Maison des Sciences de l’Homme54 Boulevard Raspail, 75270 Paris Cedex 06

Équipe expérimentale F2DS

Formalismes, Formes et Données Sensibles :recherches historiques, philosophiques et mathématiques

Textes duSéminaire de l’année 2004

Histoires de Géométries

Organisation du séminaire

Dominique FLAMENT

C.N.R.S.Fondation Maison des Sciences de l’Homme

54, Boulevard Raspail 75270 Paris cedex 06 - B. 308. Tel/fax : 01 49 54 22 54E-mail : flament@msh-paris.fr

http://semioweb.msh-paris.fr/f2ds/

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Membres de l’équipe

Responsable : Dominique FLAMENT

Bureau : Catherine HARCOUR, Charles MORAZÉ,Philippe NABONNAND, Peter STOCKINGER,Hind BEN FARES

Membres associés : Marie-José DURAND-RICHARD, Gerhard HEINZMANN,Christian HOUZEL, Michel PATY, Jean PETITOT,Roshdi RASHED, Jean-Jacques SZCZECINIARZScott WALTER

Institutions associées aux projets de l’équipeAcadémie des Sciences de Paris, Académie des Sciences de Saxe,CNPq (Brésil),Centre National de la Recherche Scientifique (CNRS, dont les unités UMR 9949,UMR 7596, UMR 7117, UMR 7062,...),CSIC (Espagne),Collège de France, Collège International de Philosophie,Ecole des Hautes Etudes en Sciences Sociales (EHESS),Ecoles Normales Supérieures de Paris et de Lyon,Ecole Polytechnique,Imperial College (Londres),Institut des Hautes Etudes Scientifiques (IHES),Institut Fourier, Institut Henri Poincaré (IHP),INPG de Grenoble, IUFM de Créteil,Maison des Sciences de l’Homme de Paris (MSH),Trinity College (Dublin),Université de Bordeaux 3,Université de Lyon 1, Université de Nancy 2,Université Denis Diderot - Paris 7 (IREM etc...),Université Pierre et Marie Curie - Paris 6,Université d’Orsay - Paris 11, Université de Provence,Université de Villetaneuse – Paris 13,Universités allemandes (Berlin, Bielefeld, Bochum, Leipzig, Hamburg...), Universitésespagnoles (Madrid, Barcelona, San Sebastián, Del Pais Vasco, Valencia...)...

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Liste des intervenants au séminaire(par ordre d'intervention)

Christian HOUZEL (Archives de la Création Mathématique, UPS2065 du CNRS)

Introduction à l’histoire de la géométrie algébrique

Aldo BRIGAGLIA (Universitá degli Studi di Palermo, Dipartimento di Matematica)

Foundations of Geometry in Italy before Hilbert

José FERREIROS (Universidad de Sevilla, Espagne)

The magic triangle : Mathematics, Physics and Philosophy in Riemann’sgeometrical work.

Jean-Jacques SZCZCECINIARZ (Université Bordeaux 3)

Chercher la géométrie : le cas d’une suite spectrale

Erwan PENCHÈVRE (Allocataire de recherche, Archives de la Création Mathématique, UPS2065du CNRS)

Le théorème de Bézout dans la Geometria Organica de MacLaurin

Philippe NABONNAND (Archives Henri Poincaré - Université Nancy 2)

La Geometrie der Lage de Von Staudt

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TABLE DES MATIÈRES

Aldo BRIGAGLIAFoundations of Geometry in Italy before Hilbert P 1-12

José FERREIROSThe magic Triangle ; Mathematics, Physics and Philosophy in Riemann’sgeometrical work P 1-12

Erwan PENCHÈVRELe théorème de Bézout dans la Geometria Organica de MacLaurin P 1-15

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Annexes :• Programme du séminaire Histoires de géométries (2005)

VI

FOUNDATIONS OF GEOMETRY INITALY BEFORE HILBERT

Aldo BRIGAGLIAUniversitá degli Studi di Palermo, Dipartimento di Matematica

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Foundations of Geometry in Italy before Hilbert

Aldo BrigagliaDipartimento di Matematica

Università di Palermo

If things had gone according to plan, it should have been an Italian to voice the newideas since no one had come closer to those ideas than the [Italian] school.… In fact it was also an Italian. I realised that only after having completed this essay. Fano

had already got there in 1892. He introduces his axiomatic system with words that resonatewith Hilbert’s own words that we have quoted above. (Freudenthal 1957)

This quotation shows that Italian school had elaborated a point of view on foundations of

geometry resonant with Hilbert’s about ten years before the publication of Hilbert’s work.

In this talk I will try to reconstruct the link between the Italian geometric tradition on the

foundations of geometry, from the work of De Paolis (1881) to the first appearance of

Hilbert’s masterpiece (1899).

Before beginning, I give a short list of the main features of Italian school:

Riccardo De Paolis (1854 – 1892) student of Luigi Cremona (1830 – 1903).

Corrado Segre (1860 – 1924)

Giuseppe Veronese (1854 – 1920)

Gino Fano (1870 – 1940)

Federigo Enriques (1870 – 1946)

Giuseppe Vailati (1863 – 1909)

Mario Pieri (1860 – 1913)

Giuseppe Peano (1858 – 1932)

1. Riccardo De Paolis

His only paper on foundations of geometry is: Sui fondamenti della geometria projettiva,

Atti della R. Acc. dei Lincei, 1882

De Paolis’ works stems directly from von Staudt’s program shown in his Geometrie der

Lage, namely: to make projective geometry free from metrical properties.

In this context the influence of Felix Klein’s Über die sogenannte Nicht-Euklidische

Geometrie, Math. Ann, 1870 – 72, is very strong:

For De Paolis, in the same way as for Klein it is vital to note that projective geometry can

be developed before solving the problem of metric determination, so his ultimate aim is to

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obtain the coordinatization of the projective space on the basis of purely geometric

considerations.

De Paolis follows mainly Möbius’ idea of rationality net:

De Paolis, as von Staudt, starts from the fourth harmonic of three points A, B, C in a line;

he calls harmonic system the minimal set of points in the line closed with respect to this

ternary operation. It is easy to proof that an harmonic set is isomorphic with rational numbers.

To extend this isomorphism to real numbers he needs some “continuity” principle to complete

the harmonic system. Therefore he introduces a new postulate similar to Dedekind’s.

The introduction of these elements is similar to the introduction of irrational numbers in

arithmetic. The aim is to permit constructions and considerations which would otherwise be

impossible.

The problem of giving a sound basis to projective geometry became more and more urgent

for Italian geometers in the because of their growing recourse to hyperspacial methods.

In this context their approach was heavily influenced by the work of Plücker and

Grassmann.

2. Plücker and Grassmann.

There is an intimate connection between the work of Grassmann and Plücker. The formerdeveloped a part of n-dimensional pure geometry, the latter showed how to regard so-calledthree-dimensional space, for example, as an n-dimensional manifold with respect to certainelements.

Thus, a space whose elements are Plücker’s line-complexes turns out to have the sameformal properties as Grassmann’s five-dimensional spaces. And conversely, each n-dimensional space of Grassmann may be interpreted in terms of the usual three-dimensionalspace when certain appropriate geometrical configurations are taken as the elements. (Nagel)

As it is well known, Plücker, in his influential paper of 1868 had clearly explored the idea

that a line in the geometry of ordinary three-space may be interpreted as a point in a four-

dimensional space. Ordinary ruled space is therefore considered as a four dimensional space,

whose elements are no longer points, but lines.

This fact has two important foundational implications: on the one hand dimension of space

is not anymore absolute; on the other hand the generating element of space, the point, was

almost definitively losing its usual meaning.

In Italy this point of view was deeply understood by a young mathematician in Turin,

Corrado Segre.

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3. Segre's thesis 1884

Segre was only twenty when he got his degree. He begins his thesis in a very bold

way:

The geometry of n-dimensional spaces has now found its place amongst the branchesof mathematics. And even if we consider it aside from the important applications toordinary geometry, that is, even when the element or point of that space is notconsidered as a geometrical element (and not even as an analytical element made ofvalues of n variable quantities), but as an element in itself, whose intimate nature is leftundetermined, it is impossible not to acknowledge the fact that it is a science, in whichall propositions are rigorous, since they are obtained with essentially mathematicalreasoning. The lack of a representation for our senses does not matter greatly to a puremathematician.

Born, as it were, out of Riemann’s famous work of 1854 ... n-dimensional geometrydevelops along two separate lines: the first one deals with the curvature of spaces andis therefore connected with non-Euclidean geometry, the second one studies theprojective geometry of linear spaces ... and in my work I am to focus on the latter. Thispath opens for keen mathematicians an unbound richness of extremely interestingresearch.

In this way Segre, since the very beginning, poses the point as an element in itself,

whose intimate nature is left undetermined. We may here clearly understand the origins

of Fano’s point of view which had surprised Freudenthal. Any link with sensible reality

was already cut in 1884: The lack of a representation for our senses does not matter

greatly to a pure mathematician.

According to this conception the new object of projective geometry is no longer

“real” space, but what we now call a vector space. Segre tries to give a definition of

linear space:

Any continuous set of elements, whose number is m-fold infinite will form an m-dimensional space formed by such elements. ... any m-dimensional space is consideredlinear when to each of its elements it is possible to attribute the numeric values (real orimaginary ones) of m quantities, so that, with no exception, to each arbitrary group ofvalues of such quantities corresponds only one element of that space, and vice versa.

These values are said coordinates of the element. If we represent them with the ratiosof m other quantities to one (m+1)-th, these will constitute the m+1 homogenouscoordinates of the element of the space considered, so that each element of this space,without exception, will be identified by the mutual ratios of these homogeneouscoordinates, and vice versa it will identify their own ratios.

And he continues giving a definition of isomorphism:

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All linear spaces with the same number of dimensions can be considered identicalbetween them, because in their study we are not considering the nature of thoseelements, but only the property of linearity and the number of dimensions of the spaceformed by those very elements. Therefore we will be able to apply, for example, thetheory of projectivity, of harmonic groups, involution theory, etc, in the forms of firstkind.

We will see that Segre’s definition was harshly criticized by Pean, but, despite of the

lack of rigour his point of view was very open to further developments. Indeed his

approach led to numerous foundational developments, and in a few years he would

submit these developments to his best students as research topics.

4. Segre's programme (1891)As far as I know no one has yet identified and discussed a system of independent

postulates that characterise the n-dimensional linear space, from which therepresentation of its points by coordinates can be derived. It would be very useful ifsome students wanted to make of this issue their research topic (which does not seemdifficult).

These words are written in the notes of Segre’s course in 1891, which dealt with the

“Introduction to geometry of simply infinite algebraic entities” and show a research

program which was took over by Fano who attended to this course.

Segre’s programme indeed aimed at demonstrating the possibility of inferring

coordinates from a set of independent postulates. His programme was therefore the

natural consequence of Staudt’s and De Paolis’ approach and represents a main thread

in the developing of Italian school.

5. Peano's point of view (1889)In the meantime Peano, who criticized the many imperfections of Segre’s definition

of linear space (e. g. he had not excluded from the coordinates of projective hyperspaces

the n-ple (0,0,0, …, 0)), elaborated, on the basis of Grassmann’s Ausdehnungslehre his

own definition of n-dimensional real linear space, which is almost identical with our

definition.

We have therefore a category of entities called points. These entities are not defined.Furthermore, given three points, we consider a relation between them, which weindicate with the expression c ∈ ab. This relation is not defined either. The reader canunderstand by the sign l any category of elements, and by c ∈ ab any relation betweenthree elements of that category.

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All the following definitions will always be true ... If a certain group of axioms is true,then all the deducted propositions will also be true.

I don’t cite in extenso Peano’s definition which is, as I already said, the same as our

own.

It is important to compare the two definitions briefly, also in the light of the debate that

shortly ensued. Both definitions were generated by a rethinking of Grassmann’s ideas, and

both aimed at establishing a solid foundation for the basic concepts of geometry. The

difference in rigour between the two definitions is only too clear, and therefore we will not

mention it any further. However, I may notice that in Peano’s case the definition, placed as it

is at the end of his work, is more of a synthesis of his preceding work than a basis on which to

build a new geometric theory. Segre’s position is quite different. He placed his definition at

the very beginning of his dissertation, and in so doing he indicated it as the conceptual basis

of the projective geometry of hyperspaces, which is in its turn the conceptual basis for the

developments of algebraic geometry.

6. Van der Waerden commentary (1986)Ce point de vue logique tout à fait abstrait est très remarquable. Dans les exposés des

géomètres allemands il est dit souvent que ce point de vue est dû à Hilbert, mais c’estPeano qui a exposé, le premier, ce point de vue logique.

It is probably worth noticing that van der Waerden here remarks that Peano had reached this

abstract point of view well before Hilbert, and at least four years before the date that

Freudenthal ascribes to Fano. As a matter of fact, however, Segre had already five years earlier

turned such point of view into an “ordinary working tool for Italian geometers”.

7. Veronese 1884 and 1891Giuseppe Veronese published his ideas about hyperspaces in 1882, in an article in the

Mathematische Annalen, but he developed them in depth only in 1891 in a massive volume

(Foundations of Geometry) I do not intend to go into too many details regarding Veronese’s

point of view, which, as it is well known, had also some influence on Hilbert (the

Foundations were translated into German in 1894). Veronese was the first to introduce the

concept of a non-Archimedean geometry – the prototype, after non-Euclidean geometry, of

the long series of non geometries that characterised the foundations of geometry in the 20th

century.

For Veronese too, the basic idea of geometry of hyperspaces is born out of the necessity of

using a more powerful tool for the synthetic study of the geometrical objects of ordinary

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space. Conics, for instance, are typically studied in a more efficient and easier way as

projections of the circle (or, which is the same thing – as sections of a cone). Similarly, the

study of complicated surfaces in ordinary space can be simplified by considering them as

projections or sections of hypersurfaces. It is in this framework that in 1884 we see the

appearance (thanks to Veronese and Segre independently) of what is probably the most

famous hypersurface in the history of mathematics: the varietà di Veronese, a four-

dimensional surface in a five-dimensional space, amongst whose sections and projections

some of the most important surfaces of ordinary space are to be found (for example the so-

called Roman surface studied by Steiner).

The method of projection and section introduced by Segre and Veronese obtained very

important results, and it was helpful in clarifying and systematising a great number of results

by providing new techniques. Veronese’s work of 1891, with its exceptional timing with

respect to Segre’s research programme announced in the same year, expressed some teaching

projects in a somewhat confused way, as well as trying to give conceptual, axiomatic and

almost philosophical conceptual foundations to an instrument that had already proved to be

highly effective for the study of definite problems.

The charm of Veronese’s text is the combination of philosophical observations on the

nature of geometry with practical and foundational problems of mathematics. However, this is

also a limitation. On the one hand his approach is extremely advanced, but on the other his

presentation is very old fashioned. It is almost impossible, for instance, to find a clearly

defined pattern in the axioms he proposes. They are divided into the geometrical and the

practical, and they alternate with philosophical observations that certainly do not help to

clarify the presentation. However, despite the difficulty of the style, it is possible to identify

some central points that justify the admiration shown to this work.

In any case Veronese does not think about points in hyperspace as numbers or as objects in

the ordinary space (in Plücker’s way):

Here the point is not defined as a system of numbers, nor as an object of whatevernature, but as the point exactly how we imagine it in ordinary space; and objects madeof points are objects (figures) to which we continually apply both spatial intuition andabstraction, and therefore the synthetic method.

Veronese, who had been one of Klein’s students, was still anchored to the empiricist

conception of geometry that had been at the forefront of research in the two decades that

preceded his work. Therefore, just like Pasch, Veronese attributed an empirical nature to

geometrical axioms, without contradicting the abstract nature of the discipline:

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Geometry is the most exact experimental science, because the objects outside thought,that we need for the formulation of axioms, are replaced in our mind by abstract forms,and therefore the truths of the objects can be demonstrated with the combination of theforms independently obtained from what happens outsideBut such purely empiricist conception was definitely not enough for Veronese and it did not

allow him to organise the rich material that he has himself produced in a conceptual way.

Experience provides, as it were, only the nucleus of the system of axioms. In the case of

arithmetic, the study of real numbers led “naturally”, so to speak, to the study of complex

numbers, which are indispensable for the study of “real” problems independently of their

existence - or their non-existence - “outside thought”. Similarly in geometry it is correct to

apply “empirical” axioms to “imaginary” objects in order to create new and powerful tools for

the study of concrete problems - just as the hyperspaces had turned out to be powerful tools

for the study of ordinary space. Such expansion of the system of axioms is achieved, in

perfect pre-Hilbert style, in a totally arbitrary way. The only principle taken into account is

the principle of non-contradiction:

In the field of mathematics it is possible to have a well determined definition, apostulate, or a hypothesis, whose terms do not contradict each other and do notcontradict the principles, the operations and the truths from which they are derived ... Ahypothesis is mathematically false only when it establishes a property that is or that canbe demonstrated to be in contradiction with the preceding truths, or with those that canbe inferred from them .... Once the characteristics of the mathematical forms have beenestablished, mathematical possibility is regulated by the principle of non-contradiction... And a possibility becomes a mathematical reality, albeit an abstract one.

8. Peano's criticism 1891The dispute between Segre and Veronese and Peano broke out shortly after and was

extremely ferocious. The main point of contention was the question of rigour, but we must

not be led to believe it was the only one. The reasons of the three opposing parties are easily

understood. Every one was involved in important and far-reaching scientific studies, to which

they were dedicating all their energies. Peano had achieved extraordinary results in his effort

to give rigour to the fundamental body of mathematics of his time, and he was also receiving

international recognition in this field. Any compromise on the issue of rigour could not but

appear a compromise with the old and careless way of “doing mathematics”.

On the other hand Segre and Veronese envisioned the foundations of a building that was

both new and fascinating, and in comparison an excessive request for rigour seemed like tying

one’s hands behind one’s back and give up the challenge of climbing peaks. In this

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perspective the two approaches mirrored necessities that were both real and deep, and that

were destined to be constantly debated in years to come. But there are other aspects to

consider. Peano was still very much linked to an empiricist conception of mathematics.

Mathematics is “a perfected logic” and therefore:

each author is allowed to accept the experimental laws that more appeal to him, andhe can also propose any hypothesis he likes ... [but] ... If an author starts fromhypotheses that are contrary to experience, or from hypotheses that cannot be verifiedby experience, nor their consequences, he will be able, for sure, to infer some wonderfultheory that will lead other to cry: what gain if the author had applied his reasoning topractical hypotheses!

On this basis Peano could not accept hyperspace geometry and non-Archimedean

geometry, not only because he considered them less rigorous (if this was the only

obstacle, he could have endeavoured to make them more rigorous), but because he

thought them “useless”! Obviously, such an attitude would not contribute a great deal

to the study of the foundations of geometry. Mario Pieri, to whom we will come back

later in this article, succeeded in amalgamating Peano’s rigour with the intellectual

audacity and the pre-formalism of both Veronese and Segre.

9 Fano's approach 1892A manifold [set] of entities of whatever nature; entities that, for brevity’s sake, we

will call points, however we are obviously leaving out any consideration of their verynature. ... I even prefer to keep .... the definition of postulates [... for those propertiesthat] will give us the prime properties of the entities or points of our manifold; those(carefully chosen) properties that we will have to accept in order to characterise thosevery entities and to be able to infer new properties of them.This is the citation of Fano’s work that impressed so much Freudenthal. Fano followed

strictly Segre’s programme. His main aim was to find “independent postulates” for projective

geometry.

Following De Paolis he begins to build the “harmonic system”. But, he asks: if we have

three points on a line, may we prove that the fourth harmonic is a new point, different from

the three generating points?

His answer is no. He builds a model of geometry in which the axioms of projective

geometry hold, while the fourth harmonic is one of the generating points.

In this geometry we have only seven points and seven lines, as shown by the following

diagram.

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7 Points: A, B, C, D, E, F, G

7 Lines: ABC, ADE, CDG, BDF, BGE, CEF, AGF

We have here the first example of finite geometry and the full development of an abstract

point of view.

It is important however to notice how such a point of view was now taken for granted in the

milieu of Italian mathematicians, so much that a young twenty-year-old did not even feel

pressed to justify it in any particular way. By now, however, this totally abstract approach has

become a common ground among the Italian mathematicians. In a way, Freudenthal had no

reason to be amazed at recognising in Fano a precursor of Hilbert. For at least a decade such

abstraction had been a topic of discussion among the Italian mathematicians, and in the Turin

school of Segre and Peano in particular. Naturally, the question of the philosophical origin of

the postulates is an altogether different matter. Just as, at least on a formal level, there is a

difference between going from abstract objects (points), considered capable of numerous

determinations, to the idea of model of a purely abstract structure, made of pure symbols.

10. Federigo EnriquesThe problem of the philosophical origins of the postulates was one of the main

interests of Federigo Enriques.

In his wok of 1894 he wrote:

The direction that they (i. e.Fano and Amodeo) follow is quite different from the onewe intend to follow, especially in that, while the clever authors set out to establish anarbitrary system of hypotheses capable of defining a linear space to which it is possibleto apply the results of ordinary geometry, we try here to establish the postulatesinferred from an experimental intuition of space which are easier to work with in orderto define the object of projective geometry

As to those intuitive concepts, we do not intend to introduce anything other than theirlogical relation, so that a geometry thus founded can still be given an infinite number of

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interpretations, where to its element called “point” an arbitrary meaning [our italic] isascribed. We think that the experimental origin of geometry should not be forgottenwhile researching those very hypotheses on which it is founded

The same standing was expressed by Enriques in his influential book Projective

Geometry.

Projective geometry can be considered as an abstract science and hence receivedifferent interpretations from the intuitive one, assuming that its elements (points, lines,planes) are concepts determined in whichever way, and that entertain the logicrelations expressed by the postulatesIn 1902 Enriques’ book was translated into German. Klein's foreword shows clearly the

principal peculiarities of Italian approach:

Over the last two decades Italy has been the true centre of advanced research in the

field of projective geometry. Among the specialists, this is well known.... But the Italian

researchers have gone far beyond also on a practical level: they have not disdained to

draw some didactic conclusions from their own studies. The remarkable textbooks for

middle and secondary schools that have been born out of this attitude, can be made

accessible to wider audiences by means of adequate translations. And this is all the

more desirable in Germany since our didactic literature has lost all touch with recent

research achievements. Therefore, both the translator and the publishing house that

offer here a German translation of Enriques’ projective geometry, can count from the

very beginning on our unfailing support. ... We are not lacking in stimulating works that

would be adequate for an introduction to projective geometry, but I do not know of any

one in particular that offers a systematic construction of this theory, in an up-to-date

form, and in an equally clear and exhaustive way. Moreover, the presentation is always

intuitive, but completely rigorous, as it could only be after the clever researches on the

foundations of projective geometry presented in earlier essays by the same author. I

would like to draw particular attention to the presentation of the metric: the clear and

explicit treatment of its foundation by means of the absolute (in the plane by means of a

circle with a known centre)

I will not discuss here the axiomatic approach of Enriques, and I want to conclude my

talk quoting Enriques who gives a very clear idea of Italian geometric school and the

deep links between foundations of geometry and algebraic geometry:

Thanks to Klein and Lie, the concept of abstract geometry made great progress, and(after Segre) it became an ordinary tool for the contemporary Italian geometers. Indeednothing is more fertile than the multiplication of our intuitive powers operated by thisprinciple: it is as if besides the mortal eyes with which we examine a figure, we have

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thousands of spiritual eyes that complete manifold transfigurations; all this while theunity of the object shines in our enriched reason, and it enables us to easily go from oneform to another.

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in Italy from De Paolis to Pieri, Arch. Hist. Exact Sci., 56, 363 – 425.

De Paolis R., 1880-81, Sui fondamenti della geometria projettiva, Atti della R. Acc. deiLincei, (3), 9, 489-503.

Enriques F., 1894, Sui fondamenti della geometria proiettiva, Rend. Ist. Lombardo, (2),27, 550-567.

Enriques F., 1895, Sui postulati fondamentali della geometria proiettiva (corrispondenzacon G. Fano), Rend. del Circolo Mat. di Palermo, 9, 79-85.

Enriques F., 1898, Lezioni di geometria proiettiva, Zanichelli, Bologna (II ed. Revised,1903; reprint 1996)

Enriques F., 1903, Vorlesungen über projektive Geometrie, translated by H. Fleisher,Teubner, Leipzig, 2 Auflage, 1915.

Enriques F., 1922, Per la storia della logica, Zanichelli, Bologna, (reprinted, Zanichelli,Bologna, 1987).

Fano G., 1892, Sui postulati fondamentali della geometria proiettiva in uno spaziolineare a un numero qualunque di dimensioni, Giorn. di mat., 30, 106-132.

Fano G., 1895, Sui postulati fondamentali della geometria proiettiva (due lettere al prof.F. Enriques), Rend. del Circolo Mat. di Palermo, 9, 79-82 and 84-85.

Freudenthal H., 1957, Zur Geschichte der Grundlagen der Geometrie, Nieuw Archiefvoor Wiskunde, (4), 5, 105-142.

Klein F., 1871, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann., 4.

Klein, F., 1873, Über die sogenannte Nicht-Euklidische Geometrie (Zweiter Aufsatz),Math. Ann., 6, 135-136

Klein, F., 1874, Nachtrag zu dem “zweiter Aufsatz über die sogenannte Nicht-Euklidische Geometrie”, Math. Ann., 7.

Nagel E.,1939, The formation of modern conceptions of formal logic in the development of

geometry, Osiris, 7, 142-223.

Peano G., 1888, Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann,Bocca, Torino.

Peano G., 1889, I principii della geometria logicamente esposti, Bocca, Torino.

Peano G. 1894, Sui fondamenti della geometria, Torino

Plücker J., System der Geometrie des Raumes, 1846.

Plücker J., Neue Geometrie des Raumes, 1868.

Segre C., 1884, Studio sulle quadriche in uno spazio lineare a un numero qualunque didimensioni, Mem. della R. Acc. delle Scienze di Torino, (2), 36, p. 3-86.

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Segre C., 1890-91, Introduzione alla geometria sugli enti algebrici semplicementeinfiniti, Torino, manuscript in the Archive of the Mathematical Department of theUniversity of Turin.

Segre C., 1891, Su alcuni indirizzi nelle investigazioni geometriche, Rivista diMatematica, 1, 42-66

Van der Waerden B., 1986, Les contributions de Peano aux théories axiomatiques de lagéométrie, in Celebrazioni in memoria di Giuseppe Peano, Torino, 61-71

Veronese G., 1882, Behandlung der projektivischen Verhälnisse der Räume vonverschieden Dimensionen durch das Princip des Projicirens und Schneidens,Math. Ann., 19, 161-234.

Veronese G., 1891, Fondamenti di geometria a più dimensioni e a più specie di unitàrettilinee, esposti in forma elementare, Tip. del Seminario, Padova.

THE MAGIC TRIANGLE :MATHEMATICS, PHYSICS ANDPHILOSOPHY IN RIEMANN’SGEOMETRICAL WORK

José FERREIROSUniversidad de Sevilla, Espagne

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The 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 befreely 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 conceivablemathematical ideas. I am convinced that we can discover by means of pure mathematical constructions theconcepts and the laws connecting them with each other, which furnish the key to the understanding of naturalphenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot bededuced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematicalconstruction. 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?

2

Riemann’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 inextremely profound mathematical researches … but had also considered, throughout, the question of the innernature of our idea of space, and had touched upon the topic of the applicability of his ideas to the explanation ofnature. (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.

*

Let me first offer to you a brief summary of events.1 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

3

the 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:

∑ complete rejection of Kant’s theory of a priori intuition;2 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

he said nothing about it in his lecture.3 And Herbart limited space to 3 dimensions, while

Riemann broke with this completely and very early on, apparently in 1847.4 One may

assume that this happened under the influence of Gauss, and perhaps also of

Grassmann?5 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 sosuperseded from the very beginning!

3

This is understandable, because the ideas of their hyperbolic geometry were not relevant to the main line of thoughtwhich Riemann developed.

4

Schmalfuss wrote (1866): “His abstractions concerning spatial dimensions do not correspond to the time of theGymnasium, 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 probablyaware of Grassmann’s papers in the Ann. Phys. Chem. I thank Emili Bifet for calling my attention to these.

4

certain: 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

offered a “decisive refutation” in a few sentences of his 18316

∑ 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

Gauss himself led to results related to non-Euclidean geometry7

∑ 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.

Many of the investigations about geometry in the 19th 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 hithertobeen justified by experience, and which Newton assumed as a foundation, and by making in this conception thesuccessive changes required by facts which it cannot explain. Researches starting from general notions, like theinvestigation we have just made, can only be useful in preventing this work from being hampered by too narrowviews, 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 fartheir connection is necessary, nor a priori, whether it is possible.

8

(Riemann 1854, 272)

6

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

7

In a letter of 1825, commenting on his work on differential geometry, Gauss said that it was taking him into “anunpredictible 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!

5

Then 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, andconsequently that space is only a particular case of a triply extended ma[nifold]. But hence flows as a necessaryconsequence that the propositions of geometry cannot be derived from general notions …, but that the propertieswhich distinguish space from other conceivable triply extended ma[nifiolds] are only to be deduced fromexperience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations ofspace 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.

6

The 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

posed two foundational problems for Riemann.9 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

7

as 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”,

that is to say, “their expression by means of different basic concepts”.10 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 1 8 5 3 , 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, Ferreirós2000.

8

electromagnetism, 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 questionof the inner ground of the metric relations in space. In this last question, which can still be counted among thosepertaining to the theory of space, is found the application of the remark that was made above; that in a discretemanifold the principle of its metric relations is already given in the concept of this manifold, while in acontinuous one, the ground must come from outside. Either therefore the reality which underlies space isconstituted by a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forceswhich act upon it. (Riemann 1854, 285–286)

If we know come back to the question, how much of 20th-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

9

hypothesis that real space has more than 3 dimensions; an idea that was also dear to

Gauss’s heart. So, again, this confirms how tight the connections between physics,

mathematics, and philosophy had been back in 1854, and how they continued evolving

together in time.11

* * *

As Weyl emphasized long time ago, the question of the infinitesimal-

geometrical expression of physical laws was a veritable leitmotiv in Riemann’s work. In

the course of his first university lectures (1854/55), devoted to partial differential

equations and their application in physics, he stated:

“Truly elementary laws can only occur in the infinitely small, only for points in space and time.” (quoted inArchibald 1991, 269; from Hattendorff’s 1869 edition of his lectures)

It seems likely that the combination of reflections on manifolds and their foundational

role, on the one hand, and on the local-infinitesimal physical laws, on the other, was the

key to a fundamental insight that must have come in 1853. The Leibnizian viewpoint,

that space is not independent from natural phenomena, but rather one of the expressions

of natural relations, found an expression in the idea that physical forces determine the

(local) metrics of space. This insight was made possible by a rather mature

understanding of the topological viewpoint, enabling the strict differentiation between

the topology of space and its metrical properties.

With this move, manifolds opened up new geometrical worlds, as a single

topological structure could support many different metric structures. This was

vehiculated by the differential geometry of Riemannian manifolds.12 With these

developments of 1853, all of the key ideas of the lecture were in place, and Riemann

only needed Gauss to set him to the task of writing the lecture. (The manuscripts that

Scholz found and published (1982) offer further confirmation that they key ideas of

Riemann’s conception of geometry, as presented in the 1854 lecture, were not yet

available in 1852.)

conception in Riemann 1854, 28611

Likewise, on a different plane, the use of differential methods in the treatment of the Paris question on heat, in 1861,confirms the constant interrelations between the circle of ideas contained in the 1854 geometry lecture and physicalquestions. See Riemann’s “Commentatio mathematica…” in his Gesammelte Werke.

12

Riemann was explicit in suggesting the possibility of more general manifolds than the ones we presently call“Riemannian.” There is thus little need to call the manifolds of GRT “semi-Riemannian,” as they are so much in the spirit ofRiemann’s lectures as the “Riemannian” ones.

10

Riemann’s great efforts to develop a new, more adequate, and more unifying

physical theory remained unpublished until 1876, and (so far as I know) they did not

influence other crucial developments in physics. Our estimation of the impact of his

philosophical ideas should perhaps be also negative: published in 1876, we do not know

of important impacts upon significant trends in philosophy (although perhaps here we

should be cautious, as the topic has not been seriously studied)13. With all due caution,

we may conclude that the main positive outcome and impact of Riemann physical and

philosophical ideas was in his revolutionary mathematical ideas – in the 1854 lecture on

geometry and space.

References.

Thomas Archibald

1991 ‘Riemann and the Theory of Nobili’s Rings,’ Centaurus 34 (1991).

Richard Dedekind

1876 ‘Bernhard Riemanns Lebenslauf,’ in Riemann’s Gesammelte Werke.

Pierre Dugac

1976 Richard Dedekind et les fondements des mathématiques (Paris, Vrin).

José Ferreirós

1999 Labyrinth of Thought: A history of set theory and its role in modern mathematics

(Basel, Birkhäuser).

2000 Riemanniana Selecta, bilingual edition with an introductory study (Madrid, CSIC,

Colección Clásicos del Pensamiento).

2003 ‘Kant, Gauss y el problema del espacio,’ in J. Ferreirós, A. Durán (eds.), Matemáticas y

matemáticos (Universidad de Sevilla / SAEM Thales / Real Sociedad Matemática Española,

2003), 105–133.

Carl F. Gauss

1831 ‘Anzeige der Theoria residuorum biquadraticorum [1832],’ Göttingische gelehrte

Anzeigen. Reprinted in Gauss’ Werke, vol.2, 169–178.

13

The names of Georg Cantor, Heinrich Hertz –who in turn influenced Wittgenstein–, and Gottlob Frege are naturalcandidates for a more serious analysis of the possible impact of Riemann’s philosophy.

11

1863/1929 Werke, 12 vols. (Göttingen, Dieterich; Reprint Hildesheim, Olms, 1973).

Felix Klein

1926 Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (Berlin,

Springer).

Detlef Laugwitz

1998 Bernhard Riemann, 1826–1866. Wendepunkte in der Auffassung der Mathematik

(Basel, Birkhäuser; English translation also in Birkhäuser).

Greg Nowak

1989 ‘Riemann’s Habilitationsvortrag and the Synthetic A Priori Status of Geometry,’ in D.

Rowe & J. McCleary, eds., The History of Modern Mathematics. Vol. I: Ideas and their

reception (Boston · London, Academic Press 1989), 17–46.

Bernhard Riemann

1851 Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen

complexen Grösse (Inauguraldissertation), in Riemann’s Gesammelte Werke, 3–45.

1853 ‘Neue mathematische Principien der Naturphilosophie (Gefunden am 1 März 1853),’ in

Riemann’s Gesammelte Werke, 520–524.

1854 ‘Über die Hypothesen, welche der Geometrie zu Grunde liegen (Habilitationsvotrag),’

Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868); in

Riemann’s Gesammelte Werke, 272–287. English translation by W. K. Clifford, in his

Mathematical Papers (New York, Chelsea, 1968).

1991 Gesammelte mathematische Werke und wissenschaftlicher Nachlass, ed. H. Weber y R.

Dedekind, revised and extended by R. Narasimhan (Berlin · New York, Springer).

Ernst Schering

1866 ‘Zum Gedächtniss an B. Riemann,’ in Riemann’s Gesammelte Werke (ed. Narasimhan),

828–844.

F. Constantin Schmalfuss

1866 ‘Letter to Schering,’ reproduced in Narasimhan’s edition of Riemann’s Gesammelte

Werke, 851–854. [Schmalfuss was Riemann’s devoted mathematics teacher, Rector of the

secondary school that he attended, and the letter contains very interesting observations.]

Erhard Scholz

12

1980 Geschichte des Mannigfaltigkeitsbegriff von Riemann bis Poincaré (Basel, Birkhäuser).

1982 ‘Riemanns frühe Notizen zum Mannigfaltigkeitsbegriff und zu den Grundlagen der

Geometrie’, Archive for History of Exact Sciences 27 (1982), 213–32.

Roberto Torretti

1984 Philosophy of Geometry from Riemann to Poincaré (Dordrecht, Reidel).

Norton Wise

1981 ‘German concepts of force, energy, and the electromagnetic ether: 1845–1880’, in

Cantor & Hodge, eds., Conceptions of ether (Cambridge University Press, 1981), 267–307.

LE THÉORÈME DE BÉZOUT DANSLA GEOMETRIA ORGANICA DEMACLAURIN

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SÉMINAIRE HISTOIRES DE GÉOMÉTRIES

Programme de l'année 2005

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUEMAISON DES SCIENCES DE L’HOMME

ÉQUIPE F2DS & CENTRE CHARLES MORAZÉ

SÉMINAIRE HISTOIRES DE GÉOMÉTRIES

Année 2005

Les séances auront lieu à la

Maison des Sciences de l’Homme54, bd. Raspail 75006 Paris. 2e étage, Salle 214

Les Lundis de 10h à 12h

07 mars Jean-Pierre BOURGUIGNON (CNRS-IHÉS) «La géométrie kählérienne comme exemple d’intégration de différents domaines dela géométrie différentielle au cours du XXe siècle, et l’ouverture vers d’autresbranches de la géométrie»

4 avril Daniel BENNEQUIN (Université Denis Diderot - Paris 7)«Constructions cohomologiques des symétries»

23 mai Sébastien GAUTHIER (Université Pierre et Marie Curie - Paris 6)«La géométrie en théorie des nombres : l'exemple de la géométrie des nombres»

30 mai Philippe LOMBARD (Archives Henri Poincaré - Université Nancy 1)«Un siècle autour du théorème de Morley»

06 juin Sébastien GANDON (Université Blaise Pascal - Clermont-Ferrand II) «L'arrière-plan grassmannien des Principles of Mathematics de Russell»

13 juin Igor LY (Archives Henri Poincaré, CNRS) «L'espace est-il un concept ? Mesure et géométrie dans l'œuvre philosophique dePoincaré»

20 juin Joël SAKAROVITCH (Université Paris V et École d’Architecture Paris-Malaquais) «La géométrie dans les projets pédagogiques de Monge»

Pour tout renseignement : Dominique FLAMENTMSH (F2DS) - CNRS (Archives Henri Poincaré, UMR 7117, Nancy)Équipe F2DS54 bd Raspail, 75270 Paris cedex 06 - Bureau 308Tél./Fax : 01 49 54 22 54 E-mail : flament@msh-paris.fr