On the mathematical structure for discrete and continuous metric point sets (Q2907614)

Motivated by the need to provide some alternative to the differential manifold as ``the arena in which physics takes place'', the authors propose to develop the basic notions of geometry in the framework of metric spaces that may be discrete, enabling the modeling of relativistic quantum field theory. This is, in itself, a highly laudable initiative, to which several authors have made contributions since the 1930s (to the authors' list of references the reviewer would add \textit{A. Schild} [Can. J. Math. 1, 29--47 (1949; Zbl 0038.40402)], \textit{H. S. M. Coxeter} and \textit{G. J. Whitrow} [Proc. R. Soc. Lond., Ser. A 201, 417--437 (1950; Zbl 0041.47202)] and \textit{A. Das} [Nuovo Cimento, X. Ser. 18, 482--504 (1960; Zbl 0094.42703)]).NEWLINENEWLINEHowever, the authors do not seem to be aware that the subject of geometry in metric spaces has a venerable history, going back to Menger's work in the 1920s and 1930s, and that subtle geometric aspects have been the topic of sustained research by A. D. Alexandrov and his school, of Busemann, and that the results obtained by these geometers are not hidden in some long-forgotten papers, but are the subject of textbooks currently in print, such as [\textit{D. Burago}, \textit{Yu. Burago} and \textit{S. A. Ivanov}, A course in metric geometry. Graduate Studies in Mathematics. 33. Providence, RI: American Mathematical Society (2001; Zbl 0981.51016)] and [\textit{A. Papadopoulos}, Metric spaces, convexity and nonpositive curvature. IRMA Lectures in Mathematics and Theoretical Physics 6. Zürich: European Mathematical Society Publishing House (2005; Zbl 1115.53002)], both of which should have been, with their emphasis on the synthetic development of concepts traditionally thought of as belonging to differential geometry, of particular interest to the authors of this papers.NEWLINENEWLINEThe authors do not cite any work in metric geometry, and attempt to develop themselves several geometric notions in metric spaces, and then to prove for those notions the five postulates from Book I of Euclid's \textit{Elements}, which they provide with a certain modernizing interpretation. They succeed only in defining lines, which are introduced by means of the notion of metric betweenness in the manner of Menger. Already the next step, the attempt to define angles must be considered irreparably flawed, in both its ``continuous'' and its ``discrete'' version. Formal mistakes start creeping in on page 673, where a segment is multiplied by a number \(\lambda <1\), and they become irreparable by page 676, where the authors introduce the ``continuous'' version of the definition of the angle \(\widehat{qpr}\) by means of the arccosine of a limit of a certain poorly (incomprehensible to the reviewer) defined ratio of contractions by \(\lambda<1\) of \(pq\) to those of \(pr\), which cannot, regardless of the way one may need to interpret definition (2.11), express anything of geometrical significance. This, however, is not the heart of the matter, for the main aim is to define the ``discrete'' notion of angle, and that definition (2.12c) is hopeless as well, for defining the angle as the arccosine of \((a^2+b^2-c^2)/(2ab)\), where \(a\) and \(b\) are the lengths of the distances from the vertex of the angle to the ``nearest discretely located neighboring point'' (p.\ 677) along the legs of the angle, and \(c\) is the third side of the triangle whose two sides are \(a\) and \(b\), would, in general, produce four different angles created by two intersecting lines. The remainder of the paper, which largely consists of definitions for ``arrows'', ``coordinates'', ``force'', and ``parallel transport'', is marred by the same carelessness, rendering this a clear-cut case of a paper that should have been rejected.