Galileo: Mathematizing Science

Galileo Galilei

Introduction: the mathematization of science

It was the habit of medieval scientists who worked in the Aristotelian rhetorical tradition of philosophical speculation to propose theories for the ways in which the world worked based on how well they agreed with the claims of the most respected philosophers, or with how well one could marshal explanations to support one's theory. The sixteenth and seventeenth centuries saw a serious challenge to this way of learning about the world in the new methods of experimental science undergirded by mathematical justification which were pioneered by Galileo Galilei (1564 - 1642).
The new methods approached reality as knowable through rational argumentation based on mathematical principles and agreement of theory with experimentation, what is now identified as the scientific method. Instead of speculation as to why phenomena occurred as they did, the new scientist described precisely what the phenomena were and measured their effects on other phenomena quantitatively in an effort to isolate the essential ingredients that composed the observables. Once these underlying components were identified, then rational deduction was employed, as in mathematics, to formulate theories about the phenomena. Galileo formulated these new principles in The Assayer (1610):

[Natural] philosophy is written in that great book which ever lies before our eyes--I mean the universe--but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

    Galileo, born into a family of nobility, had at first planned on a career of medicine, but became enamored of mathematics after attending the lectures of Ostilio Ricci, a student of Tartaglia's, on Euclid's Elements at the University of Pisa. At age 25, he obtained a lectureship at the university, and it is said that he gave public demonstrations at the campanille (the famous Leaning Tower) on how bodies of different weights fall in equal times, in contrast to the claims of Aristotle. Rather than convincing the Aristotelian philosphers, he was accused of sorcery, and was forced to give up his position at Pisa after only three years. He later moved to the University of Padua where he taught as professor of mathematics for 18 years. While at Padua, he learned of the manufacture of a telescope in 1609 by Hans Lipperhey and built one for himself. He spent the next years observing the night sky, discovering the Galilean moons of Jupiter (which he named the Medicean moons after his patrons, the wealthy Medici family), mountains on the Earth's Moon, and the makeup of the Milky Way as millions of individual stars. These discoveries, and a defense of the Copernican theory of heliocentrism (that the Earth and the planets revolved about the Sun) were published in The Starry Messenger (Sidereus Nuncius) in 1610, bringing him instant notoriety. His controversial stances forced him to leave his position at Padua under a barage of criticism to return to the University of Pisa and the patronage of the Grand Duke of Tuscany, Cosimo de Medici.
    Galileo came under increasing fire for holding views contrary to Sacred Scripture. (A commonly cited instance was the story in the Book of Joshua (Joshua 10:7-14) in which a prayer to the Lord God stops the sun in the course of its trek across the sky to aid the Israelites in battle, "evidence" that the sun does revolve about the earth.) He was summoned to Rome in 1615-1616 to answer these charges, and in a public hearing before the great Cardinal Robert Bellarmine, was formally censured. In 1623, Maffeo Barberini, a friend of Galileo's, became Pope Urban VIII; Galileo was granted a dispensation to publish on the Copernican theory under the restriction that it be presented as hypothetical and that the Ptolemaic system be presented alongside it. He then began work on Dialogo sopra due massimi sistemi del mondo (Dialogue Concerning Two Chief World Systems), written in Italian rather than Latin to achieve a wide readership, and in the form of a dialogue amongst three characters: Salviati, the scholar-scientist; Simplicio, the dull Aristotelian; and Sagredo, the layman and moderator. Galileo's enemies were incensed, and even Urban VII considered the book a humiliation. Galileo, now aged 70, was brought before the Inquisition, accused of heresy, suffered the punishment of having the Dialogo added to the Index of Forbidden Books, and was put under house arrest. (Galileo was finally exonerated of these "crimes" by John Paul II in 1992.) It was in the years after his trial that he wrote the Discorsi e dimonstrzioni mathematiche intorno a due nuove scienze (Discourses and Mathematical Demonstrations Concerning Two New Sciences), in which he laid out the theory of the motion of objects in free fall and of projectiles.
    In the Discorsi, published in 1638, he mathematizes the concepts of speed and acceleration, noting that the former measures the rate of change of distance over time and that the latter measures the rate of change of speed over time. He also formulates and proves geometrically a principle known to Medieval scientists and called the mean-speed law: an object subject to uniform acceleration covers a distance equal to that of an object moving over the same interval of time at a constant speed equal to the speed of the original object at the halfway point in its motion. These results were significant in that they provided the motivation to explore further geometrical ideas with an end to provide applications to physical phenomena, in particular the study of mechanics. Galileo placed himself squarely in the tradition of Archimedes and Ptolemy, who applied mathematical ideas to the real world, rather than Euclid, Plato, and, of course, Aristotle, who saw the benefit of mathematics as describing an idealized world.

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