Whitehead on the Riemann-Helmholtz-Lie Problem of Space Intervention lors de Mathematics in Philosophy, 25 novembre 2014, Université de Liège, International Workshop. Organisé par: François Beets, Emeline Deroo, Bruno Leclercq, Stany Mazurkiewicz, Vesselin Petrov. Avec le soutien de l’Académie bulgare des sciences, le Centre Nationale de Recherches en Logique, la Faculté de Philosophie et Lettres de l’ULg. À paraître dans le Balkan Journal of Philosophy, édité par Vasselin Petrov, Institute for the Study of Societies and Knowledge, Bulgarie. * * * It is well-known that the uniformity of space, implying a constant degree of the Gaussian curvature, was required according to Whitehead, as a necessary condition in order to satisfy the possibility of measurement. Anyway, such a demand compelled him to hold the flatness of the Euclidean space. This point is obvious, reading the preface of The Principle of Relativity with application to Physical Science (1922): “It is this uniformity which is essential to my outlook, and not the Euclidean geometry which I adopt as lending itself to the simplest exposition of the facts of nature. I should be very willing to believe that each permanent space is either uniformly elliptic or uniformly hyperbolic, if any observations are more simply explained by such a hypothesis.”1. Even a merciless opponent of Whitehead’s natural philosophy, Adolf Grünbaum, willingly granted it2. The argument that, without spatial uniformity, it would be impossible to export our measurement standards from one part of the universe to the whole, plays of course the major part against the use of Riemannian manifolds in Einstein’s theory of general relativity. However, the objection was rather reenacted than provoked for this occasion. Whitehead’s claim of spatial uniformity was already formulated in Russell’s disowned book An Essay on the Foundations of Geometry (1897) further renewed in his article “Geometry, Non-Euclidean”, written for the tenth edition of the Encyclopaedia Britannica “If our coordinates are to represent any kind of spatial magnitudes, we must assume the possibility of equal quantities in different places, and hence, it will be found, we shall be compelled to regard the measure of curvature as constant. Let us examine the consequences of supposing it variable. In the first place, the method of superposition would have become impossible, so that measurements could no longer be affected. Thus a metrical coordinate system would become impossible. Moreover, geometry would become akin to geography; it would not consist of general theorems, but of descriptions of various localities. The variation of the space-constant would be not quantitative merely, but qualitative, and quantities in different places would be of different kinds. Thus the constancy of the measure of curvature is a 1 2 2004, p. v. 1963, p. 426. 2 precondition of any metrical coordinate system, and cannot be held doubtful while such a system is retained.”1 Henceforth we could be stricken with the contrast between, on one hand, Russell’s capacity of evolution, beginning with this thesis of an a priori knowledge of the spatial uniformity towards the main stream of the theoretical physics, when he will write The Analysis of Matter (1929); on the other hand, Whitehead’s stubborn refusal that the geometry of the real space actually become something like a geography of space-time. Here, our hypothesis will be that the so-called Riemann-Helmholtz-Lie problem of space constituted for Whitehead an intangible structuring frame of though, explaining his reception of the Einsteinian physics. I – What was the “Helmholtz-Riemann-Lie” problem of space? At the end of the XIXth century, grounding metrical geometry apart from the physical notion of movement seemed hopeless. The metrization of space implied to determinate the conditions of congruence or superposition; and the very possibility of congruence itself led to draw the necessary and sufficient conditions for moving a rigid body. What did the meaning of the opposition between hypothesis and facts, supposed at the basis of the geometrical metrization of space, involve? 1. Riemann: the starting point of the problem of space. In the aim of characterizing the nature of physical space, Riemann takes the issue at the highest level of generality, namely, from the concept of magnitude with any number of dimensions, in his famous Habilitationsvortrag lectured in 1854. Magnitude divides two kind of manifold depending on whether it’s discrete or continuous. We obtains quantum of a magnitude as soon as a limit is introduced. In order to compare various quanta, the operation of counting on the side of the discrete manifolds corresponds to measuring on the side of the continuous manifolds. Riemann trusted in the differentiability of a continuous manifold to express the simplest way of spatial metrization2. His leading twofold hypothesis bears on the sufficient and necessary conditions for the determination of the metric relations of space3. The first hypothesis would grant the possibility of comparing any parts of space in the case of continuous manifolds, initially deprived of any relation to numbers. Indeed the possibility of measurement is guaranteed with the transference from place to place of the same standards 4. If the geometer agrees that “measure-determinations require that magnitudes should be independent from position”, the first hypothesis following from amounts to: “[…] that according to which the length of lines is independent of their position and consequently every line is measurable by means of every other.”5. 1902, « Geometry, Non-Euclidean », Encyclopædia Britannica, tenth edition, 28, pp. 664-674 (also The Collected Papers of Bertrand Russell, vol. 3, Towards the “Principles of Mathematics”, 1900-1902, Routledge, London and New York, part IV, « Geometry », § 18, pp. 474–504). The same arguments were put forth in (Russell, 1897), pp. 152-153. – We underline. 2 “On the Hypotheses which lie at the Basis of Geometry”, part II § 1, translated by William K. Clifford, Jürgen Post editor, Birkhäuser, 2016, p. 34-35. 3 Ibid. part III § 1, p. 38. 4 Ibid. part II § 5, p. 38. 5 Ibid. part II § 1, pp. 34-35. 1 3 A continuous manifold allows Riemann to divide any line in infinitesimal lines, letting their length unaltered after moving. According to a second hypothesis, any of these infinitesimal lines may be expressed by the square root of a quadric differential (expression of the second degree). Among the wide generality of metrical structures, Riemann focused on a privileged class (without explanation) nowadays called “Riemannian manifolds”, with a measure of constant curvature. The infinitesimal length ds of an arc between two points infinitely neighbour is given with a homogeneous quadratic function. With the coordinates of the point P being (x1, x2 … xn), and of P′ in its neighbourhood being (x1 + dx1, x2 + dx2 … xn + dxn), this function is: ds2 = ∑𝑛𝑖,𝑗=1 𝑔𝑖𝑗 𝑑𝑥𝑖 𝑑𝑥𝑗 This is with the metrical tensor 𝑔𝑖𝑗 expressing the distance between two points infinitely neighbour as a function of the second degree of the differentials of the coordinates. Riemann’s concept of distance is represented by a function linking P with the coordinates (x1, x2 … xn) to P′ with the coordinates (x1 + dx1, x2 + dx2 … xn + dxn). This function associated with a manifold of any dimensions has continuous second derivatives, and attains its minimum value, zero, namely at the point P. Manifolds where the square of the linear element may be reducible to the sum of the squares of differential expressions constitutes the simplest mode of metrization. Riemann could choice another root than the square root. Hermann Weyl knew that the Pythagorean metric would include a more general theory of space (due to Finsler, 1918). The insuperable difference between a discrete and a continuous manifold is that in the later, there is no intrinsic metric. Space must be metrized in many different ways on the basis of various extrinsic standards. If distance is expressed with the so-called “fonction de Poincaré”: 𝑑𝑥 2 + 𝑑𝑦 2 ds = √ 𝑦2 this formula deals with geometry of the hyperbolic kind. Hence, that what is congruent for one system will be no more congruent according to the other. If physical space is only deduced from the concept of a continuous manifold, it is so amorphous that another metric may be applied to the same space than the Euclidean one. 4 From Analysis situs toward space as a Riemannian Manifold Manifoldness (Mannigfaltigkeit) discrete continuous (measuring) (counting) metrical structures Riemannian manifolds non metrical structures determined by other functions than quadratic positive or negative degree of the Gaussian curvature flat or manifolds locally isometric to the Euclidean space (cone, cylinder) 5 At the end of a dichotomy, the final hypothesis bears on the probability that physical space may be relevant to the Euclidean mode of metrization characterized by the nullity of the degree of spatial curvature. So Riemann started with a wide hypothesis granting the possibility of measurement, for narrowing to a terminal limited hypothesis about the nature of physical space, thanks to the Gaussian concept of curvature. However, such a progressive restriction is not able to give the last evidence of the Euclidean nature of physical space. Since natural philosophy remains in the area of probability, a conventionalist interpretation of Riemann’s Habilitationsvortrag anyway would be irrelevant. Indeed, if from the a priori analysis of the concept of magnitude stems a wide range of spatiality, experience suggests among them to opt for the class of spaces of constant degree of curvature. In every space of constant curvature, “figures” may be moved without any distortion, stretching or tearing, so that any translation or rotation from or around any point, keeps the same metrical relationships, only depending of the coefficient of curvature. While Riemann deals with some general considerations about the free mobility, he used the word “figure” (Figur): “Manifolds whose curvature is constantly zero may be treated as a special case of those whose curvature is constant. The common character of those continua whose curvature is constant may be also expressed thus, that figures may be viewed in them without stretching. For clearly figures could not be arbitrarily shifted and turned round in them if the curvature at each point were not the same in all directions.” 1. In the following paragraph (III, § 1), the word “surface” (Fläsche) is associated to the whole spatiality which comes under the measurement of curvature. So “figure” is able to work as a generic term for some geometrical loci then specified as rod or segment or as bodies. It is worth noting that the condition for a rod to be moved around any point of a manifold without changing its length is a weaker hypothesis than the assumption that a body can be transported without harm. As the condition of the free-mobility bears only on segments, geometries on surfaces are not necessarily of constant curvature. But as soon as this condition concerns bodies, it is linked with a curvature which remains of the same degree in all directions for any point of the manifold. “ […] if we assume with Euclid not merely an existence of lines independent of position, but of bodies also, it follows that the curvature is everywhere constant; and then the sum of the angles is determined in all triangles when it is known in one.” 2. Is this outcomes drawn from the two starting hypothesis are limited to the space of null degree of curvature? Not at all, as we saw, scrutinizing part II, § 4. In the last part III, § 2, Riemann asked “to what extent these assumptions are borne out by experience”. Yet he didn’t manage to check empirically his hypotheses by experience. He still held at a high level of generality, underlying the demarcation between his own conceptual enquiry and the content of physical science. So rather sceptically than in an empiricist way, he concluded pointing out according to which conditions experience would be satisfied, in order to have some confidence concerning the nature of space. ― Firstly, he brings to the fore a new distinction, then famous thanks to general relativity theory, between infinity and unboundedness. The experimental confirmation of the unbounded spatial manifold of a three-dimensional has the benefit of a high level of empirical certitude. However Riemann claims that the 1 2 Ibid. part II § 4, p. 37 (we underline). Ibid. part III § 1, p. 38. 6 unboundedness extent doesn’t imply a spatial infinity, as it is obvious that, endowed with a positive curvature, space would be bounded. ― Secondly, the importance of the knowledge of the infinitesimally small metric relations is emphasized, considering the role of causality in modern physics. If we come back to the hypothesis that bodies exist independently from a position, the degree of spatial curvature is constant. Riemann goes on: and considering astronomical measurements, spatial curvature couldn’t be different from zero, probably alluding to the fact of the sum of the sides of triangles equal to two right angles. “If we suppose that bodies exist independently of position, the curvature is everywhere constant, and it then results from the astronomical measurements that it cannot be different from zero; […]” 1. Let’s notice that he does not plead for the Euclidean or at least for the constant curvature of the physical space on the ground of observing the free-mobility of existent bodies. However he cannot grant that this hypothesis is the good one, given that the opposite is taken into account: “But if this independence of bodies from position does not exist, we cannot draw conclusions from metric relations of the great, to those of the infinitely small; in that case the curvature at each point may have an arbitrary value in three directions, provided that the total curvature of every measurable portion of space does not differ sensibly from zero.” ― First implicit consequence (turning around the first negative sentence in an assertion): if we suppose that bodies exist independently of position, we can draw conclusions from the great level to the infinitely small. Thus space exhibits uniformity. In all spaces of constant curvature, not only Euclidean, it turns out that “the sum of the angles is determined in all triangles when it is known in one”2. It doesn’t matter if these sums are equal to the sum of two right angles, as in the Euclidean geometry, less than this sum as in the Lobatschewski-Bolyai hyperbolic geometry or more, as in the Riemann elliptic geometry. ― Second explicit consequence: Clifford translated the German coordinating conjunction “wenn” by “provided that”, which put the possibility of arbitraries values in three directions, under the condition of the nullity of the spatial curvature. Unless I mislead, “wenn” may have a weaker translation in order to merely mean “when”. So the second part of the last quotation may signify that it could be the case that the measurement of curvature would be null at the great level, while the metric relations in the infinity small are no more uniform. In his conclusion, Riemann didn’t deal with the relation of the tangent space with the global spatial form of the universe. He gave a direction for a physical enquiry, suggesting that at the microscopic level, metric relations could be different form the observational level. Concerning the shape of physical space, he let us in front of different hypothesis, recalling that, even at the top of the dichotomy, any decision is asserted. “The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. In this last question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete manifoldness, the ground of its metric relations is given in the notion of it, while in a continuous manifoldness, this ground must come from outside. Either therefore the reality which underlies space must form a discrete manifoldness, or we must seek the ground of its metric relations outside it, in binding forces which act upon it.”3 Ibid. part III, § 3, p. 39 (we underline). Ibid. part III § 1, p. 38. 3 Ibid. part III § 3, p. 40. 1 2 7 His prominent aim at the end of the Habilitationsvortrag seems again to widely extent the scale of possibilities likely to furnish some alternative forms to the customary space. Riemann attempted to uninhibit the physicist about the subject-matter of space on the occasion of a mere conceptual exercise, superseding the allegiance to the Euclidean one. Without making up his mind, he opened the problem of space, highlighting, for the use of his followers, two main ideas. From the philosophical side: if space is not a discrete manifoldness, metrization must come from outside. Here lies the ground for a conventionalist solution of the problem of space. From the mathematical side: the conditions of free-mobility of bodies, joined to the constancy of the Gaussian curvature, are distinguished, even if not strictly explicated, from the ones relevant to a rod. 2. Helmholtz’ preliminary conditions for grounding the free-mobility of bodies For alleging a more empiricist approach of the problem of space by Helmholtz, it is often noticed the substitution of “facts” to “hypothesis” in the title of his two papers (1868): “Über die Tatsächlichen Grundlagen der Geometrie”1. This was followed by “Ueber die Thatsachen, die der Geometrie zum Grunde liegen”, with more analytical geometry2. Nevertheless, he independently made up his mind about the nature of space, as they stemmed from his works on colours and the determination of distance in the visual field. Among the geometrical assertions, some comes from facts and other are mere definitions. Helmholtz tried to grasp which one come from experience. As some geometers did, he had doubts concerning the completeness of Euclid’s axioms. Further, having read Riemann’s Habilitationsvortrag before publishing, Helmholtz saw his own outcomes as a reversal of the Riemannian method: “Riemann starts by assuming the above-mentioned algebraical expression, which represents in the most general form the distance between two infinitely near points, and deduces therefrom the conditions of mobility of rigid figures. I, on the other hand, starting from the observed fact that the movement of rigid figures is possible, in our space, with the degree of freedom that we know, deduce the necessity of the algebraic expression taken by Riemann as an axiom.” 3 As we saw, Riemann didn’t exactly airdropped the algebraic expression ds = √∑(𝑑𝑥)2 as an axiom: he recognized, in this function of the coordinates of two distant points, the simplest mode of spatial metrization. After determining the flat curvature of space as a particular case of spaces of constant curvature, it is plain that, given the sameness of the space curvature at any point and direction, figures are able to move arbitrary without distortion. Granted that some geometrical assertions come from experience, needing no more help with a special intuition, he could achieve thanks to analytical tools only. Indeed, Helmholtz’ starting point was not exactly mere facts, but the observability of a fact, depending of an “Anschaulichkeit” (ability of being plainly imaginable if not actually perceived). The free movement of a rigid body has itself for necessary and sufficient conditions, the possibility of occupying an infinite series of congruent positions through space. So the possibility of a judgement of congruence constitutes the fundamental hypothesis for granting all spatial measurement. 1 2 1868a, pp. 197-202 (lectured 22th, May, 1866). 1868b, pp. 193-221 8 The ideas of continuous manifold, rigid body and free-mobility are linked together under four conditions. As a result, Helmholtz thought he could performed the hard task of concluding the Pythagorean generalised theorem from the conditions (Voraussetzungen) of the free-mobility of a rigid body. (1)Condition concerning continuity and dimensions: Helmholtz just quoted Riemann about the determination of space as a differentiable manifold, for directly focusing on the analytical considerations about space. The physiologist he was noticed that a system of colour could be another example of a threefold extended manifoldness. In any space of n-dimensions, the position of points are determinated by n magnitudes mutually independent. During any movement, each coordinate shifts continuously and independently form the others, with at least one of them not remaining constant. 2) Condition granting the existence of moving rigid bodies: supporting the measurability of space, some rigid bodies must exist, so that at least two points let something invariant during each movement, namely, their distance. From an analytic point of view, the equations linking the 2n coordinates of each couple of points belonging to a moving body remain independent from this movement. Consequently a movement is an isometric transformation of space into itself, namely a kind of automorphism. Assuming that at least two points keep a same spatial relation, Helmholtz holds a concept of distance and was able to give a metrical geometry. In a second version of this postulate in “Uber den Ursprung und die Bedeutung der Geometrischen Axiome” (1870), Helmholtz beforehand avoided the criticism of having committed in a vicious circle1. The objection is that without presuming a concept of distance, one cannot defining rigidity. In order to get around, it is strategic to postulate the sameness of the equations linking all congruent couple of points. (3) Condition concerning the free mobility of rigid bodies: any point may be translated from a position to another. However, linking this assertion to the previous axiom, some restrictions limit this freedom in the case of the movement of a solid body. Each couple of points may continuously come at the place of any other congruent couple of points, in order that their function of distance between them still remains invariable. In our experience of space, the movements of a rigid body occurs according to six degrees of 𝑛 (𝑛+1) freedom, that is, with n = 3 as the number of spatial dimensions: 2 = 6. These six degrees are shared out to three degrees for the group of translations and to three others for the group of rotations around each axis of translation. (3) Condition of invariance of the form of rigid bodies by rotation (monodromy): then Helmholtz put forwards a last axiom, assuming that the three previous would be compatible with the rotation of a point around a central axis determined by two fixed points, with the distance from the axis of rotation growing more and more, describing a spiral on the plane, or a screw in space. (4) Condition of monodromy : a fourth axiom, then at length discussed by Klein, Lie and Engel, Killing, Poincaré, gives the assurance that at the end of any movement of rotation, a body carries out its track and comes back at the starting. Thus the rotation may be periodic instead of spiral or helicoid. 1 1884, pp. 1-31 (about the second hypothesis, H. von Helmholtz, Epistemological Writings, Cohen and Elkana ed., 1977, p. 16 sq.). 9 However Helmholtz made a mistake, ascribing to Riemann the assertion that: if space would be infinite, this would imply a flat curvature. Beltrami was so upset reading Helmholtz that he thought he has himself misled and wrote to him (24th April 1869), who recognized that the property of infinity further pertains to spaces with a negative curvature1. Under the name of pseudo-spherical spaces, Beltrami has conceived in his 1868 paper “Saggio di interpretazione della geometria non-euclidea”, a first Euclidean model for the LobatchevskyBolyai geometry2. As the expression of a remorse, in the 1870 lecture (translated in Mind, 1876), Helmholtz devoted a substantial analysis to Beltrami’s work. Strikingly, from now on he involved the non-Euclidean geometries in his axiomatization of rigid bodies. Right away, he put into perspective Kant’s philosophy of geometry in relation to the unsuccessful attempts for demonstrating the axiom of parallels. Not taking the geometries others than with flat curvature into account, the demonstration of the year 1868 could support the necessity of the Euclidean space. Two years later, he anyway didn’t give up this previous intuition, that at the basis of: “[…] all proof by Euclid’s method consists in establishing the congruence of lines, angles, plane figures, solids, etc.”3 Henceforth, in order to weaken the Kantian idea that space is a necessary form of our representations, Helmholtz conceived a famous fiction, afterwards taken over by Poincaré, telling the story of flat people conceiving an alternative geometry in a flat world. As the dwellers live in a sphere, they would ignore the notion of parallelism as well as all concept of similitude between figures, so that the relation of similitude clearly appears like a specificity of the Euclidean geometry. “Of parallel lines the sphere-dwellers would know nothing. They would declare that any two straightest lines, sufficiently produced, must finally cut not in one only but in two points. The sum of the angles of a triangle would be always greater than two right angles, increasing as the surface of the triangle grew greater. They could thus have no conception of geometrical similarity between greater and smaller figures of the same kind, for with them a greater triangle must have different angles from a smaller one. Their space would be unlimited, but would be found to be finite or at least represented as such.”4 Helmholtz ’empiricism upholded that the type of geometry we built depends on the kind of world we inhabits. Geometry of positive curvature arouses a spherical geometry. So it seems that spaces of negative curvature and of zero curvature share the conjoined properties of infinity and unboundedness at the opposite of the spherical one. Indeed “pseudo” J.-D. Voelke, Renaissance de la géométrie non euclidienne entre 1860 et 1900, pp. 226-227. For an analysis of Beltrami’s essay, see L. Boi, Le problème mathématique de l’espace, 1995, chapter 6. 3 Mind, vol. 1, n° 3 (July, 1876), p. 303. 4 Ibid., p. 305. 1 2 10 could be supersede by “anti”, Beltrami’s model of a negative constant curvature being a negative sphere without limit, exactly representing the interior in a sphere. The surface aabb (see figure 1) results from a rotation around the symmetry axis AB. At the top and the bottom of the two arcs of circle ab, namely the two edges, the curvature bends more and more, becoming perpendicular to the axis AB, and finally reaches an infinite curvature. Actually, this model cannot represent exactly the Lobatchevsky-Bolyai geometry (Beltrami’s original model takes the shape of a “Kelche”, that is a champagne glass becoming at the bottom infinitely thin: figure 2). Degree of curvature constant Axiom of parallelism Existence of a geodesic Possibility of extending a line infinitely Similitude of figures Sum of the angles on a triangle Plane geometry (Euclid) Pseudo-spherical geometry (Lobatchevsky-Bolyai) zero negative one parallel crossing a point without crossing a given straight line Between two points, exist one and only one straight line which is the shortest way yes yes Equal to two right angles Exists an unlimited number of straight lines, infinitely extended, inside a sheaf, without crossing a straight line passing by a given point Spherical geometry (Riemann, but “Riemannian manifolds”) positive any parallel Idem in Euclid Between two points, exist an infinity of line which are the shortest ways yes no Inferior angles no to two right no Superior to two right angles Updating his geometrical knowledge after 1868, Helmholtz didn’t give up his previous angle of attack of the analytical geometry, as far as some notions of magnitudes are obviously implied in our synthetic table. Did the awareness, in the 1870 lecture, of the non-Euclidean geometries modify the data of the problem of space, compared to the anterior version provided in 1868? Helmholtz saw an endorsement from Rudolf Lipschitz’ recent work (“Untersuchungen in Betreff der ganzen homogenen Funktionen von n Differentialen”, 1869 and “Untersuchungen eines Problems der Variationsrechnung, in welchem das Problem der Mechanik enthalten ist”, 1872) about the quadratic differential forms into mechanics1. Helmholtz noticed the applicability of the main principle of dynamics, namely Hamilton principle, on the spherical and pseudo-spherical spaces. Hamilton provided classical mechanics with a particularly strong principle according to which a mechanical system moves from a configuration to another, in order that a function of the Helmholtz quoted Journal für die reine und angewandte Mathematik, 1869, first ruled by Crelle, according to the editor’s name then: Borchardt’s Journal für Mathematik, and substituted “über die” to “in Betreff”). The Helmholtzian 1870 lecture having been published not before the vol. iii of Populären wissenschaftlichen Vorträge, Braunschweig, 1876, it was possible to add the reference to Journal für die reine und angewandte Mathematik, 1872. 1 11 magnitudes, namely action, between the initial state and the final one, generally takes a minimum value. So not only Helmholtz took into account other spaces of constant curvature as Euclidean, but moreover the fact that the transposition of the laws of dynamics into nonEuclidean spaces led to any contradiction. These obvious conclusions are drawn from the reformed Helmholtzian outlook: (1) the particular determinations of the actual space as flat are not enrolled into the general concept of an extended three-dimensional magnitude; (2) the fact of the free-mobility turns out to be unable to discriminate one kind of space as the actual one, because the two other types of constant curvature space than the Euclidean one are compatible with the laws of dynamics. The characterization of our homely space cannot stem neither from the a priori concept of spatial continuous magnitude nor from the free-mobility of rigid bodies contained inside. So this characterization, no more relevant from “necessities of thought”, must arise from an empirical origin, mainly from: (1) the possibility of comparing similar bodies but of different sizes, granted only in the Euclidean space; (2) The fact that in our universe, the parallax of stars infinitely distant stars is null (despite of that the enlargement of the considered distances could weaken the argument, following a Riemann’s caveat). II - Whitehead facing the Riemann-Helmholtz-Lie problem of space in The Axioms of Descriptive Geometry (1907 ) The place where Whitehead gives his full attention to the Riemann-Helmholtz problem of space is situated in the §§ 42-45 of the Cambridge tract The Axioms of Descriptive Geometry (1907, after ADG). Joined to The Axioms of Projective Geometry (1906, after APG), this set more often was seen by some historians of mathematics as a mere appendix of the part VI of Russell’s The Principles of Mathematics (1903). Probably are they somewhere disappointed by the missing of a Hilbertian formalization perhaps promised under the two titles? Recently, Sébastien Gandon and Ronny Desmet focused on the brilliant definition of geometry we can read as the “science of crossed-classification” at the beginning of the first tract, from which geometrical space is recognized to be a structure of incidence1. The three points I intend to bring in the fore in the following are: Firstly, the problem of space, as re-enacted by Whiteheadian, integrates the solutions brought by Sophus Lie and Friedrich Engel at the third volume of Theorie der Transformations-gruppen (1893) for the Riemann-Helmholtz problem of space. Secondly, we have to explain the reservation expressed in the foot-note (*): “But Lie’s line of thought was not that suggested above”. 1 Gandon, 2012, § 1.5, pp. 44-46; Desmet, 2010a, pp. 135-136. 12 Whitehead’s The Axioms of Descriptive Geometry (1907) about Lie’ solutions Thirdly, in these paragraphs, Whitehead deals with the crucial topic of congruence. Concerning the definition of congruence, are Helmholtz and Whitehead in accordance? (1) Lie has learnt around 1875 the problem of space from Felix Klein. Being aware that this problem amounted to look for a class of transformation group, he expressed his doubts on Helmholtz’s axioms to Felix Klein in 1883, then publicly three years later, in his Berliner lecture intitled “Bemerkungen zu Helmholtz’ Arbeit über die Thatsachen, die der Geometrie Grunde liegen”1. Then his introduction of the chapter V of the magnum opus, third volume, highlighted the shortcomings he found both in Riemann both in Helmholtz. For summing up very quickly Lie’s approach, the key was given by the study of the group structure of spatial transformations, which are continuous in this case. On the plane, rotations constitute a group named SO (2), which is a Lie continuous finite group with one parameter, because of the angle of rotation. In a three-dimensional space, we hold the SO(3) group ― as three parameters are necessary for describing this group ― losing the property of commutativity displayed by SO(2), as Rowan Hamilton was aware of with his discovery of quaternions. As we know, Hamilton vainly struggled to build a theory of algebraic triplets, after performing those of algebraic couples, failure from which the algebra of quaternions was born. Given that in the Euclidean space, three degrees of freedom are SO(3) non-commutative group counted for translations and equally for rotations, the problem of space, stated in the group theory language, becomes: finding all the transitive sub-groupes with six parameters for which two points will have only one invariant, systematizing the relation of a couple of points, without the need of the solidity of 1 1886, pp. 337-342. 13 rigid bodies. As Whitehead emphasized: a finite group is called transitive if each of its points may be transformed into another through at least one transformation of the group. Lie detected a mathematical mistake in Helmholtz’ axioms of the free-mobility, resting upon a defective inference from that what is valid in finite spatial areas, to a set of points which are infinitely neighbour. So it will be two solutions for the Riemann-Helmholtz problem, distinction which was retrieved in Whitehead’s tract on the axioms of descriptive geometry (§§ 43-44): Firstly, for the case of infinitesimal areas (1893, chap. XXII, §§ 97-100, summed up in ADG, § 43). The Lie algebra connected to the group of transformation arose with the study of the angles of infinitesimal rotations in the neighbourhood of identity. The export of an invariant between some finite distances to infinitely small isn’t at all obvious. For putting the problem on the right track, a new concept of free-mobility in the infinitesimal was required. This concept of freemobility in the infinitesimal is defined first on the plane (§ 97), then in the three-dimensional space (§ 98), finally in space of any number of dimensions (§ 99). - As a result, if some movement belongs to a real continuous group of transformations with the free-mobility in the infinitesimal, thus the groups of Euclidean and non-Euclidean movements are completely determined. This consists in a purely mathematical reasoning without the help of any dynamical consideration. Congruence lies only on the transformations of points through space. - Secondly, for the case of finite areas (1893, chap. XXIII, §§ 101-103, summed up in ADG, § 44). The problem becomes more difficult according to Lie and Engel themselves, who gave only a complete solution for the case of the three-dimensional space. However, the second solution of the space problem kept on dealing equally with Euclidean as non-Euclidean movements. (2) Why did Lie’s twofold solution brought to the problem of space let Whitehead quite unsatisfactory, given that he finally chose Peano’s axioms about congruence? Indeed, Whitehead compares Peano’s axioms of congruence with the first solution of the space problem expressed under the language of the group theory, to the detriment of Lie and Engel. “The conception of a finite continuous group, though it is simple enough analytically, does not seem to correspond to any of the obvious and immediate properties of congruencetransformations as presented by sense-perceptions. The following set of axioms conforms more closely to the obvious properties of congruence-transformations; they are based upon, and are modifications of, a set of congruence-axioms given by Peano.”1 Even if he underlines the adjective “finite”, he manifestly enjoyed to give up the whole expression “finite and continuous”, as we read further, in favor of “a set of congruence-axioms given by Peano. The new axioms Whitehead found more useful came from Peano’s 1894 memoir “Sui fondamenti della geometria”, supposed to be more convenient for supporting the “[…] obvious and immediate properties of congruence-transformations as presented by sense-perceptions”. We must, of course, pay close attention to the very astonishing sudden entrance of a vocabulary pertaining to the natural philosophy phase (1919-1922) of Whitehead’s epistemological development, a little early at the time of the axiomatic tract. 1 ADG, § 45, pp. 47-48. 14 This forward-looking arising didn’t escape to Desmet in his 2010 study on “A Refutation of Russell’s Stereotype”, who adds this commentary about Whitehead’s refusal of Lie group theory for defining congruence: “Whitehead takes his distance from Poincaré’s preference for the language of group theory, used by Felix Klein and Sophus Lie to express all types of geometry. Indeed, the groups involved are groups of imagined transformations of one figure into the other, and those transformations – reflections, translations, rotations, etc. – are not directly given in sense perception, but are the result of a complex process of thought. The latter account invites us to argue that Whitehead’s vision can indeed be projected back to the period in which he was a Cambridgian mathematician, for in 1906, Whitehead publicly takes his distance from the group theoretical approach of metric geometry in terms of Lie’s congruence groups in favour of a more direct sensory approach in terms of Peano’s axioms of congruence.” 1 Just at the beginning of the chapter V of ADG, § 41, Whitehead quoted Moritz Pasch’s axioms of congruence. Peano in 1894 had expressed Pasch’s axioms in the symbolic language of his Formulario mathematico. We can compare the four first axioms belonging to Peano, through Pasch, with the four first axioms linked by Whitehead with a conception closer to perception. ADG § 43 - From Lie’s first solution of the problem of space : ADG § 45 - From a modification of Peano’s closer to sense-perception : Obviously, Whitehead didn’t give up the group theory approach, and more probably, has he displayed a group-theorical way for explaining a stuff quite far from original Peano’s exposition of Pasch’s axioms of congruence, as we can see under. 1 2010b, p. 164. 15 ADG § 41 (from Moritz PASCH, Vorlesungen Giuseppe PEANO, “Sui fondamenti della über neuere Geometrie, 1882, § 13) : geometria” (Rivista di Matematica, vol. IV, 1894, pp. 76-77) : Perhaps Whitehead seems as much as likely to keep his distance from Lie than bringing Peano closer to group theory. Through this operation, he provides us a hybrid formulation between an empiricist assertion concerning the topic of congruence announcing the future natural philosophy, and the advancement towards more mathematical abstractions thanks to the theory of group-congruence. His mathematical goal was indeed in any way trivial. That was at stake amounted to find a theory-group expression of the Cayley-Klein theory of distance in projective geometry. As explained in A Treatise of Universal Algebra (1898) Book VI Chapter 1, distance becomes proportional to the logarithm of the cross-ratio of four points ranged on a line cutting a fundamental conic on the plane, or a quadric in space, called “the Absolute”. And when Whitehead reached his goal (ADG, §§ 65-66), he could no more conceal, renewing with Lie’s language, that a congruence group in a three dimensional space consists in a sixlimber (i. e. with six parameters) finite continuous group1. Further, the congruence groups are divisible intro three types: - If the Absolute is an imaginary quadric, such congruence group is called Elliptic. - If the Absolute is a real convex quadric, such a congruence group is called Hyperbolic. - If the Absolute is the limit-case between the latter, or degenerates towards an infinite plane, groups of this type are called Parabolic. In other words, Whitehead has recovered spaces with a constant degree of curvature, into where the free mobility of a rigid body is equally allowed, as it was upholded by Riemann and Helmholtz. (3) 1 At the end of his lecture “On the Origin and Meaning of Geometrical Axioms”, Helmholtz let us ion the feeling that it was no more possible to consider alone the axioms of geometry, in isolation from the mechanical behaviour of solid bodies. ADG, § 65, p. 66. 16 This entails the actuality of real experiences so that in advance, he rejected a transcendentalist understanding of his rigid body. From a Kantian perspective, it would surely be tempting to see Helmholtz rigid body as a transcendental condition of every possible experience of measurement. Thus the axioms of geometry would recovered the status of a priori propositions given by intuition, neither likely to be refuted by experience nor confirmed by it. Linking the mechanical properties of bodies to the geometrical axioms, these propositions gain an empirical content. However, we must here remember Lipschitz and attach to his result involving the laws of dynamics all the importance it deserves, laws which are under-determined by respect to the three types of constant curvature degree. Justifying the choice of the Euclidean space, that is, of the congruence group of transformation in space allowing the similarity of bodies, Helmholtz remained on an Empiricist position. “For the rest, I do not, of course, suppose that mankind first arrived at space intuitions in agreement with the axioms of Euclid by any carefully executed systems of exact measurement. It was rather a succession of every day experiences, especially the perception of the geometrical similarity of great and small bodies, only possible in flat space that led to the rejection, as impossible, of every geometrical representation at variance with this fact. For this no knowledge of the necessary logical connection between the observed fact of geometrical similarity and the axioms was needed, but only an intuitive apprehension of the typical relations between lines, planes, angles, &c., obtained by numerous and attentive observations — an intuition of the kind the artist possesses of the objects he is to represent, and by means of which he decides surely and accurately whether a new combination which he tries will correspond or not to their nature. It is true that we have no word but intuition to mark this; but it is knowledge empirically gained by the aggregation and reinforcement of similar recurrent impressions in memory, and not a transcendental form given before experience.”1 Thus Helmholtz looks to be very close to Whitehead’s treatment of congruence relations, emerging during the controversy between Russell and Poincaré concerning the possibility to decide empirically the geometrical shape of space, some years before the time he will write the Cambridge tracts on the axioms of geometry. In a letter to Russell dated 21th December 1899, Whitehead warned that “space is the true arena on which to fight the battle with nominalism”, namely targeting Poincaré’s conventionalist epistemology2. For quoting the article “Axioms of Geometry” first published in the 1910 edition of the Encyclopaedia Britannica: “[…] we have, in fact, presented to our senses a definite set of transformations forming a congruence group, resulting in a set of measure relations which are in no respect arbitrary. Accordingly our scientific laws are to be stated relevantly to that particular congruence-group. Thus the investigation of the type (elliptic, hyperbolic or parabolic) of this special congruence-group is a perfectly definite problem, to be decided by experiment.”3 Hence Whitehead ratified the solution provided by Lie-Engel to the Riemann-Helmholtz problem of space, which we will again find in the preface of The Principe of Relativity quoted above. However, in 1910, he asserted, following Helmholtz, that it is a relationship towards experience that makes the problem of space likely to be “perfectly” resolved. But he brought this outcome to the fore without any explicit commitment into the Euclidean congruence-group. All constant curvature spaces shares the property of uniformity which was called in French “isogénéité”4. This neologism created by J. Delbœuf corresponds to the concept of “uniformity of texture of experience” required by Whitehead since 1915 as a general principle of our natural knowledge, according to which without postulating a spatial uniformity, the export of our 1 Helmholtz, 1876, p. 320. Desmet, 2010b, p. 157. 3 1910 (also Essays on Science and Philosophy, 1947, p. 265). 4 Louis Couturat, « Note sur la géométrie non euclidienne et la relativité de l’espace », Revue de métaphysique et de morale, vol. 1, 1893, pp. 302-309. 2 17 standards of measurement from local to global levels turns out impossible1. General theory of relativity was rejected for the very reason that, dismantling such spatial uniformity which forms the thread of all possible experiences, Einstein moved physics towards some geography of space-time. To the uniformity, the Euclidean determination of space adds another particular property, which is the reciprocal independence of magnitude and form. As recognized above following Riemann, spaces of constant curvature only show the property of independence of the magnitude with regard to the place. That gives only uniformity but not necessarily similitude, that is to say: in a non-Euclidean space, magnitude is not independent of form. Into the Euclidean space, one can enlarge or, at the opposite diminish, a figure as you like it, keeping the same shape: magnitude is variable, however not the form of figure. Let’s think to some spherical space: enlarging a curvilinear triangle, you change not only its shape, but also the value of the angles. The same holds in a pseudospherical space: given the angles of a nonEuclidean triangle, one settles the absolute magnitude of the triangle, proportional to the square root of the spatial parameter representing the degree of curvature in this space. * * * Firstly, Whitehead upholded that the problem of space was fully determined: mathematically “isogène”, our concept of space expects of a long historical experience the criterion able to teach humanity what is the shape of the three possibles constant-curvature spaces. Lastly, alluding to the problem of space in 1922, he opened the possibility that space would be spherical or pseudospherical or flat, so long as remaining uniform. However, he didn’t not bring to the fore, as Helmholtz did with strength, the experience of similarity in order to underline that our space very probably would be Euclidean. At the chapter III of The Principle of Relativity, the Euclidean parallelism was quite discreetly introduced, thanks to an axiom of elementary geometry (Playfair’s) concerning planes rather than lines, as in Euclid’s fifth postulate. Whitehead emphasized that the spatial structure, namely uniformity, more exactly here Euclidean flat uniformity, must be granted before introducing any concept of congruence. Playfair’s axiom allowed him to involve parallelism without the presupposition of congruence. “An additional factor of structure can be that of ordinary Euclidean parallelism. By this I mean that through any point outside a plane there is one and only one plane which does not introduce some presupposition of congruence, either or length or angles. I draw your attention to the absolute necessity of defining our structure without the presupposition of congruence. If we fail in this respect our argument will be involved in a vicious circle.” 2 The introduction of Euclidean parallelism constitutes “an additional factor of structure” (with regard to uniformity) not justified by any empirical reference to experience, but implicitly highlighted for a matter of convenience and logical coherence. With its main resulting effect, “Space, Time, and Relativity”, Proceedings of the Aristotelian Society, vol. XVI, p. 123. The Principle of Relativity, with application to Physical Science (1922), Dover reprint 2004, p. 51. Above, Sophus Lie was quoted for having demonstrated that there exists an indefinite number of qualifying classes leading to mutually exclusive sets of congruence relationships (p. 49). 1 2 18 the possibility of spatial measurements, congruence is drawn from the properties of parallelograms and from symmetry of perpendicularity1. We are in the position to get in what extent Whitehead’s empiricism can be understood among the Helmholtzian legacy. Helmholtz’ solution of the problem of space was, in this study, divided into two successive versions. If the original one completely ignored the non-Euclidean geometries, the awareness of Lipschitz’ works about non-Euclidean dynamics allowed him to renew with the Riemannian outcome that free mobility may be intelligible in the threefold types of constant-curvature spaces. To the sceptical conclusion of Riemann concerning the impossibility of determining what the actual structure of our common space is, he substituted the idea that we live in a world where our concrete geometry corresponds to perception which shows us the independence of the form with respect to magnitude. However such an argument was anyway not used by Whitehead. We may formulate the hypothesis that, if any judgement of congruence results from an immediat recognizance of affine structures or patterns from experience itself in the lapse of a “specious present”, the derivation of the Euclidean structure of the spatial experience from the independence of the form regarding to the magnitude could not upholded to provide an elementary datum of the sense-awareness. Our judgements of spatial and temporal congruence derive from a more fundamental congruence of events2. In this regard, the immediate experience of correspondance precedes a more elaborated invariance of form during a homothetic shifting. It would necessary to remember unequal magnitudes between different unities of specious present, that introducing memory further than mere recognizance3. In his 2013 suggesting paper, “Collision of Traditions. The Emergence of Logical Empiricism between the Riemannian and Helmholtzian Traditions”, Marco Giovanelli has upholded that the transition from special to general relativity amounted of escaping to the conceptual resources of the problem of space, such as arisen in Helmholtz’s in-depth study of the Riemannian Habilitationsschrift, finally resolved with a mathematically flawless way by Lie. From this outlook, Whitehead’s rejection of a “geography of space-time” with manifolds of variable curvature looks like a last attempt for understanding the XXth century physics in the frame of the Helmholtzian tradition. Whitehead could be the founder of an analytic natural philosophy: If he will created “free and savage concepts” in the synthetic cosmology in Process and Reality, his approach in the London period was analytical because of the quest of the ultimate constituents of the physical world. The ultimates of the Newtonian worldview, space, time and matter, must to be dissected under the scalpel of the natural philosophy. As the ultimate entites, i. e. events before “actual occasions” are not given in isolation, Whitehead seemed to try something new in metaphysics on the basis of his “panphysics”: applying an organic perspective to a realistic ― by distinction with neo-Hegelianism ― conception of nature. Jean-Pascal ALCANTARA Centre Georges Chevrier Université de Bourgogne-Franche-Comté 1 Ibid., 2004, p. 9. Ibid., 2004, p. 58. 3 For the needs of his charateristica geometrica, Leibniz upholded something close: supposing an eye deprived of retina, reduced to a mere point, “on ne saurait retenir les grandeurs” (see J.-P. Alcantara, Sur le second labyrinthe de Leibniz: mécanisme et continuité au XVIIe siècle, pp. 416-417). 2 19 Quoted Works and References ALCANTARA Jean-Pascal (2003) : Sur le second labyrinthe de Leibniz: mécanisme et continuité au XVIIe siècle, Paris, L’Harmattan, 2003. BOI Luciano (1995) : Le problème mathématique de l’espace. Une quête de l’intelligible, Heidelberg-Dordrecht-New York, Springer Verlag. DESMET Ronny (2010): “A Refutation of Russell’s Stereotype”, chap. IV of Desmet Ronny and Weber Michel eds., Whitehead. The Algebra of Metaphysics, Louvain-la-Neuve, les éditions Chromatika. GANDON Sébastien (2012) : Russell’s Unknown Logicism. A Study in the History and Philosophy of Mathematics, Basingstoke, Palgrave Macmillan ed. GIOVANELLI Marco (2015): “Collision of Traditions. The Emergence of Logical Empiricism. Between the Riemannian and Helmholtzian Traditions”, The Journal of the International Society for the History of Philosophy of Science (forthcoming). Displayed: http://philsciarchive.pitt.edu/10157/. GRÜNBAUM Adolf (1963): Philosophical Problems of Space and Time, New York, Alfred A. Knopf. HELMHOLTZ Hermann von (1883): Wissenschaftliche Abhandlungen von Hermann Helmholtz, vol. I-III, Barth, Leipzig. HELMHOLTZ Hermann von (1868a): “Über die Thatsächlichen Grundlagen der Geometrie”, Verhandlungen des naturhistorisch-medicinischen Vereins zu Heidelberg, n°4, pp. 197-202; Wissenschaftliche Abhandlungen, II, pp. 610-617. HELMHOLTZ Hermann von (1868b): “Ueber die Thatsachen, die der Geometrie zum Grunde liegen », Nachrichten von der Königlischen Gesellschaft der Wissenschaften zu Göttingen, pp. 193-221, Wissenschaftliche Abhandlungen, II, pp. 618-639. HELMHOLTZ Hermann von (1876): “Ueber den Ursprung und Sinn der geometrischen Sätze”, Populäre Wissenschaftliche Vorträge von H. Helmholtz. Drittes Heft. Braunschweig, Vieweg und Sohn. S. 21-54], p. 21-54; Wissenschaftliche Abhandlungen, II, pp. 640-660. English translation, “On the Origin and the Sense of Geometrical Axioms”, Mind, vol. 1, n°. 3 (Jul. 1876), pp. 301–321. HELMHOLTZ Hermann von (1977): Epistemological Writings, newly translated by Malcolm F. Lowe, edited, with an introduction and bibliography, by Robert S. Cohen and Yehuda Elkana, Boston’ Studies in the Philosophy of Science, vol. XXXVII, D. Reidel, Dordrecht-Boston. LIE Sophus (1886): “Bemerkungen zu Helmholtz’ Arbeit über die Thatsachen, die der Geometrie Grunde liegen”, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, 38, Supplementheft, pp. 337–342. 20 LIE Sophus with ENGEL Friedrich (1893): Theorie der Transformations-gruppen, vol. III, Leipzig, B. G. Teubner. LIPSCHITZ Rudolf (1869): “Untersuchungen in Betreff der ganzen homogenen Funktionen von n Differentialen”, Journal für die reine und angewandte Mathematik, Bd. LXX, S. 71-102. LIPSCHITZ Rudolf (1872): “Untersuchungen eines Problems der Variationsrechnung, in welchem das Problem der Mechanik enthalten ist”, Journal für die reine und angewandte Mathematik, Bd. LXXIV, S. 116-149. PEANO Giuseppe (1894) : “Sui fondamenti della Geometria”, Rivista di Matematica, Torino, vol. IV, pp. 51-90. RIEMANN Bernhard : On the Hypotheses Which Lie at the Bases of Geometry, translated by William K. Clifford, Jürgen Post editor, Birkhäuser, 2016. VOELKE Jean-Daniel (2005) : Renaissance de la géométrie non euclidienne entre 1860 et 1900, Peter Lang International Academic Publishers, Berne/Berlin.. WHITEHEAD Alfred North (1898): A Treatise on Universal Algebra, with Applications, Cambridge, Cambridge University Press. WHITEHEAD Alfred North (1906): The Axioms of Projective Geometry, Cambridge, Cambridge University Press, Cambridge Tracts in Mathematics and Mathematical Physics IV. WHITEHEAD Alfred North (1907): The Axioms of Descriptive Geometry, Cambridge, Cambridge University Press, Cambridge Tracts in Mathematics and Mathematical Physics V, 1907. WHITEHEAD Alfred North (1922): The Principle of Relativity, with application to Physical Science, Cambridge, Cambridge University Press. Reprint: Dover Phoenix edition, Mineola, 2004. WHITEHEAD Alfred North (1947): Essays in Science and Philosophy, New York, Philosophical Library, Inc.
