La prospettiva e l'infinito

Perspective is perhaps the only mathematical tool that enables us to deal with infinity in finite terms. The vanishing point is, apparently, the image of a point located at an infinite distance; in reality, a close study of the history of this geometric concept, from Guidoubaldo del Monte to Poncelet, and including the intuitions of Desargues and Taylor, leads us to draw very different conclusions. In the first place, the vanishing point I’is not a point, but a region of the picture plane, as tiny as can be but, where necessary, measurable. In a perspective drawing, the vanishing point therefore represents an area of space I undetermined in extension, in which the objective line and the projecting line behave in a way that is no longer controllable: they may or may not have a common point; this way of conceiving the undetermined area does not mean that we cannot determine its starting point (depending on the quality of our instruments), it only excludes that we can measure its extension. At the same time, the vanishing point also represents the undetermined area located on either side of the line. Bearing in mind what we know about the direction determined by two points, we can say that the vanishing point is the image of the direction of the line. It must be clear, however, that, whereas the points I belong to the family of geometric concepts defined by Euclid, the direction is simply a quality of the line and, as such, it has no locus, because it is neither near nor distant, either in physical space or in mental space. The vanishing point is, by definition, neither the projection of a point to infinity nor an improper point, but simply the image of an inaccessible and therefore undetermined position. Similar considerations, of course, can apply to the vanishing line of a plane understood as the image of its laying. Subsequent developments of this concept are related to the effective possibility of operating in general that is, beyond the perspective, without using points at infinity but, on the contrary, exclusively using finite geometric concepts.