Hilbert's Synthesis on Foundation of Geometry

The relationships between intuition, axiomatic method and formalism in Hilbert's foundational studies has been discussed several times, but geometrical ones still have unclear sides and there is not a commonly held opinion.In this article we try to frame Hilbert’s geometrical works within a historical context. The aim is to show that intuition and nature of the axioms in \emph{Grundlagen der Geometrie} do not derive from a mature philosophical awareness of the author, but from the development of a historical path of the idea of geometry and of its foundations. The path begins with the discovery of non-Euclidean geometry and finds in Hilbert’s work its final and definitive synthesis for Euclidean geometry

On a Theorem of Cooperstein

A theorem by Cooperstein that partially characterizes the natural geometry An,d(F) of subspaces of rank d − 1 in a projective space of finite rank n over a finite field F, is somewhat strengthened and generalized to the case of an arbitrary division ring F.Moreover, this theorem is used to provide characterizations of An,2(F) and A5,3(F) which will be of use in the characterization of other (exceptional) Lie group geometries

Abstract convexity spaces

The principal question discussed in this dissertation is the problem of characterizing the linear and convex functions on generalized line spaces. A linear function is shown to be a convex function. The linear and convex functions are characterized, that is, a function f:[right arrow] X— is linear [convex] if and only if f[subscript l] is linear [convex] in the usual sense on each line of a generalized line space X. We prove that if a function has at least one support at each point on its graph, then it is a convex function. In the first chapter the basic concepts of abstract convexity spaces are introduced. The next chapter is concerned with join systems which are shown to be examples of abstract convexity spaces. On the other hand, a domain-finite, join-hull commutative abstract convexity space with regular straight segments satisfies the axioms of a join system. Consequently, such abstract convexity spaces satisfy the separation property. In Chapter III, the linearization of abstract spaces is done using a linearization family. The following chapter is on generalized line spaces and graphically it is shown that Pasch’s and Peano's axioms do not hold in a certain generalized line space. It is also proved that the separation property may not hold, in general, in a generalized line space. Finally, the convex and linear functions are studied on generalized line spaces. The linearization of generalized line spaces is done by means of the properties of a linearization family

Recent developments on absolute geometries and algebraization by K-loops

AbstractLet (P,L,α) be an ordered space. A spatial version of Pasch's assertion is proved, with that a short proof is given for the fact that (P,L) is an exchange space and the concepts h-parallel, one sided h-parallel and hyperbolic incidence structure are introduced (Section 2). An ordered space with hyperbolic incidence structure can be embedded in an ordered projective space (Pp,Lp,τ) of the same dimension such that P is projectively convex and projectively open (cf. Property 3.2). Then spaces with congruence (P,L,≡) are introduced and those are characterized in which point reflections do exist (Section 4). Incidence, congruence and order are joined together by assuming a compatibility axiom (ZK) (Section 5). If (P,L,α,≡) is an absolute space, if o∈P is fixed and if for x∈P,x′ denotes the midpoint of o and x and x̄ the point reflection in x then the map o: P→J; x→xo≔x̃′ satisfies the conditions (B1) and (B2) of Section 6, and if one sets a+b≔ao∘0o(b) then (P,+) becomes a K-loop (cf. Theorem 6.1) and the J of all lines through o forms an incidence fibrtion in the sense of Zizioli consisting of commutative subgroups of (P,+) (cf. Property 7.1). Therefore K-loops can be used for an algebraization of absolute spaces; in this way Ruoff's proportionality Theorem 8.4 for hyperbolic spaces is presented

Linearization of an abstract convexity space

Axiomatic convexity space, introduced by Kay and Womble [22] , will be the main topic discussed in this thesis. An axiomatic convexity space (X,C), which is domain finite and has regular straight segments, is called a basic convexity space, A weak complete basic convexity space is a basic convexity space which is complete and has C-isomorphic property. If in addition, it is join-hull commutative then it is called (strong) complete basic convexity space. The main results presented are: a generalized line space is a weak complete basic convexity space, a complete basic convexity space is equivalent to a line space; and a complete basic convexity space whose dimension is greater than two or desarguesian and of dimension two, is a linearly open convex subset of a real affine space. Finally, we develop a linearization theory by following an approach given by Bennett [3]. A basic convexity space whose dimension is greater than two, which is join-hull commutative and has a parallelism property, is an affine space. It can be made into a vector space over an ordered division ring and the members of C are precisely the convex subsets of the vector space

The completeness axiom of Lobachevskian geometry

This paper gives a proof that the Completeness Axiom of Lobachevskian geometry -- as formulated in the second English translation of David Hilbert's Foundations of Geometry (tenth German edition)--is a theorem in the three dimensional Poincare model. An explicit canonical isomorphism between all models of Lobachevskian space is given. This, together with the work of William Lee Zell (A Model of Non-Euclidean Geometry in Three Dimensions, Master's Thesis, Oregon State University, 1967) and Robert W. Eschrich (A Model of Non-Euclidean Geometry in Three Dimensions, II, Master's Thesis, Oregon State University, 1968), establishes that the three dimensional Poincar4 model is a model of Lobachevskian geometry based upon Hilbert's axioms