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Frigyes Riesz's Approach to Hilbert's Problem of Continuity of Space. A

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Frigyes Riesz's Approach to Hilbert's Problem of Continuity of Space. A CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE SÉMINAIRE HISTOIRES DE GÉOMÉTRIES FONDATION MAISON DES SCIENCES DE L’HOMME Année 2007 ÉQUIPE F2DS & CENTRE CHARLES MORAZÉ 54 bd Raspail 75006 Paris Frigyes Riesz's approach to Hilbert’s problem of continuity of space. A chapter in the history of general topology Laura RODRIGUEZ (Johannes Gutenberg – Universität, Mainz) Lundi 23 avril de 10h à 12h, Salle 214 The question we will be dealing with is: to which extent is the concept of topological space historically related with the concept of space in geometry? For some time it was believed that Felix Hausdorff came to his definition of the concept of topological space starting from Hilbert’s notion of the plane as a two-dimensional manifold 1. It was Hermann Weyl, Hilbert’s best student, who set on the germ of this legend in an obituary dedicated to his master 2. In recent years Erhard Scholz showed that this assertion does not hold 3 . According to Scholz, Hausdorff developed his system of axioms for neighbourhoods when he was preparing a lecture on Riemannian surfaces in the spring of the year 1912. As for Hilbert’s work on the foundations of geometry, Hausdorff had known it for sure since some years before 1912 and so he had probably knew also Hilbert’s definition of the plane, and yet Scholz argues convincingly against any direct influence. In this way Scholz proved that Hilbert’s work on the foundations of geometry did not play any essential role in the development of Hausdorff’s concept of topological 1 “With Hausdorff general topology as it is understood today starts. Taking up again the notion of neighbourhood, he knew how to choose, among the axioms of Hilbert on neighbourhoods in the plane, those that could give to his theory at the same time all the precision and the generality that were desirable.” Nicolas Bourbaki. Elements of the History of Mathematics . Berlin: Springer 1994 (Original French edition from 1984), p. 143. A similar statement can be found in a text by C. Chevalley and A. Weil called “Hermann Weyl (1885-1955)” that appeared in L’Enseignement Mathématique tome 3, fasc. 3 (1957) and was partially reprint in: Hermann Weyl Gesammelte Abhandlungen , K. Chandrasekharan (ed.), Heidelberg: Springer 1968, 2 Hermann Weyl. David Hilbert and his mathematical work , Bulletin of AMS 50 (1944), 612-654. Here p. 638. 3 Erhard Scholz. Logische Ordnungen im Chaos: Hausdorffs frühe Beiträge zur Mengenlehre. In: Egbert Brieskorn (Hrsg.) Felix Hausdorff zum Gedächtnis. Band I: Aspekte seines Werkes. Wiesbaden, Vieweg 1996 space. This is true for Hausdorff’s concept. But some years before Hausdorff’s discovery another mathematician was working on the development of an abstract point set theory based on a general concept named “mathematical continuum”, which happens to be very closely related to Hausdorff’s notion of topological space. I am talking of Frigyes Riesz, who 1906 was a young Hungarian mathematician that had spent some years as a student and a research fellow in Göttingen and in Paris. Contrary to Hausdorff, Riesz did indeed attempt to contribute with his theory to Hilbert’s problem of the continuity of space. This is the story I want to tell you: how Riesz’s abstract point set theory was connected with Hilbert’ work on the foundations of geometry. The aims of this talk are: To present Hilbert’s problem of continuity of space in the context of his work on the foundations of geometry To show how Riesz came to develop his notion of continuous space And to evaluate the role that Riesz’s ideas played in the further development of mathematics Hilbert and the foundations of Geometry By 1900 David Hilbert had emerged as the leading mathematician in Germany. In 1899 Hilbert published his book The foundations of geometry, with which he succeeded in building up the Euclidean geometry axiomatically and in a strict systematic way, namely starting from a group of axioms and adding on subsequently the other four sets of axioms. The last group he added was the group of axioms of continuity, showing in this way how far the geometry can be built up without any assumptions of continuity. In 1902 Hilbert pursued an approach to the foundations of geometry entirely different from the one followed in his book. The background was given by the so-called Riemann-Helmholtz-Lie problem of space. From the standpoint of mechanics Hermann von Helmoltz had asked in 1868: in which of all those possible spaces embraced by Riemann’s notion of a three dimensional manifold can the task be fulfilled of describing the mobility of a solid. This is known as Helmholtz’s postulate of the free mobility of rigid bodies. Helmholtz succeeded in limiting the class of those geometries to the now so-called simply connected three-dimensional manifolds with constant curvature. These are the Euclidean, the non-Euclidean by Bolya- Lobatschewski, the spherical, and the elliptical geometry. The question had been taken up by Sophus Lie in the light of his general theory of continuous groups. Lie’s approach depends on certain assumptions of differentiability; to get rid of them is the task asked by Hilbert in his fifth Paris problem from 1900. In the paper ”Ueber die Grundlagen der Geometrie” from 1902 Hilbert does get rid of any assumptions of differentiability as far as Riemann-Helmholtz-Lie’s problem in the plane is concerned, that means for the two-dimensional space. The proof is difficult and laborious. Very important here is that the notion of continuity is now the foundation and has therefore to be assumed from the beginning. So continuity is no more the last keystone of the building as it had been in Hilbert’s Grundlagen book. Hilbert’s fifth mathematical problem I would like to say some words on Hilbert’s fifth mathematical problem. In 1900 David Hilbert gave in the second International Congress of Mathematicians in Paris an invited paper. He spoke on The Problems of Mathematics and presented a list of unsolved problems that had become since quite famous. The so-called Hilbert's problems came in four groups. The first group concerned foundational questions and consisted of six problems. From them, the fifth one was related with the foundations of geometry. Hilbert described it in the following terms: On sait qu’en employant la notion des groupes continues de transformations, Lie a établi un système d’axiomes pour la Géométrie, et a démontré au moyen de sa théorie des groupes continus de transformations que ce système d’axiomes suffit pour édifier la Géométrie. Or dans l’exposition de sa théorie, Lie suppose toujours que les fonctions définissant les groupes sont susceptibles de différentiation; alors rien dans ces développements ne nous dit si, dans la question des axiomes de la Géométrie, cette hypothèse relative à la différentiation est de tout nécessité, ou si elle ne serait pas plutôt une conséquence du concept de groupes ainsi que des autres axiomes géométriques employés. 4 4 David Hilbert. Sur les problèmes futures des mathématiques. Traduite par M. L. Laugel. Deuxième Congrès International des Mathématiciens, Paris 1900. From this quotation I would like to point out two things: First, the terms used by Hilbert suggest an interpretation of Lie’s contribution into a question of providing a foundation of geometry axiomatically. This is easily understood considering Hilbert’s own achievements at that time, namely his book on the foundations of geometry from 1899, which was precisely a very accurate attempt of building up the geometry systematically on the base of a system of axioms. But I have to stress that this was Hilbert’s very personal interpretation. Lie himself didn’t claim to have attempted an axiomatic foundation of geometry –and most surely not in the sense meant by Hilbert. Second, in the fifth problem Hilbert wondered whether the assumptions of differentiability imposed by Lie had really to be included within the axioms. He wondered whether the properties of differentiability might actually follow as a consequence of the continuity of the transformations defining the rigid motions, together with the group property and the other axioms of geometry. So, looked from the perspective of his axiomatic approach, the task Hilbert was setting is actually that of finding a system of axioms of geometry and a more appropriated definition of the continuity of group of transformations from which the properties of differentiability could be proved to be a consequence. It is the question of the mutually independence of the elementary assumptions. The significance Hilbert saw in the independence of the elementary assumptions, as he stressed in his Foundations book, was to establish which of them were really unavoidable for the proof of a geometric elementary truth 5. This lecture of Hilbert’s fifth mathematical problem is by no means new but I find it important to recall you in which way this problem was meant to clear also a foundational question, namely the question: how far can the geometry be built up using a group theoretical approach and starting alone from assumptions of continuity? Hilbert’s problem of continuity of space from 1899 to 1902 For Hilbert that question was a double one: How shall continuous space be defined? How shall continuous group be defined? This last question will not be of our further 5 „In der Tat sucht die vorstehende geometrische Untersuchung allgemein darüber Aufschluß zu geben, welche Axiome, Voraussetzungen oder Hilfsmittel zum Beweise einer elementargeometrischen Wahrheit nötig sind.“ D. Hilbert, Grundlagen der Geometrie, Stuttgart: Teubner 1956 (8th ed.), p. 125. interest. Suffice it to say that its solution is known as the solution of Hilbert’s fifth mathematical problem and its importance is associated with the emergence of a whole new branch of mathematics: the theory of topological groups.
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