THE MAGIC TRIANGLE :

MATHEMATICS, PHYSICS AND

PHILOSOPHY IN RIEMANN’S

GEOMETRICAL WORK

José FERREIROS

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

1

For further biographical information, see Schering 1866, Dedekind 1876, Laugwitz 1998, Ferreirós 2000.

2

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!

3

This is understandable, because the ideas of their hyperbolic geometry were not relevant to the main line of thought

which Riemann developed.

4

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”.

5

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

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their connection is necessary, nor a priori, whether it is possible. (Riemann 1854, 272)

6

I have discussed this matter in Ferreirós 2003. In my opinion Gauss’s refutation works, although it is so unknown to

Kant experts.

7

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).

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

triangle.

* *

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

9

I have discussed this interpretation of Scholz’s documents in more detail elsewhere: Ferreirós 1999, 57–60, Ferreirós

2000.

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

hypothesis.

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

10

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

8