nLab synthetic geometry

Idea

To speak of points of space, as if they constituted the positive element of space, is inadmissible (PdN§254b)

That the line does not consist of points, nor the plane of lines, follows from their concepts (PdN§256b)

Synthetic geometry is about formalization of geometry by axioms that directly speak about the fundamental concepts of geometry – such as points and lines – instead of about a backdrop for such objects – such as Cartesian spaces.

In the 19th century, after noneuclidean geometries were found (Lobachevski, Bolya, Gauss), mathematicians reexamined the foundations of geometry. Some schools concentrated on the axiomatic construction of geometry, independent from the tools which were offered by the models with concrete coordinate algebras. Synthetic geometry in this sense referred to doing geometry without recourse to algebras of functions and analytic computations. This was especially successful in projective geometry, see synthetic projective geometry.

The main part of synthetic geometry is the study of incidence structures in geometry, sometimes also called incidence geometry. A modern point of view on incidence geometry is through the theory of buildings.

Comparison to synthetic differential geometry

This perspective of synthetic geometry is somewhat different from the more modern subject called synthetic differential geometry, although the terminology has the same origin. Indeed, in view of the fact that in the axiomatic approach the emphasis is on the relationship between the elements of geometry (points, lines, curves) and that in modern geometry these are viewed as subvarieties, hence subobjects, in his original axiomatization of synthetic differential geometry William Lawvere has reinterpreted the meaning of synthetic geometry as being concerned with the categorical relationship between geometrical objects. However, he was not concerned with working without analytic and algebraic calculations (which do appear in his axioms for infinitesimally thickened points, the Kock-Lawvere axioms); he just wanted to introduce the foundations internally. (But this is different for his later development of cohesive geometry which includes the original synthetic differential geometry via differential cohesion. Here no analysis enters the axioms.)

Lawvere also notes the failure (for physical applications) of the correspondence between evolution paths and corresponding maps in differential geometry, and wanted to work in closed monoidal categories (having the exponential law for mapping spaces). In synthetic differential geometry, which he created with Kock, Dubuc and others, this enabled also intuitive, pre-Cauchy reasoning with differentials, what also reminded him of 19th century mathematics what probably also influenced on choosing the name. But still, the subject of synthetic differential geometry is non-synthetic in the sense of traditional synthetic geometry. A genuinely synthetic axiomatization of differential geometry is cohesion.

Related concepts

synthetic geometry
Euclidean geometry
hyperbolic geometry
elliptic geometry

• parallel postulate

• synthetic mathematics

• geometric homotopy type

• cohesive homotopy type

References

• Wikipedia, Synthetic geometry

• Alfred Tarski, What is elementary geometry?, in The axiomatic method. With special reference to geometry and physics Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958 (ed. L. Henkin, P. Suppes, and A. Tarski), Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland (1959), pp. 16–29. (pdf)

Textbook accounts of the axiomatization of Euclidean geometry includes these:

• Wolfram Schwabhäuser, Wanda Szmielew, Alfred Tarski, Mathematische Methoden in der Geometrie, Springer 1983

• Anton Petrunin, Euclidean Plane and its Relatives; a minimalistic introduction, (arXiv:1302.1630)

Full formalization of the first part of this book in Coq (as synthetic geometry but following Tarski’s work) is discussed at

• Gabriel Brau, Pierre Boutry, Julien Narboux, GeoCoq, La géométrie de Tarski en Coq