The Copernican revolution was consummated by Galileo; but even more significant was the Cartesian revolution which created a new physics… They [Galileo, Descartes and those that followed them] were led to abandon the Aristotelian view of Nature as a hierarchy of different types of objects each striving to fulfill its purpose of attaining perfection in its own way, and to substitute for it the sublime faith that Nature is a great harmonious and mathematically ordered machine.
Confident in that faith, they proved by actual experiment to their skeptical opponents that such was indeed the case. As Galileo says, “Ignorance has been the best teacher I have ever had, since in order to be able to demonstrate to my opponents the truths of my conclusions, I have been forced to prove them by a variety of experiments, though to satisfy myself alone I have never felt it necessary to make many.” This last clause well illustrates how modern science was born of a faith in the mathematical interpretation of Nature, held long before it had been empirically verified…
It was left for Galileo to translate this faith into an exact mathematical procedure. His fundamental principle of method he recorded when he wrote: “To be placed on the title- page of my collected works: Here it will be perceived from innumerable examples what is the use of mathematics for judgment in the natural sciences, and how impossible it is to philosophize correctly without the guidance of Geometry.” Mathematics best expresses the natural structure of things. “Philosophy is written in that great book which ever lies before our eyes; but we cannot understand it if we do not first learn the language and characters in which it is written. This language is mathematics, and the characters are triangles, circles, and other geometrical figures.” …A verbal definition of the essence of gravitation brings no new knowledge; a mathematical formulation of the measurable relations of falling bodies does. Whatever can be said of Nature in mathematical discourse is truly said…
Galileo
was preeminently the investigator rather than the speculative system-builder…
In his greatest work, the Mathematical Demonstrations of Two New
Branches of Science. Proudly he summarizes his achievements:
“My purpose is to set forth a very new science dealing with
a very ancient subject. There is, in nature, perhaps nothing older
than motion, concerning which the books written by philosophers
are neither few nor small; nevertheless I have discovered by experiment
some properties of it which are worth knowing and which have not
hitherto been either observed or demonstrated...
It has been observed that missiles and projectiles describe a curved
path of some sort; however no one has pointed out the fact that
this path is a parabola. But this and other facts, not few in number
or less worth knowing, I have succeeded in proving; and what I consider
more important, there have been opened up to this vast and most
excellent science, of which my work is merely the beginning, ways
and means by which other minds more acute than mine will explore
its remote corners. ... The theorems set forth in this brief discussion
will, when they come into the hands of other investigators, continually
lead to wonderful new knowledge. It is conceivable that in such
a manner a worthy treatment may be gradually extended to all the
realms of nature.”
That Galileo had set forth the outlines of dynamics is remarkable enough; but for human beliefs his work had an even greater significance. He had turned men from the science of perfections, ranks, and purpose to the conception of a universal law in nature…
Two momentous changes had occurred. The world was perceived to be a mechanical order, and simultaneously it became amenable to human control. “Nature,” says Galileo, “is inexorable and immutable, and never passes the bounds of the laws assigned her”