THE MAGIC TRIANGLE : MATHEMATICS, PHYSICS AND PHILOSOPHY IN RIEMANN'S

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THE MAGIC TRIANGLE : MATHEMATICS, PHYSICS AND PHILOSOPHY IN RIEMANN'S GEOMETRICAL WORK José FERREIROS Universidad de Sevilla, Espagne

• riemann's lecture
• cauchy-riemann equations
• mathematical history
• guides riemann's brilliant
• euclidean geometry
• certainly riemann
• cannot explain
• riemann
• his

Sujets

• Dissertation
• Ferreiros
• Relativity
• Theodor
• Sevilla
• Riemann
• Cauchy
• Guidisme
• Mathematics
• Geometry
• English modal verb

Informations

• Publié par: kepap
• Nombre de lectures: 81
• Langue: English

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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 ques