Frigyes Riesz's approach to Hilbert's problem of continuity of space ...

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUEFONDATION MAISON DES SCIENCES DE L’HOMMESÉMINAIRE HISTOIRES DE GÉOMÉTRIESAnnée 2007ÉQUIPE F 2 DS & CENTRE CHARLES MORAZÉ54 bd Raspail 75006 ParisFrigyes Riesz's approach to Hilbert’s problem of continuityof space. A chapter in the history of general topologyLaura RODRIGUEZ(Johannes Gutenberg – Universität, Mainz)Lundi 23 avril de 10h à 12h, Salle 214The question we will be dealing with is: to which extent is the concept of topologicalspace historically related with the concept of space in geometry? For some time itwas believed that Felix Hausdorff came to his definition of the concept of topologicalspace starting from Hilbert’s notion of the plane as a two-dimensional manifold 1 . Itwas Hermann Weyl, Hilbert’s best student, who set on the germ of this legend in anobituary dedicated to his master 2 . In recent years Erhard Scholz showed that thisassertion does not hold 3 . According to Scholz, Hausdorff developed his system ofaxioms for neighbourhoods when he was preparing a lecture on Riemanniansurfaces in the spring of the year 1912. As for Hilbert’s work on the foundations ofgeometry, Hausdorff had known it for sure since some years before 1912 and so hehad probably knew also Hilbert’s definition of the plane, and yet Scholz arguesconvincingly against any direct influence.In this way Scholz proved that Hilbert’s work on the foundations of geometry did notplay any essential role in the development of Hausdorff’s concept of topological1 “With Hausdorff general topology as it is understood today starts. Taking up again the notion ofneighbourhood, 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 weredesirable.” Nicolas Bourbaki. Elements of the History of Mathematics. Berlin: Springer 1994 (OriginalFrench 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. Herep. 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’sdiscovery another mathematician was working on the development of an abstractpoint set theory based on a general concept named “mathematical continuum”, whichhappens to be very closely related to Hausdorff’s notion of topological space. I amtalking of Frigyes Riesz, who 1906 was a young Hungarian mathematician that hadspent 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 toHilbert’s problem of the continuity of space. This is the story I want to tell you: howRiesz’s abstract point set theory was connected with Hilbert’ work on the foundationsof geometry.The aims of this talk are: To present Hilbert’s problem of continuity of space in the context of his workon 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 developmentof mathematicsHilbert and the foundations of GeometryBy 1900 David Hilbert had emerged as the leading mathematician in Germany. In1899 Hilbert published his book The foundations of geometry, with which hesucceeded in building up the Euclidean geometry axiomatically and in a strictsystematic way, namely starting from a group of axioms and adding on subsequentlythe other four sets of axioms. The last group he added was the group of axioms ofcontinuity, showing in this way how far the geometry can be built up without anyassumptions of continuity.In 1902 Hilbert pursued an approach to the foundations of geometry entirely differentfrom the one followed in his book. The background was given by the so-calledRiemann-Helmholtz-Lie problem of space. From the standpoint of mechanicsHermann von Helmoltz had asked in 1868: in which of all those possible spacesembraced by Riemann’s notion of a three dimensional manifold can the task befulfilled of describing the mobility of a solid. This is known as Helmholtz’s postulate ofthe free mobility of rigid bodies. Helmholtz succeeded in limiting the class of thosegeometries to the now so-called simply connected three-dimensional manifolds with

interest. Suffice it to say that its solution is known as the solution of Hilbert’s fifthmathematical problem and its importance is associated with the emergence of awhole new branch of mathematics: the theory of topological groups.The question we will deal with is: how shall continuous space be defined? That iswhat I call “Hilbert’s problem of continuity of space”. How shall continuity becharacterised in order to provide a foundation on which geometry can be built up? SoHilbert’s fifth mathematical problem as well as his paper on the foundations ofgeometry from 1902 concerned the problem of continuity of space.In his paper from 1902 Hilbert explained his new approach to the foundations ofgeometry as follows [my translation]:There [in his book from 1899, LR] the axioms have been ordered in a way that thecontinuity is assumed after all the others axioms as the last one, so that the nextquestion follows up naturally: to what extent are the known theorems and results ofthe elementary geometry dependent of any assumptions of continuity?In the present paper the continuity is, on the contrary, assumed before any otheraxiom, from the very beginning, via the definition of the plane and that of the motion,so that the main task becomes to find out the least possible amount of assumptionsfrom which (and making extensively use of the continuity) the elemental objects ofgeometry (circle and line) can be obtained as well as those properties of them thatare necessary for building up the geometry. 6In his paper from 1902 “On the foundations of geometry” the space he studied wasthe two-dimensional one. He set the property of continuity from the beginning in thedefinition of the concept of the plane and in the definition of the notion of the motion.He took a topological approach: he applied the methods of the point-set theory andthose of the analysis situs, From the point-set topology he used elementary conceptslike those of open and closed set, but first of all he made extensively use of thenotion of convergence of a sequence in the plane and of the concept of accumulation6 „Dort ist eine solche Anordnung der Axiome befolgt worden, bei der die Stetigkeit hinter allen übrigenAxiomen an letzter Stelle gefordert wird, so daß dann naturgemäß die Frage in der Vordergrund tritt,inwieweit die bekannten Sätze und Schlußweisen der elementaren Geometrie von der Forderung derStetigkeit unabhängig sind. In der vorstehenden Untersuchung dagegen wird die Stetigkeit vor allenübrigen Axiomen an erster Stelle durch die Definition der Ebene und der Bewegung gefordert, so daßvielmehr die wichtigste Aufgabe darin bestand, das geringste Maß an Forderungen zu ermitteln, umaus demselben unter weitester Benutzung der Stetigkeit die elementaren Gebilde der Geometrie(Kreis und Gerade) und ihre zum Aufbau der Geometrie notwendigen Eigenschaften gewinnen zukönnen.“ D. Hilbert. “Über die Grundlagen der Geometrie“ Math. Ann. 56 (1902). Reprint in D. Hilbert.Grundlagen der Geometrie. Stuttgart: Teubner 1956 (8th ed.) 178-230. Quotation p. 230.

point in the plane. From the analysis situs the notion of Jordan curve and relatedresults like the theorem of Jordan and its inverse were going to play an essential role.Hilbert defined the plane as a two-dimensional manifold and by doing so he providedat the same time the most accurate definition for this concept of the time:According to Hilbert, the plane E is a point-set in which to each point p corresponds afamily of so-called neighbourhoods, formed by sets U ⊂ E containing p. The systemof neighbourhoods fulfils six axioms. One of them postulates the existence of a oneto-onemap onto a Jordan region, that means onto the inside of a closed Jordancurve in IR 2 . We call these maps coordinate functions. Furthermore, when aneighbourhood has two different images, then Hilbert demanded the correspondingcoordinate change to be continuous. That means: let f: UV and g: U W be twocoordinate functions of the same neighbourhood U onto the Jordan regions V and W,then the coordinate change g∗f -1 : VW is a continuous map of V onto W in IR 2 .The other four axioms state:1. For each Jordan domain V’⊂ V and containing the point ϕ(p), its counterimageϕ -1 (V’) is also a neighbourhood of p.2. A neighbourhood U of p containing the point q is also a neighbourhood of q.3. Each two neighbourhoods U and U’ of p contain another one U’’⊂U∩U’4. To any two points p, q of the plane there exists a common neighbourhood.Jordan regions: a basis of neighbourhoods for the planeThese are the neighbourhoods axioms from which it was once thought Hausdorffdeveloped his concept of topological space. True is that the system of Jordan regionsHilbert used determine what in modern terms is called a neighbourhood basis for thetopology of the plane. So Hilbert provided the plane with a topology but not havingthis notion he could not take full advantage of it. So for instance, he could notpostulate that the coordinate functions shall be continuous mappings from the planeinto the Cartesian space IR 2 . He did not have the means for defining this notion.Transferring Cantor’s point-set theory of IR 2By defining the coordinate functions in that special way Hilbert introduced a series ofconcepts on the plane that depended of the natural topology of the Cartesian planeIR 2 . He transferred point-set theoretical concepts valid for IR 2 onto the plane using

“The origin of the concept of space” from 1906 he wrote in the introduction a similarpassage but there he pointed directly to Lie’s and Hilbert’s notion of manifold.Back to the quotation of his letter: Riesz was criticizing the fact that Lie as well asHilbert had defined the notion of a n-dimensional manifold using the Cartesian spaceIR n and so they had defined it in terms of the real numbers. First of all he pointed tothe consequences for the power of the set of points forming the manifold, namely thatit could not be bigger that the power of the set of real numbers. This Idea of Riesz isquite impressive then it means that he was approaching the notion of continuity ofspace from a quite general point of view. He continued his letter as follows:In my opinion the only essential features of IR n are the n-tuply ordering, thebeing dense everywhere and the continuity (the being perfect and connected)as these clearly appear in your definition of the plane although with thementioned limitation. According to this, the [space, LR] IR n is to be understoodas a set of n-tuply order type.[Riesz to Hilbert, 18th November 1904]Thus, Riesz was searching an abstract characterisation of the Cartesian space IR n interms of the point-set theory and multiply order types. Following Cantor, hesuggested that a set shall be said to be continuous if it is perfect and connected 8 , butRiesz was thinking of course in applying the notions of perfect set and connected sethe had developed in his own theory of multiply order types --Cantor studied onlylinear order types.Why was Riesz suggesting an abstract characterisation of IR n in terms of point-settheoretical properties for order types? Because he thought it possible to give a moregeneral definition of the concept of n-dimensional manifold using those properties buthe did not want to have the limitations Hilbert did have as a consequence of thedirect relationship he postulated between the plane and the Cartesian space IR 2 .Apart from the already mentioned implications on the power, Riesz was concernedwith the problem of defining continuity in an abstract general way and independentlyof IR n in order to avoid using implicitly the point-set theoretical properties of thisspace.8 In Cantor’s theory a set is said to be perfect if it is closed and dense in itself; the set is said to beconnected if for every two points of the set and for every ε>0 there exists a finite subset of T of pointst 1 ,…,t n , such that the distances d(t 1 , t 2 ), d(t 2 , t 3 ), etc. are all smaller than ε.

M’ symbolizes the set of all accumulation points of the set M.X is a mathematical continuum, if for every element x of X and for every subset Mof X holds: either x is not an accumulation point of M or if x is an accumulation pointof M and the following conditions are satisfied:1. Let M be a finite set, then M has no accumulation point;2. Let x be an accumulation point of M and let M be a subset of N, then x is anaccumulation point of N too;3. Let M be a subset of X that can be divided into two disjoint subsets P and Q andlet x be an accumulation point of M, then x is an accumulation point of P or Q;4. Let x be an accumulation point of M and let y be a different element of x, thenthere exists a subset P of M, such that x is an accumulation point of P but y is notan accumulation point of P.This is the abstract concept of space that Riesz introduced in his paper „The origin ofthe concept of space“ and which is closely related with the modern concept oftopological space. Examples of mathematical continua are the spaces IR n with theirstandard concept of accumulation point, but also any set of functions whenconsidered as a metric space.Riesz used his concept of mathematical continuum as the basic condition that makespossible to investigate the properties of continuity of an abstract set. That means, thecontinuity of a set that is a mathematical continuum can be further characterized. Butif the set is not a mathematical continuum to investigate its continuity is nonsense.With regard to mathematical continua the interesting question is not if they arecontinuous but how good their continuity structure is.Riesz demanded as the basic property space to be a mathematical continuum. Forthe further characterization of its continuity he introduced the concept of similarlycondensed (ähnlich verdichtet'). According to Riesz two sets are similarly condensed,if their structures of mathematical continua are isomorphic to each other. Besides hedefined the concept of condensation type (Verdichtungstypus) as „the totality of allproperties common to all mathematical continua, which are similarly condensed toeach other”. Furthermore he defined a general concept of neighbourhood based onthe notion of accumulation point. Applying these theoretical tools Riesz chosefollowing characterization of continuous geometrical space:

In this way, one achieves a characterisation of space as a connectedmathematical continuum, of which every element has a neighbourhood, which,considered as an independent continuum, has the same condensation type asthe three-dimensional space of numbers. So one arrives at Hilbert‘s moregeneral conception of continuous space. (Riesz 1907, 349)That means: for Riesz the space is a connected mathematical continuum locallysimilarly condensed as IR 3 .Historical meaning of Riesz’s contribution to Hilbert’sproblem of continuity of spaceNow the question I would like to discuss is: what happened with this contribution ofRiesz?With relation to the research work on the foundations of geometry: there were not anyreception of these Riesz’ ideas, as far as I know. There are several factors that mighthave played a negative role. First: the task set by Hilbert drew more urgently theattention to the problem of getting rid of Lie’s assumptions of differentiability in theconcept of group of continuous transformations. On the other hand, Riesz presentedhis ideas as a part of a more ambitious project that went beyond the frame of puremathematics. A third factor might be the fact that the paper appeared in anHungarian Journal and not in a Journal of the Rang of the Mathematischen Annalen.The historical meaning of Riesz’ theory of mathematical continuum is related with thehistory of general topology and functional analysis. Riesz’ general concept ofmathematical continuum was the basic concept with which he set out to develop anabstract point set-theory. This development preceded Felix Hausdorff’s theory oftopological space. When Hausdorff’s theory appeared in 1914 Riesz’ ideas were notdiscarded. On the contrary, until the beginning of the 1920ies Riesz’ conceptscompeted with those from Hausdorff in the work of young set-theoreticalmathematicians. Since 1911 there were different efforts to develop an abstract pointset-theory based on Riesz’ concept of the mathematical continuum. They took placein the USA in the work of Ralph Root, Edward Chittenden and Robert Lee Moore.Once Hausdorff introduced his concept of topological space in 1914 other pioneers ofthe general topology like Maurice Fréchet and Leopold Vietoris considered bothRiesz’ as well as Hausdorff’s ideas as a starting point. That means: Riesz’ ideas were

considered as a serious alternative to Hausdorff’s theory of topological space at leastuntil the beginning of the 1920ies, something, it seems to me, which has not beenproperly acknowledged in most of the studies on the history of general topology. Infact, it can be said that the history of Riesz’ concept of mathematical continuumconstitutes a chapter in the history of both modern branches of mathematics: generaltopology and functional analysis.