Toward a New Science: Axiomatization and a New Foundation

The chapter discusses Galileo’s attempt to provide his new results concerning the pheno-kinematics of naturally accelerated motion—his new propositions concerning the regular relations between spaces traversed and the corresponding times elapsed in motions of this type—with a foundation. Providing his new propositions with a deductive structure rooted in fundamental principles or assumptions would allow Galileo to publish them as a new science. He, in particular, systematically explored the question of which of his new statements could serve as a minimal yet strong enough set from which all remaining propositions could be derived. This search for an axiomatic foundation disclosed that to root the deductive tree, at least two of his propositions needed to be equipped with a proof from basic principles. Further considerations deprived the only such fundamental proof he had found so far, and which was based on a dynamical argument, of its explanatory power. This brought Galileo’s search for a foundation temporarily to a halt. It is discussed how in 1604, as evidenced by a letter written to Paolo Sarpi, Galileo picked up work on the problem of a foundation once again, availing himself of a new approach based on representing accelerated motion as characterized by the change of degrees of velocity.