854–1912), (Paty 1999) was it possible to claim the existence of a geometry different from that conceived by Euclid (Pisano, 2006b; Pisano & Casolaro, 2011); a geometry where the first four axioms were valid but not the fifth, a so–called non Euclidean geometry, but still perfectly coherent and logical. To sum up, the hypothesis of a geometrical space conceived differently from the Euclidean plane contributed to the pattern of the mathematical continuum, dealing with the interpretation of an ideal physics. It is interesting to recall Poincaré’s view, warning that
[…] but it would be a mistake to conclude from that that geometry is, even part, an experimental science. if it were experimental, it would only be approximative and provisory. And what a rough approximation it would be! Geometry would be only the study of the movements of solids bodies ... (Poincaré, [1905] 2007, p. 70, line 8)3.
2. On geometry and physics
When geometry is used in the study of ideal images4, a body (a virtual one) is the result of a process of idealization of reality, naturally elaborated by our mind; and the empirical performance in geometry, as Poincaré thought, represents but the occasional stimulus required to make the image materialize (Poincaré, 2003, p.117). According to Poincaré the geometrical axioms are neither synthetic previous comments nor experimental facts, but they are conventions. Therefore, it would be futile to wonder about the truth of such statements as they are already set as conventions. As well as being nonsense – according to Poincaré –wondering whether the metrics system is true and the old measurement system is false, an element which comes to be essential in the study of Poincaré’s geometry is the concept of group –meant either as a file containing information and properties valid under determined conditions or as a new way of conceiving geometry. In this sense, it is not possible to choose one type of geometry rather than another, because there is no such thing as a better geometrical group. It is well–known that in classical physics space represents a system where the natural phenomena are described and exist independently from the occurring events. In theory, this way of perceiving the physical sciences permits one to interpret the movement of a punctual–form body compared to an absolute space (Aristotle), even in the event that the description of the phenomenon is not anticipated by an experiment. In fact, classical physics assumes the mathematical sciences as a control element of reality, so that within the Cartesian reference model a (material) point comes to be associated unequivocally with a real number; for example, according to Newtonian theory any variation (in continuum5) of a physical magnitude is also a mathematical function of space and time. Concepts of space and time (Jammer. 1957, 1960. 1961) have, on the whole, been set as the foundation for subsequent theoretical physics. Evidently, due to the power of calculations and the authority of the Newtonian tradition, which gave birth to the physical–mathematical tradition, they were instituted as a theoretical pattern, influencing most of the physicists until the beginning of the 20th century: the continuum is an indisputable means of giving a mathematical interpretation to physical phenomena. It should be maintained that physics itself in 1900 often dealt with the interpretation of some physical effects (through the mathematics of continuum) perceived not always by direct observation, but by the study of built–up mathematical models, so to speak, built–up in papers in order to interpret both the micro and macro reality; also the scientific instruments used for the experiments were arranged according to those models. Let us just consider the operating principle underlying the detection of particles rather than the kinetic model of gases in the homonymous theory; or let us recall the problems encountered with the use of the standard model in modern physics. The opinion was supported by some mathematicians of the time that the mathematical issues proving so helpful in physics theory were the most reliable and relevant part of mathematics as a whole. No alternative to the close connection between physics and mathematics was considered possible, yet this was disputed.6 The context, though containing many underlying uncertainties, undoubtedly produced interesting scientific innovations, such as the birth of special relativity (1905) and general relativity (1916–19), even suggesting a total change in the physical (Newtonian paradigm) categories and in the (Kantian) philosophical categories of time and space. In this perspective, Einstein’s relativity theories have completely changed the foundations of classical Newtonian science, highlighting that, from a physical point of view, the acknowledgment of space, even in the absence of visible phenomena, is nonsense (as well as the absence of absolute time); that is why it is appropriate to imagine a space whose properties are related to the observer and to the body’s motion. Furthermore, as the speed of a body is the result of the variation of its position instant by instant, it is incorrect to consider space as being independent from time. This idea was properly developed by Einstein (Einstein, 1905, 891–921; D’Agostino, 1995, pp. 167–178) through the hypothesis of a fourth dimension “t” which, with the coordinates of space, should describe a phenomenon by quadrivector. Fundamental problems7 emerged: Hermann Minkowski (1864–1909) regarding the choice of space, more than twenty mathematical problems set by David Hilbert (1862–1943), the debate about logic in quantum mechanics, the introduction of differential geometry and the mathematics of the tensors8 in general relativity.
3. On spherical geometry and special relativity
In the first decades of the past century, space–time and mass–energy concepts became very important in distancing physics from the past but seemed to be ahead of their time. Most notably, these concepts appeared to be a real problem for the traditionalist physicist (Hirosighe, 1976). A new adventure in physics followed but this perturbed the scientific authorities of Newtonianism and Descartesian space. According to Einstein, a plane Universe (Euclidean Modelling) is an (materially) empty universe, devoid of matter since the presence of matter causes a curvature of the space. Regarding the inner phenomena of the solar system, it is well-known that general relativity brings light variations (which are rather remarkable with regard to the applicability of Newtonian theory) and space is slightly deformed. Yet for ca. 109 light year distances, the matter in the universe dramatically affects the geometrical nature of space, meaning that the Euclidean theory is not more basically applicable over huge distances. In fact, Einstein’s theory of general relativity is merely based on an elliptic model analogous to Bernhard Riemann’s (1826–1866) non–Euclidean geometry model. Half a century earlier, in his works (Smith, 1958) on k–variety space surfaces (n>k dim.), Riemann concluded that the physical space is 3–dimensioned with a constant curvature. From an exclusively geometrical point of view, the problem consists of the rigorousness of a model with regard to non–plane geometry by means of the theory of vectors. The perpendicular vectors to a plane at any point all have the same direction. Thus, in succession, performing on plane spaces, of three, four dimensions and so on, the perpendicular vectors to the space all have the same direction. Whereas, for a curved space, a direction of a perpendicular vector to the space changes at any given point, so the components of a normal vector to the space at one point (precisely at the plane tangent to the surface in that very point) result as functions of the coordinates. One can approach the non–Euclidean geometries from various points of view (Boi, Flament & Salanskis, 1992):
Elementary approach (Gauss, Bolyai and Lobacewski), developing by observations analogous to those involved within the Elements, except for some variations inside the postulates corresponding to the hypothesis of the acute angle.
Metrics–differentia approach, (Riemann) that consists of the extension to a variety of three or more dimensions of the differential geometry of the surfaces introduced by Gauss, with particular regard to the surfaces having constant curvature.
Grouping approach which, exploiting (as much as possible) the free mobility of figures in space, studies those particular transformations, the motions keeping all distances unaltered.
Projective approach, that states the metrics properties of the figures in a projecting form.
Riemann assumed that the properties singling out physical space from other kinds of space can be deduced by experiment only; consequently, he thought that the axioms of Euclidean geometry could be approximately shown and related to physical space. Moreover, his elliptic model (1854), which can be considered an extension of Gauss’ differential geometry, stated that within a space, where the curvature changes from one place to another, owing to the presence of matter, and instant by instant (owing to the matter motion) the Euclidean geometry laws can no longer be applied. Therefore, in order to determine the adequate nature of a physical space, space and matter (consequently, time) should be related to each other. Assuming an analytical approach9to the local behaviour of space, Riemann used the metrics–differential method. For the linear elements, he defined the distance between two points as follows:
have ranks ??2
A? and represent, respectively, a space 2S (analogous to the Euclidean plane) as a hyperplane of3 S , and a space 1 S (analogous to the Euclidean straight line) as a hyperplane of2S . According to Einstein’s approach, the functionsgij, that, in (2), represent the coefficients of the variables x, y, z, t, include the effects of gravitational masses in the space. Such effects (causing a curvature at any point) are the source of the orientation change of the normal vector to the surface. By introducing time as coordinate the analysis is combined with the 4–dimensional space–time à la Minkowski. The set of the points–events in (x, y, z, t) defines a continuum by 4–dimensions and represents a geometrical space4S . In fact, the space 3 S of the points (x, y, z) of the Euclidean model, assumed as subspace of4S , can be considered as a hyperplane of space–time à la Minkowsky. If we analyse the structure of the subspaces of 4S and the analytical Euclidean pattern, we can deduce:
By t=0 (or t=const.), we have the hyperplane3 S , which is the 3–dimensional Euclidean geometrical space where the laws of the classic kinematics are still valid;
Whereas, if one of the coordinates is constant, we have hyperplanes 3 S of 4S which featuring kinematics and relativistic models over 2S is analogous to the Euclidean plane.
Let us see an example. A set of the points P(x, y, 0, t) ? P(x, y, t) is a 3 S space, whose coordinates (x, y, t) mark the points–events of the plane 2S (x, y) extended to kinematics by the introduction of the time coordinate. In such a context, it should be taken into account that metrics is defined by
ds???? (3)
Such a relation, presenting a minus sign in the fourth term of the quadratic expression, can be geometrically justified, by adding to the real term (O,x,y,z) of the reference S3 a forth imaginary orthogonal axis at S3 where the time coordinate is multiplied by the imaginary unit i. According to a dimensional
4. Some final remarks
Between the 16th and 18th centuries, modern science reached a peak. Many factors, social and political
were part of this process. A crucial transformation of scientific theories characterized the development of
physics and mathematics in astronomy (Pannekoek, 1961).
Table 1
Paradigm lost?
Astronomy should not question religion or philosophy and did not aim to confront the absolute power of
the past. In this sense, geometry cannot theoretically be abstracted from the physical evolution. As a matter of fact, according to the classical conception, geometry expresses a set of properties with regard to the motion of bodies and the law of the propagation of light, sometimes obtained through the abstraction from time and forces. Thus, by the application of geometry to kinematics (that is the theory of movement in space–time) and, later on, to dynamics (by the introduction of forces) interpretative approximations closer to the physical phenomenon, shall result. This should all be further clarified thanks to the theory of general relativity. Furthermore, we notice an attempted although unaccomplished axiomatic formulation of special relativity: even by means of isotropy of the space or by the law of light propagation, one always assumes a constant to be determined or a parameter specified. Now, as a result of historical writing, it follows that the physical universe does not include Euclidean geometry only, as a privileged paradigm, but additionally includes other models.
In conclusion, the foundations of astronomy seem to be partially recovered by means of other new theories based on mechanics: should we have to wait a long time to see the foundations of astronomy replaced? Finally, since 1800 the history of astronomy has not been merely a collection of discoveries and tables, so should it, as an interpretative history, be integrated into the history of physics? (Pisano, 2012) A debate upon the role played by non–Euclidean geometries and their relationship to both the Riemann studies and modern conjectures on the expansion of the hyperbolic universe seems necessary.
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