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|Historical Context=Although Newton’s curriculum at the University of Cambridge in the 1660’s would have consisted of Aristotelian-scholastic science, Newton is known to have distanced himself from classical metaphysics and instead studied the works of Réné Descartes, who’s work conceived the Cartesian mosaic of science that dominated much of 17th century European natural philosophy.[[CiteRef::Janiak (2016)|pp. 13,55]]
Both Newton’s physics and philosophy were heavily influenced by Descartes’ ideas. Although he disagreed with many of the theories about the natural world adopted in the Cartesian mosaic, it was clear that Newton viewed the Cartesian mosaic as a step forward from the preceding Aristotelian-scholastic one.[[CiteRef::Janiak (2016)|p. 55]] When structuring his view of the natural world, Descartes based his model on a Copernican view of the universe, as opposed to the classical geocentric understanding. Geocentrism was an important axiom to theories of motion developed under Aristotelian-scholasticism.[[CiteRef::Disalle (2004)|p. 37]] With the Earth at the centre of the universe, all motion could be explained causally according to whether the moving object in question existed in the terrestrial or celestial realm, which in that mosaic were thought to be fundamentally different.[Aristotle’s Natural Philosophy, Stanford[CiteRef::Bodnar (2016)]]
Once Descartes had adopted Copernican heliocentrism, the causal theory of motion as understood by Aristotelian-scholastic natural philosophers had to be replaced along with its cosmological model. [Cambridge 1. P. 48.] Cartesian mechanics was developed around a radical comprehension that the source of motion was the same for all bodies in the universe. This idea acted as a pillar upon which a new, mechanical philosophy was constructed. According to this philosophy, the source of all motion of material objects is direct, physical contact with other material objects. The mechanical philosophy was adopted by Leibniz, Huygens, and many other prominent scientists who worked alongside Newton, indicating that much of later 17th century science was deeply rooted in Cartesian philosophy. [Cambridge 1.]
|Major Contributions='''Newton on Calculus'''
Whereas Descartes did not rely on mathematical reasoning for his deductions of scientific propositions, Newton believed that mathematics was an imperative part of conducting natural philosophy.[Newtons philosophy[CiteRef::Janiak (2016)]] In Newton’s physics, material objects were not required to be in direct contact with each other in order for motion to occur. Instead, objects react to each other via a force, which Newton envisioned as a quantifiable property contained in all material objects, the amount of which is directly proportional to the quantity of matter contained in the object. Quantities of force and matter were thus introduced to the mosaic as ontological entities that were measurable. By applying Newton’s three laws of motion that outlined the mathematical relations between force and matter, the motion of all material objects could be explained and predicted mathematically, thereby giving mathematics a new central role in the study of natural philosophy. In The Principia, Newton made extensive use of mathematics in his argument for the unified theory of gravity.[Newtons principia] The mathematical language used in the Principia was geometry, which was also the basis for many of the major models for celestial mechanics that were studied at the time, including the works of Ptolemy, Copernicus and Kepler.[Newtons Principia[CiteRef::Smith (2009)]]
Even though Newton published his key work in the language of geometry, as a mathematician he is primarily role in inventing integral and differential calculus. He is co-credited independently for the calculus alongside his contemporary and rival natural philosopher, Leibniz. [Cambridge 0[CiteRef::Cohen and Smith (Eds. Introduction ) (2002)|pp. 13-14]] The calculus is a mathematical technique that is capable of solving problems in physics involving acceleration, which is a quantity that lay at the heart of Newton’s theory of motion.[[CiteRef::Friedman(2002)]] It was only in the 18th century that mathematicians Jacob Hermann and Leonhard Euler expressed Newton’s laws of motion using calculus.[Newtons principia, [CiteRef::Smith (2009)|p. 29]] In preceding years, calculus became indispensable tool for scientists in the Newtonian mosaic to solve problems in physics, and to predict the behaviour of material objects with an unprecedented degree of accuracy.[Newtons principia[CiteRef::Smith (2009)]] Although geometry is still taught in schools today, the calculus I the primary mathematical technique learned and used in physics and engineering classrooms.
'''Newton on method'''
Prior to the publication of The Principia, the philosophy of motion and change in the universe was largely a theoretical and non-mathematical enterprise. The dominating methodological approach to natural philosophy both in the Aristotelian-scholastic and Cartesian mosaic, was one in which truths about the natural world were proposed as conjectural hypotheses. Newton explicitly rejected the method of hypotheses, and instead demanded that all propositions be deduced from the phenomena and then converted into general principles via induction.
In the second edition of The Principia, Newton states:[Newton’s philosophy]
“I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies and the laws of motion and law of gravity have been found by this method. And it is enough that gravity should really exist and should act according to the laws that we have set forth and should suffice for all the motions of the heavenly bodies and of our sea.” [Cohen-Whitman translation, via “Newtons Philosophy” in Stanford[CiteRef::Newton (1999)]]
Newton called his method the experimental philosophy, because theories about the behavior of empirical objects can only be refuted via experimental procedures.[ 4 The methodology of the Principia[CiteRef::Smith (2004)]] He expressed the core beliefs from which he derived his method in a set of four “rules for the study of natural philosophy,” which he stated in book III of The Principia as follows:[Bernard Cohen transl.]:]
1) “No more cause of natural things should be admitted than are both true and sufficient to explain their phenomena”
2) Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same
3) Those qualities of bodies that cannot be intended and remitted (i.e. qualities that cannot be increased and diminished) and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally”
4) “In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true nonwithstanding any contrary hypothesis, until yet other phenomena make such propositions either more exact or liable to exceptions”[[CiteRef::Newton (1999)]]
Out of these four rules a new, active method for conducting science emerged that stood in stark contrast the previous passive and theoretical Cartesian and Aristotelian-scholastic methods. Propositions are born from natural sources and placed back into the natural world to be tested empirically.[Cambridge, chp. 4[CiteRef::Smith (2004)]] As the four rules were absorbed into the ensuing mosaic, the calculus became deeply incorporated in the experimental method, as it was used to mathematically calculate from natural laws an empirical prediction, and then evaluate how exactly the prediction matched the observed reality.
Using these principles, Newton was able to derive the law of universal gravity in the context of his method. In the Cartesian mosaic, the centripetal force had already been defined as the agent that pulled the moon towards the Earth, keeping its orbit circular rather than linear. Newton applied rules 1 and 2 to determine that the centripedal force, and the force that compelled objects to move downwards towards the Earth, were merely two different expressions of the same thing. Newton then went on to apply the third rule, and argue that this force, which he called gravity, must be a universal property of all material objects. From here, he went on to argue for the unification of superlunary and sublunary phenomena. [Cambridge, chpt. 5 pp. 183-184]
