Time and spaces from physics to biology: Ontogenesis, phylogenesis and the formation of “sense”

1 – The mythical and theological origin of physico-mathematical spaces; the geometrization of time Abstract There is no mathematical plane nor space in Euclid's geometry. Lines are traced, extended, intersected on a plane, an “apeiron” (it has no boundary), which is “practiced” but not mathematized. These lines have no thickness, they intersect in a point, that is a sign (“semeion”); they are objects of a mythical, ideal realm. Infinity is only potential: lines in the plane or endless sequences of numbers can be extended with no limit. Actual infinity will be fully conceptualizedThere is no mathematical plane nor space in Euclid's geometry. Lines are traced, extended, intersected on a plane, an “apeiron” (it has no boundary), which is “practiced” but not mathematized. These lines have no thickness, they intersect in a point, that is a sign (“semeion”); they are objects of a mythical, ideal realm. Infinity is only potential: lines in the plane or endless sequences of numbers can be extended with no limit. Actual infinity will be fully conceptualized much later, in the theological debate of late middle age, as an attribute of God. How does actual infinity relate to or how can it be represented in the finite? The Renaissance Italian painters will show that this is possible: the projective limit of the newly invented linear perspective, first used in XIV century paintings of the Annonciation, shows the infinite in the finite and joins the infinity of God to the bodily, three dimensional presence of a human being, the Madonna. We will critically analyze paintings from Giotto to Piero della Francesca that show this joint invention of actual infinity and of mathematical spaces, which allows as well the expression of a new, corporeal humanity. Later, this theological invention will become the mathematics of Descartes' and Desargues' spaces, the science of Newton's infinities. In the XIX century, physics will extend this mystical creation of infinite space and time to the mathematical “phase space” of its new scientific rigor, based on an increasing mathematical unity of space and time.