A point (as a point in space ) is a fundamental element of geometry . One imagines an object underneath it without any expansion. With the axiomatic approach to geometry ( synthetic geometry ), there are other classes of geometric objects, such as straight lines, in addition to points . In contrast, in analytical geometry and differential geometry , all other geometric objects are defined as sets of points. In functional analysis, functions can be viewed as points in a function space. In higher geometry , for example, planes of a three-dimensional projective space are understood as points of the associated dual space.
Ancient geometry to synthetic geometry
After Proclus , Pythagoras was the first to offer a definition of a point, as a unit ( monas ) that has a position. The Greek mathematician Euclid describes around 300 BC In his work The Elements in the first definition the point as "something that has no parts" and uses the term semeion ( ancient Greek σημεῖον actually "sign", in mathematics especially "point"). It is an abstract designation that is probably to be understood as an answer to the difficulties discussed extensively in the Platonic school of grasping the connection between points that have no extension and the lines that are composed of them and that have extension; for example in Aristotle's De generatione et corruptione .
For theorems and their proofs in synthetic geometry, however, the true nature of points and lines does not matter, only the relation of the objects to one another determined by axioms. David Hilbert is credited with saying that instead of “points, straight lines and planes” one could also say “tables, chairs and beer mugs” at any time; all that matters is that the axioms are fulfilled.
A point in this case is a term to which the individual axioms refer. An example is the first axiom from Hilbert's system of axioms :
- Two different points P and Q always determine a straight line g.
The meaning of the term point results from the totality of the axiom system. An interpretation as an object without extension is not mandatory.
In the projective plane , the terms point and straight line are even completely interchangeable. This makes it possible to imagine a straight line as infinitely small and a point as infinitely long and infinitely thin.
In analytic geometry, the geometric space is represented as a -dimensional vector space over a body . Each element of this vector space is called a point. A base defines a coordinate system and the components of a vector with respect to that base are called the coordinates of the point. One point has the dimension zero.
All other geometric objects are defined as sets of points. For example, a straight line is defined as a one-dimensional affine subspace and a plane as a two-dimensional affine subspace. A sphere is defined as the set of points that are a certain distance from the center.
In differential geometry, the elements of a manifold are called points. In this case these are not vectors, but a point can be provided with coordinates using a local map.
The following comment comes from Oskar Perron :
"A point is exactly what the intelligent, but harmless, uneducated reader imagines."
- Manon Baukhage: The point. Admittedly, it doesn't look much - as small as it is. In fact, it is one of the great riddles of the world ; in: "PM - Peter Moosleitners Magazin No. 2/2005 (Munich: February 2005); pp. 58–65.
- Definition of the point according to Euclid (Leipzig 1549) 
- Wilhelm Gemoll : Greek-German school and hand dictionary . G. Freytag Verlag / Hölder-Pichler-Tempsky, Munich / Vienna 1965.
- Leslie Kavanaugh: The architectonic of philosophy: Plato, Aristotle, Leibniz . Amsterdam University Press. 2007.
- Oskar Perron: Non-Euclidean elementary geometry of the plane , Stuttgart 1962