Geometry began as a systematic study of physical space and the shapes that move in it. The space in which we move is for everyone one of the first experiences we have from the first months of life. Our senses determine the sensations that allow us to recognize the shapes of objects and their movements. However, the geometric notions such as point, line, rectangle, cube, sphere, etc., do not find a perfect match in physical reality. In physical space, in fact, there are no points and lines as described by geometry, nor figures with only two dimensions, nor perfect cubes or spheres. Geometry therefore aims to provide an ideal “model” of physical reality, both for the shapes of objects and for the properties of space.
Until the second half of the nineteenth century, mathematicians and philosophers were essentially in agreement in considering geometry as the science that rationally described the properties of physical space.
Beginning in the second half of the nineteenth century, mathematicians became convinced that geometry does not exactly describe physical space, that more than one equally true geometry is possible from a logical and mathematical point of view. The mathematical study of geometry then became differentiated from the study of physical space and from the study of psychological space as perceived by humans with their senses. Mathematicians have accepted the existence of different mathematically possible geometries, they have been content to build abstract models and left to physicists the “choice” of the model that best describes physical phenomena from infinitely small to infinitely large. Geometry then became a branch of mathematics to which mathematicians have tried to give an exclusively logical foundation, independent from physical experiences.
The link between physics and mathematics has never been interrupted. With the passing of the centuries, we have realized more and more how much the “geometry” of the physical world is very complex and how some new geometries are able to better describe phenomena that with the old Euclid’s geometry was not able to explain.
The axiomatic method, primitive concepts and definitions
Geometry, since the time of Euclid, has been organized axiomatically, that is, starting from the foundations. In mathematics, these foundations consist of the primitive concepts and axioms. The primitive entities are the notions that we decide not to define. One can easily realize, in fact, that not everything can be defined, since in every notion that is defined one must have recourse to other notions, which in turn must be defined by means of other notions and so on backwards without theoretically this process ever having an end, necessarily arriving at some notions so primitive that they cannot be defined by other more elementary notions.
It is neither necessary nor possible to associate any explicit meaning with these notions; instead, it is essential to express their properties exclusively through axioms, that is, through unprovable properties that indicate, however, how the primitive entities should and can be used. The mathematician Hilbert uses three primitive entities – point, line and plane – and 21 axioms. All definitions of geometric entities are derived from the primitive entities.
- Euclidean geometry
- Non-Euclidean geometry
- Solid geometry