There were of course philosophical and cultural reasons for the development of the (linear) perspective during the Renaissance.
It cannot be considered simply a change of tastes of patrons of art or a phenomenon out of curiosity.
It was linked to profound cultural changes and different attitudes toward the role of man in the world and toward religion which took place during the so-called humanism ( also called Renaissance humanism).
And it was also linked to a renewed interest in sciences, as physics and mathematics: consider, first of all Leonardo Da Vinci, who was at the same time one of the greatest artists of all time and a scientist.
In short words, it can be said that perspective can be considered a consequence and an expression of the modern anthropocentrism, which arose with humanism and Renaissance.
By the term Renaissance humanism historians denote a vast cultural renewal movement, in arts and in philosophy, which characterized European civilization, conventionally established between the end of the XIV century and the end of XV century, which had an attitude of sharp break with Medieval thought, above all by setting the man at the center of the vision of life and of the world, and therefore substituting anthropocentrism for theocentrism, which was typical of the previous era.
Renaissance humanism was characterized by an intense study of ancient classical authors, and an acknowledgment of the privileged position of man in nature.
In figurative arts, perspective is the expression of this new anthropocentrism:
In perspective the anthropocentric culture of modern era finds expression.
The new painting style which we call Renaissance, based on perspective and on the rationalization of space, is the sign of a new attitude of man toward the world, of a spiritual, cultural and social transformation […]. The concept of space which emerges is not an expression of eternal ideas, immutable and abstract[…] The perspective image confers to the observer a privileged position, that acknowledges the primacy of the subject in the area of knowledge, and their centrality in historical agency. For this reason, perspective is the sign of the constitution of an anthropocentric thought, which abandons the Medieval theocentric attitude.
[…] The space that is represented is no longer the divine and absolute
space, but the projection of a subjective point of view, both
individual and universal: individual because it is supposed to be
occupied by a unique observer, with only one open eye; universal as
any other observer could stay on that point and reconstruct the space
which from it takes form according to the universal laws of Euclidean
geometry and optic. $^1$
The literature about Renaissance and perspective is very vast. Some of the most important historians of art dedicated efforts to the subject.
With regard to perspective, specifically, there are two books of some of the most well-known scholars of art:
Panofsky, Erwin, Perspective as a symbolic Form
Florensky, Pavel, Beyond Vision
Florensky was a Russian philosopher, a mathematician and an art historian, and he wrote also about art and mathematics.
Perspective is at the center of many connections between art and sciences, in particular, of course, mathematics and physics.
Boyer, A History of Mathematics, and Kline, Mathematic Thought from Ancient to Modern Times, devote some chapters to Renaissance and mathematics, and in particular to the relationships between mathematics, perspective and art.
Boyer recalls Italian and European artist who developed relatively new links between mathematics and figurative arts, as Filippo Brunelleschi, Leon Battista Alberti, Piero Della Francesca, Leonardo Da Vinci, Albrecht Dürer.
In particular it is interesting that he mentions a treatise, which has been lost, by Leonardo Da Vinci, about perspective, Trattato della Pittura (Treatise of Painting), which begins with the admonition. “Don’t read me, if you are not a mathematician!” (Boyer, Chap. 15, sec. 20) $^2$,
Morris Klein devotes a chapter to the mathematical contribution of Renaissance, and recalls that the Renaissance artists were ‘universal’ men, as they had to do any kind of jobs, so that
they were obliged to learn mathematics, physics, architecture,
stonecutting, engineering, metalworking, anatomy, woodworking, optics,
statics, and hydraulics. They worked manually and at the same time
they coped with the most abstract problems. At least in the XV
century, they were the best mathematical physicists. [emphasis mine] (Kline, Chap. 12, section 1) $^3$
Therefore, they contributed to the development of geometry, for instance Alberti introduced the concepts of projection and section.
Kline, moreover, points out that at the origin of the projective geometry there were some problems that arose in the theory of perspective:
Some geometers of the XVIII century tried to give an answer to these
problems. They considered the methods and the results they obtained as
a part of Euclidean geometry. Actually, these methods and results [..
.] proved to be the origin of a new branch of geometry , which in the
XIX century became known as projective geometry.
Obviously, the studies about perspective were linked not only to mathematics but also to physics.
An interesting study available in English is
Camerota, Filippo, Linear Perspective in the Age of Galileo.
The author here discusses the work of Ludovico Cigoli, La prospettiva pratica (The Practical Perspective), which is an extraordinary scientific document of the role of the representation of the visible world in Galilean research. Cigoli was a painter very close to Galileo, and his treatise illustrates important aspects of the perspective painting, as orthogonal projections of shadows or the use of mechanical instruments.
Another book by Filippo Camerota is La prospettiva del Rinascimento. Arte, architettura, scienza.
(Perspective of Renaissance. Art, Architecture, Science)
The book is a history of perspective, beginning with the studies about vision based on the geometric model of Euclid, through Giotto, Brunelleschi, Alberti, Leonardo, with its connections with mathematics and science.
But I don’t know if an English translation exists.
You can see also:
Samuel Y. Edgerton,
The Mirror, the Window, and the Telescope: How Renaissance Linear Perspective Changed Our Vision of the Universe
$^1$ https://finestresuartecinemaemusica.blogspot.com/2018/09/la-prospettiva-come-espressione.html (my translation). The literature about Renaissance humanism and anthropocentrism is broad, for some link in English see https://www.archaeology.wiki/blog/issue/the-anthropocentric-cosmotheory-of-the-renaissance-and-the-cult-of-man/
$^2$ p. 343 Italian edition
$^3$ p. 271 Italian edition.