In the Quattrocento and Cinquecento the rise of linear perspective caused many polemics which opposed the supporters of an artificial geometrisation of sight to those who were praising the qualities of the drawing according to nature, or were invoking some arguments on a physiological basis. These debates can be grouped according to the four alternatives that form their central concerns: restricted vs. broad field of vision; ocular immobility vs. mobility; curvilinear vs. planar picture; monocular vs. binocular vision. By retaining the first terms of these four alternatives, the history of perspective eliminated many heterodox constructions. From the viewpoint of mathematisation the interest of these debates is that they succeeded, rather than preceded, the adoption of a perspective system defined by the intersection of the visual pyramid. Thus the history of linear perspective constitutes a genuine case of a posteriori justification, or, put differently, it gives us a case of upside down mathematisation.