Swinging and Rolling pp 317-358 | Cite as

# Toward a New Science: Axiomatization and a New Foundation

## Abstract

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.

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