Ueber die Grundlagen der Geometrie.

Göttingen: K. Geselchaft der Wissenschaften, 1902. First edition, the rare offprint, of Hilbert’s partial solution to the fifth of his famous list of 23 problems, proposed in 1900. This problem asked whether certain continuous groups of transformations automatically admit a differentiable structure, to which Lie’s theory applies. Hilbert answered the problem affirmatively for groups of transformations of the plane.

First edition, the rare offprint issue, of Hilbert’s partial solution to the fifth of his famous list of 23 problems, proposed in 1900, that he viewed as important in guiding the development of mathematics in the next century. Hilbert also gives in this work the first rigorous definition of a ‘manifold,’ a notion that would play a central role in twentieth century mathematics and physics. This paper is a further development of the ideas in his great book Grundlagen der Geometrie (1899), an “influential work that led by its axiomatic method to a new thinking in all mathematical fields in the 20th century” (Michael Toepell in Landmark Writings in Western Mathematics, p. 710). The present paper was appended to the second edition of Grundlagen der Geometrie, published in 1903.

“Hilbert’s interest in the foundations of geometry turned his attention to what became known as the “space problem” of Hermann von Helmholtz (1821-1894), a distinguished physiologist and physicist who in 1868 proposed to deduce the geometrical nature of space from observed facts of experience having to do with the properties of mobile rigid bodies. Using calculus, he argued that his facts lead to the conclusion that metric relations in space correspond either to Euclidean geometry or to the geometry of Lobachevsky. In 1887 Poincaré used Lie algebra techniques to solve the analog of Helmholtz’s problem in two dimensions, and Lie (1890) did the same in n dimensions. Hilbert questioned the necessity of assuming, as was done by Helmholtz and his successors, the differentiability of the transformations they considered, a sine qua non if one were to solve the problem using Lie’s theory. In particular, the assumption that the group is generated by infinitesimal transformations did not fit in easily or naturally with the other geometrical axioms. Hilbert wondered whether this assumption 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. Thus in his famous lecture on mathematical problems Hilbert posed as his fifth problem the more general question as to what extent Lie’s theory, with its differentiability assumptions, could be recovered from seemingly weaker continuity assumptions about the transformations of the group and their multiplication. “Hence there arises the question whether, through the introductions of suitable new variables and parameters, the group can always be transformed into one with differentiable defining functions; or whether at least with the help of certain simple assumptions a transformation is possible into groups admitting Lie's methods.”

“Shortly thereafter, Hilbert himself dealt with this problem within the limited context of the analogue of Helmholtz’s problem for the plane. Here we find the first attempt to deal with continuous groups by topological means. Hilbert stressed the fact that his methods of proof were completely different from Lie’s, that he would mainly use the concepts of Cantor’s theory of point sets and the Jordan curve theorem for the plane. Of course, the context of Hilbert’s topological approach to groups was a special and limited one, but it showed that results could be achieved and suggested the possibility of doing something similar with respect to the formidable fifth problem... it should be noted that in seeking to characterize the plane as a two-dimensional manifold, Hilbert introduced the approach that eventually led (through Weyl) to the modern concept of a manifold. “These stipulations,” Hilbert wrote, “contain for the case of two dimensions, it seems to me, the rigorous definition of what Riemann and Helmholtz designated by ‘multiply extended manifold’ and by Lie ‘number manifold’ and which is at the basis of their entire investigation” (T. Hawkins, in History of Topology, I. M. James (ed.), pp. 175-6).

Hilbert’s fifth problem was finally solved in full generality by Andrew Gleason, Deane Montgomery and Leo Zippin in 1952.

Seperately paginated offprint from: Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1902, pp. 233-241. Original printed self-wrappers, pp. [1] 2-9 [10:blank]. A very fine copy.

Item #3250

Price: $650.00

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