Whitehead on the Riemann-Helmholtz-Lie

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 it
2
. 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
2004, p. v.
2
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 metrization
2
. His leading twofold hypothesis bears on the sufficient and necessary
conditions for the determination of the metric relations of space
3
.
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
.
1
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.

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 = 
This is with the metrical tensor  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”:
ds = 
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 non metrical structures
Riemannian manifolds 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
Ibid. part II § 4, p. 37 (we underline).
2
Ibid. part III § 1, p. 38.