Understanding mathematical abstraction in the formularization of Galileo's law

Abstract

Galileo's revolution in science introduced an analytical method to science that typifies the overall modern thinking of extracting, abstracting, and grasping only critical aspects of the target phenomena and focusing on “how”, which is a quantitative relationship between variables, instead of “why”. For example, to him, the question of 'why does an object fall' is of no significance; instead, only the quantitative relationship between distance from the falling object and time is important. Yet, the most fundamental aspect of his idea is that he introduced a quantified time t. When an object is projected horizontally, the distance travelled at some time in the horizontal direction is summed up as d ∝t, whereas the distance falling at some time in the vertical direction is summed up as d ∝ t². Here, the distance, which is a spatial attribute, is expressed as a function of time, t. That is, time is identified as a homogeneous amount that can be reduced to an algebraic number. It is now possible to calculate the laws of motion of things using functions of time. In this respect, mathematical time was a decisive variable in making mathematisation of physical nature practical. Because, according to atomic theory, vacuum exists between an atom and an object composed of atoms or between objects – ignoring factors that interfere with motion, such as friction – the space for absolute time, which is a mathematical time, can be geometrically defined. In order to justify this mathematical abstraction strategy, thought experiments were conducted rather than laboratory experiments, which at that time were difficult to perform.

Keywords

• galileo
• quantitative relationship
• atomic theory
• mathematical abstraction
• thought experiment