Geometries of O

Authors: Hannes Hutzelmeyer

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by Tarski in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the planar geometries of O contain only functions and no relations to start with. Essentially three quaternary functions occur: appension for line-joining of two pairs of points, linisection representing intersection of straight lines and circulation corresponding to intersection of circles. Functions are strictly defined by composition of given ones only. Both, Euclid and Lobachevsky planar geometries are developed using a precise notation for object-language and metalanguage, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them: (A) Based on a special triangle construction Euclid planar geometry can start with a less powerful ontological basis than Lobachevsky geometry. (B) Usual Lobachevsky planar geometry is not complete, there are nonstandard planar Lobachevsky geometries. One needs a further axiom, the 'smallest' system is produced by the proto-octomidial-axiom. (C) Real numbers can be abandoned in connection with planar geometry. A very promising conjecture is put forward stating that the Euclidean Klein-model of Lobachevsky planar geometry does not contain all points of the constructive Euclidean unit-circle.

Comments: 93 Pages.

Download: PDF

Submission history

[v1] 2018-12-04 07:19:40

Unique-IP document downloads: 16 times is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus