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Universidad de Sevilla, EspagneThe magic triangle :
Mathematics, Physics and Philosophy in Riemann’s Geometrical Work
… an almost incredible gift of intuition, of constructive phantasy, and at the
same time of abstractive generalization … (Schmalfuss 1866, on Riemann)
The expression “the magic triangle” has been used by historians of science in
connection with Einstein’s work on relativity theory. In his early work, philosophical
ideas played a very important role for Einstein; it was the case with the views of Hume
and Mach by 1905. Later on, Einstein’s philosophical outlook would change, due to the
experience of formulating General Relativity, and he became more and more captivated
by mathematics. This is the viewpoint captured in words he pronounced on the occasion
of the Herbert Spencer Lecture, 1933:
If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be
freely invented, can we ever hope to find the right way?
Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable
mathematical ideas. I am convinced that we can discover by means of pure mathematical constructions the
concepts and the laws connecting them with each other, which furnish the key to the understanding of natural
phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be
deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical
construction. But the creative principle resides in mathematics. (emphasis added)
And of course, everybody knows the link with Riemann: according to Einstein himself
in 1922, the basic mathematical knowledge making possible General Relativity was due
to Gauss and Riemann, and Riemann had foreseen the physical meaning of his
generalization of geometry “with prophetic vision”.
It must be said, however, that this is both an overstament and a
missunderstanding of Riemann’s views. Riemann did not envision what Einstein later
accomplished. He did not expect the emergence of a 4-dimensional space-time, but
rather an understanding of the usual three dimensions of physical space as a subsystem
of an n-dimensional space. Most crucially, he thought the main applications of his ideas
would not be found in the large, but rather in the extremely small. Perhaps in this way
he prophetized some physical theory that is yet to come?
1Riemann’s geometrical work was presented in his short inaugural lecture at
Göttingen that took place in June 1854, almost exactly 150 years ago. The story is well
known (Dedekind 1876): Riemann had finished his Habilitation thesis in Dec. 1853, and
proposed three topics for the lecture; against the usual procedure, Gauss chose the third,
and the one that Riemann was far from having prepared completely, because it was so
close to his heart. At the time, Riemann was deeply involved in mathematical physics,
and it took him a few months to start preparing the lecture, which he finally wrote in
some 5 weeks. The lecture “superseded all of [Gauss’]s expectations and left him most
astonished”, he spoke to Weber “with an excitement that was rare in him, about the
depth of Riemann’s ideas” (Dedekind 1876).
A similar reaction would come after publication of the lecture in 1868 by
Dedekind. A young and particularly gifted witness, Felix Klein, would later reminisce:
This lecture caused a tremendous sensation upon being published … For Riemann had not just embarked in
extremely profound mathematical researches … but had also considered, throughout, the question of the inner
nature of our idea of space, and had touched upon the topic of the applicability of his ideas to the explanation of
nature. (Klein 1926, 173)
In connection with my talk, it is also interesting to mention what the physicist Wilhelm
Weber had to say after Riemann’s death: with Gauss, Dirichlet and Riemann, Göttingen
had became “the plantation of the most profoundly philosophical orientation in
mathematical research” (quoted in Dugac 1976, 166).
The title of Riemann’s lecture was “On the hypotheses upon which geometry is
founded” (Sur les hypothèses qui sont au fondement de la géométrie). A closer look at
the circumstances in which the ideas were developed reveals that, indeed, the interaction
between mathematics, physics and philosophy was most intimate in Riemann’s mind. I
believe the example is certainly stronger than that of Einstein, and perhaps the most
impressive one to be found in the history of human thought. Riemann was then 27 years
old, and it is certainly astonishing what he was able to accomplish in his 20s.
1Let me first offer to you a brief summary of events. In 1851 Riemann presented his
dissertation on function theory introducing Riemann surfaces, and the evidence suggests
that he had problems with the justification of this move (see below), which led him to
2the concept of a continuous manifold; by 1853 he had found this concept and developed
ideas on n-dimensional topology. Then in 1853/54 he became Weber’s assistant, and in
1853 he embarked in “an almost exclusive” study of natural philosophy which
prolonged into 1854 (writing the Habilitation thesis at this time seems to have been
subsidiary work!). It was also (most likely) in 1853 that he arrived at the breakthrough
of seeing how the concept of a continuous manifold opened a new road into geometry.
This was indeed intimately linked with his work in natural philosophy, as becomes clear
from the lecture itself.
When it comes to forerunners, Riemann (1854, 273) mentions exactly two: his
great predecessor Gauss, and the philosopher Herbart. Taking into account Riemann’s
close and careful study of philosophy, I do not doubt to call him a philosopher and not
just a scientist. The following is a list of elements that he took from Herbart and
developed further:
2∑ complete rejection of Kant’s theory of a priori intuition; Leibnizian view of
space as an order of coexistence of phenomena (Herbart liked to give as an
example the triangle of colours, a 2-dimensional domain emerging from the
“natural” relations between colours)
∑ a philosophical conception of mathematics and its method; the need for general
concepts as a starting point and core of every single mathematical discipline.
(This helped reinforce Riemann’s turn to radically modern mathematics – his
Wendepunkt, as Laugwitz has said in his 1998.)
However, Herbart of course knew nothing of non-Euclidean geometry, while Riemann
probably knew the work of Lobatchevskii and Bolyai (at least by hearsay), even though
3he said nothing about it in his lecture. And Herbart limited space to 3 dimensions, while
4Riemann broke with this completely and very early on, apparently in 1847. One may
assume that this happened under the influence of Gauss, and perhaps also of
5Grassmann? Perhaps, but while those assumptions remain conjectural, one thing is

For further biographical information, see Schering 1866, Dedekind 1876, Laugwitz 1998, Ferreirós 2000.
It has been written that Riemann’s lecture is aimed against Kant (Nowak 1989), but this is incorrect: Kant is just so
superseded from the very beginning!
This is understandable, because the ideas of their hyperbolic geometry were not relevant to the main line of thought
which Riemann developed.
Schmalfuss wrote (1866): “His abstractions concerning spatial dimensions do not correspond to the time of the
Gymnasium, but to the first year at the University”.
Even if Riemann had not read the Ausdehnungslehre (we simply do not know), in the early 1850s he was probably
aware of Grassmann’s papers in the Ann. Phys. Chem. I thank Emili Bifet for calling my attention to these.
3certain: the problem of the Riemann surfaces, of understanding their geometrical nature,
forced Riemann to consider n-dimensional geometry.
Let us now list some of the key issues and elements that Riemann took from
Gauss and developed further:
∑ also the partial rejection of Kant, against whose doctrine on geometry Gauss had
6offered a “decisive refutation” in a few sentences of his 1831
∑ the connection between complex numbers, 2-dimensional manifolds, and
topology; likewise the word “manifold” itself
∑ development of differential geometry, the concept of Gauss curvature, which for
7Gauss himself led to results related to non-Euclidean geometry
∑ late in his life, Gauss was obsessed with n-dimensional manifolds and the
problem of physical space; Riemann probably knew of this by lectures, personal
conversation, and second-hand information, e.g., through Weber.
thMany of the investigations about geometry in the 19 century, and especially on
non-Euclidean geometry, were of a foundational character. Not so with Riemann: his
main aim was not to axiomatise, nor to understand the new ideas on the basis of
established geometrical knowledge (say, projective geometry), nor to analyse questions
of independency or consistency – rather, he aimed to open new avenues for physical
thought. Thus:
The answer to these questions can only be got by starting from the conception of phenomena which has hitherto
been justified by experience, and which Newton assumed as a foundation, and by making in this conception the
successive changes required by facts which it cannot explain. Researches starting from general notions, like the
investigation we have just made, can only be useful in preventing this work from being hampered by too narrow
views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.
… This leads us into the domain of another science, of physics … (Riemann 1854, 286; emphasis added)
Even so, he begins the lecture by criticizing traditional geometry, its nominal
definitions, and especially the uncertainty about the axioms:
The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far
their connection is necessary, nor a priori, whether it is possible. (Riemann 1854, 272)

I have discussed this matter in Ferreirós 2003. In my opinion Gauss’s refutation works, although it is so unknown to
Kant experts.
In a letter of 1825, commenting on his work on differential geometry, Gauss said that it was taking him into “an
unpredictible plane … the metaphysics of space” (Gauss, Werke, vol. XII, 8).
Nota bene! This is the question of consistency, stated for the first time in mathematical history!
4Then he points out that these obscurities can be solved by embedding the idea of
physical space under a more general concept, in particular the “general notion of
multiply extended magnitudes” or n-dimensional manifolds. This new concept is
essentially topological.
It will follow from this that a [n-dimensional manifold] is capable of different measure-relations, and
consequently that space is only a particular case of a triply extended ma[nifold]. But hence flows as a necessary
consequence that the propositions of geometry cannot be derived from general notions …, but that the properties
which distinguish space from other conceivable triply extended ma[nifiolds] are only to be deduced from
experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of
space may be determined. (Riemann 1854, 272–273)
These matters of fact are –“like all matters of fact”, adds Riemann the epistemologist–
“not necessary, but only of empirical certainty; they are hypotheses.” Here we find the
reason why Riemann spoke of “hypotheses” and not “axioms” in his lecture: he
understands “axiom” in the old sense (established by Kant among many others), while
he wants precisely to speak of axioms in the modern sense. This forces him to find a
different terminology, and axioms appear as hypotheses when it comes to physics, to
their physical application.
It is convenient at this point to present a brief summary, making clear the
structure of Riemann’s lecture. I use a slightly modernized language (compare Riemann
1854, 286–287):
I. Concept of n-dimensional manifold. 1) general ideas about manifolds –
distinction between topology and metric geometry for continuous manifolds;
2) topological notion of the dimension of a manifold; 3) parametrization,
need of n coordinates for n dimensions.
II. General differential geometry for Riemannian manifolds. 1) line element
given by positive definite quadratic differential form; 2) concept of curvature
generalizing Gauss’s, manifolds of variable curvature; 3) manifolds of
constant curvature, with geometric examples.
III. Applications to the space problem. 1) “simplest matters of fact from which
the metrics of space may be determined”; 2) properties of physical space in
the extremely large; 3) properties of physical space in the extremely small.
5The key idea that guides Riemann’s brilliant exposition is the following: departing from
the new general concept of n-dimensional manifold, to establish a series of hypotheses
(axioms) which are more and more restrictive, leading us from pure topology to the
concretion of Euclidean space. The main hypotheses are:
1. Space is a continuous (and differentiable) manifold of 3 dimensions.
2. Lines are measurable and comparable, so that their length does not
depend upon position in the manifold.
3. The length of a line element can be expressed by a positive definite
quadratic differential form.
4. Solids can move freely without metric deformation (“strechting”).
Let me now discuss in more detail some aspects of the emergence of Riemann’s ideas.
This will enable me to highlight the interaction between the three vertices of the magic
* *
As we have seen, in 1851 Riemann presented his thesis on function theory, offering new
Grundlagen [foundations] for a general theory. Part of the business was to set the whole
theory upon a new, abstract foundation, departing from the basic concept of analytic
function (Cauchy-Riemann equations). Then, as a very fruitful element for the
characterization of given functions, he introduced the “geometrical invention” (Klein)
of the Riemann surfaces, and elaborated topological ideas concerning the Betti numbers,
the “order of connection” of surfaces.
To judge from manuscripts published by Scholz (1982), the Riemann surfaces
9posed two foundational problems for Riemann. They were n-dimensional geometrical
objects, and thus n-dimensional geometry had to be elaborated – according to some,
already in 1847 he had related ideas (see above). And even worse, contrary to the
tendencies of Cauchy, Dirichlet, etc., they seemed to introduce geometry back into
analysis; this impression had to be dispelled, and Gauss had already pointed the road.
The key idea is that the concept of n-manifold, while certainly topological, does not in
the least depend on any form of spatial intuition. Moreover, Riemann and Gauss take it
6as a basic principle to fully introduce the complex numbers in analysis; and with the
complex numbers, 2-manifolds are already present. The idea of manifold is an abstract
mathematical concept that can be used in analysis, so that its introduction does not in
the least compromise the purity of method, and the autonomy of analysis as a discipline.
Thus, if my reconstruction is correct, it was the issue of Riemann surfaces, their
role in analysis and their general foundations, that led Riemann to the new concept of
manifold. Discrete and continuous manifolds now became a new basis for the
development of the most basic mathematical concepts: discrete manifolds lead to
counting numbers, continuous manifolds lead to measuring numbers, but also to
topological and metric spaces. (Interestingly, however, Riemann has no reflection on the
concept of function in the first sections of his geometry lecture, and for that matter – to
the best of my knowledge – in any of his writings.)
From 1852 to 1854, Riemann’s “main occupation” was to set up a new unifying
theory of the physical interactions: “a new conception of the known laws of nature”,
10that is to say, “their expression by means of different basic concepts”. This new
conception should make it possible to “deploy experimental data about the interaction
between heat, light, magnetism and electricity, in order to investigate their
interrelation”. To this project of a grand unification of the physical forces he was led by
the study of Newton, Euler, and again Herbart (see also Wise 1981).
In 1853 , Riemann wrote a manuscript with the ambitious title: New
mathematical principles of natural philosophy. It was an attempt to revise and modify
the theories of Newton, Ampère, and Weber, with a direct attempt to eliminate action-
at-a-distance. Riemann employs a geometrically conceived system of dynamic
processes in the ether, which “can be pictured as a physical space, whose points move
within the geometrical”. With hindsight, we see him moving towards some kind of
unified field theory, based on the assumption of an ether field. The behaviour of the
ether at a small scale was in analogy with classical elasticity theory; line elements and
volume elements “offer resistance” to dilatation.
It is most noteworthy that, in his theory of 1853, electromagnetic forces will
alter the expression of the physical line element, which is clearly related to the lecture
on geometry. The idea being that the metrics of the ether space was entangled with

I have discussed this interpretation of Scholz’s documents in more detail elsewhere: Ferreirós 1999, 57–60, Ferreirós
7electromagnetism, it seems natural to think that this is the way in which he came to
think of possibilities that are presented in §3 of the 1854 lecture. I strongly recommend
reading the full text, but will limit myself here to quoting one single passage:
The question of the validity of the assumptions of geometry in the infinitely small is bound up with the question
of the inner ground of the metric relations in space. In this last question, which can still be counted among those
pertaining to the theory of space, is found the application of the remark that was made above; that in a discrete
manifold the principle of its metric relations is already given in the concept of this manifold, while in a
continuous one, the ground must come from outside. Either therefore the reality which underlies space is
constituted by a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces
which act upon it. (Riemann 1854, 285–286)
thIf we know come back to the question, how much of 20 -century physics did
Riemann envision?, we see that the answer is not Einstein’s. Certainly Riemann sought
to explain gravity from a field-theoretical standpoint, but he remained very far from
considering a link between gravitation and the metrics of space-time. His explanation
for gravity was in the line of Euler: the constant stream of ether substance coming
towards material particles was the reason for gravitation. It is true that he entertained the
possibility of a connection between spatial metrics and physical forces, but along the
lines of electromagnetic forces. And thus his “prophetic vision” pointed more toward
Weyl’s part in his attempt to develop a unified field theory, than toward Einstein’s
revolutionary theory of gravitation.
Quite obviously, the constant streaming of “Stoff” [ether substance] into material
particles posed a problem: what happens to it? The problem was resolved in the 1853
manuscript by a very speculative hypothesis, the idea of a strong unification of physics
and psychology. This was far from unheard of at the time, since the idea to unify head-
on the mental and the physical was a leitmotiv for idealistic philosophers and for many
post-idealistic thinkers (a case in point is Gustav Theodor Fechner, the physicist-
psychologist-philosopher, whose works were read and reviewed by Riemann). But the
evidence suggests that Riemann did not remain for long with this speculative
Later developments, once again, reinforce the link with the 1854 lecture. By
1860 approx., Riemann regarded bodies as infinitely dense points in ether, or
alternatively as points at which the ether flows into an ambient n-dimensional space (see
Schering 1866). Now, the solution for the problem of the stream of ether is found in the

Manuscript quoted in Riemann’s Gesammelte Werke, 494; compare what he says about superseding Newton’s physical