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Both While both Newton’s physics and philosophy were heavily influenced by Descartes’ ideas though , they were also a challenge to what had, by then, become the new Cartesian orthodoxy. Descartes' '''mechanical natural philosophy''' was derived from ancient Greek atomism. He was the most prominent member of a community of '''corpuscularist''' thinkers, who maintained that visible objects were made of unobservably tiny particles, whose relations and arrangement were responsible for the properties of visible bodies. Particles influenced one another only by direct physical contact, which was the cause of all motion, and ultimately all change.[[CiteRef::Disalle (2004)]] Aristotle had explained the properties of visible bodies in terms of their form, rather than in terms of the arrangement of their constituent parts. He maintained that heavy objects, composed of the element earth, tended towards their natural place; the center of the universe. The concept of a sphere of earth at rest in the center of the universe was central to Aristotle's entire cosmology. Motion in the terrestrial and celestial realms were seen as fundamentally different.[[CiteRef::Bodnar (2016)]] Descartes' theories explained gravity as due to a swirling vortex of particles around the Earth, which pushed things towards its center. Celestial motions were not different in kind. In accord with Copernican heliocentrism, Descartes posited that a larger vortex surrounded the sun, with the smaller planetary vorticies caught in a larger solar vortex.[[CiteRef::Garber (1992)]][[CiteRef::Disalle (2004)]] In Newton's time, major champions of the mechanical natural philosophy included Christiaan Huygens (1629-1695) and Gottfried Wilhelm Leibniz (1646-1716), who was to become a major rival of Newton's. By the time Newton published his magnum opus, ''Philosophiae Naturalis Principia Mathematica'' (''Mathematical Principles of Natural Philosophy'')in 1687, Descartes' views had been accepted at Cambridge. The title of Newton's work suggests he intended it to be in dialog with Descartes' ''Principia Philosophiae'' (''Principles of Philosophy'') published in 1644.[[CiteRef::Janiak (2016)]] Newton contested Cartesianism as the orthodoxy he sought to overturn.
Whereas Descartes did not rely on mathematical reasoning for Newton's two most important works of natural philosophy were the ''Principia'', published in 1687, which dealt with his theories of motion and universal gravitation, and ''Opticks: or, A Treatise of the Reflexions, Refractions, Inflexions, and Colours of Light'', or simply ''Opticks'', which was published in 1704 and dealt with his deductions theories of scientific propositionslight and color. Like Descartes, Newton believed that mathematics was an imperative part of conducting natural 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, a new concept 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, material objects in the universe were compelled to accelerate through action at a distance. Additionally, the laws outlined the mathematical relations between this acceleration and the quantities of force and matter could be explained and predicted mathematically, thereby giving mathematics a new central role in the study of natural philosophy. In The the ''Principia'', Newton made extensive use of mathematics in his argument for the unified theory of gravity.[[CiteRef::Smith (2009)]] 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.[[CiteRef::Smith (2009)]]
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|Historical Context=When Isaac Newton began his studies at Cambridge University's prestigious Trinity College in 1661, more than a century had passed since Nicolaus Copernicus (1473-1543) had proposed a '''heliocentric cosmology''' in his 1543 ''De Revolutionibus Orbium Coelestium'' (''On the Revolutions of Heavenly Spheres''). It had been fifty years since Galileo Galilei (1564-1642) had published his observations with the telescope in 1610, which uncovered dramatic evidence favoring the Copernican system. His discovery of the phases of the planet Venus indicated that it revolved around the sun and was lit by reflected sunlight. His description of four moons circling Jupiter indicated that Earth, with its own moon, resembled this planet. Finally, his discovery of surface features on the moon indicated that it was another world, as expected under the Copernican system, but not by Aristotelianism. Around the same time, Johannes Kepler (1571-1630)published his laws of planetary motion, indicating that the planets revolved around the sun on elliptical paths, replacing the circular motion and complex epicycles of Copernicus and Ptolemy.[[CiteRef::Westfall (1980)|pp. 1-7]] According to Westfall, "by 1661 the debate on the heliocentric universe had been settled; those who mattered had surrendered to the irresistible elegance of Kepler's unencumbered ellipses, supported by the striking testimony of the telescope, whatever the ambiguities might be. For Newton, the heliocentric universe was never a matter in question".[[CiteRef::Westfall (1980)|p. 6]] A planetary Earth that rotated on its axis and revolved around the sun was incompatible with the accepted Aristotelian physics. The community of the time was engaged with the question of how it could be that the Earth itself was in motion through space, and with the question of how one could hope to gain reliable knowledge in the face of the failure of Aristotelian scholastic knowledge accepted for centuries.
Newton’s education at Cambridge was classical, focusing on Aristotelian rhetoric, logic, ethics, and physics. Bound to Aristotelian scholasticism by statutory rules,[[CiteRef::Christianson (1984)|p. 33]] the curriculum had changed little in decades.[[CiteRef::Westfall (1980)|pp. 81-90]][[CiteRef::Smith (2009)]] Like many of the more ambitious students, Newton distanced himself from classical metaphysics and instead studied the works of the French natural philosopher [[René Descartes]](1596-1650) on his own. By 1664, Newton is known to have read the 1656 Latin edition of Descartes' ''Opera Philosophica'', a one volume compilation of Descartes' major works.[[CiteRef::Smith (2009)]] Descartes had died just over a decade prior, and these works had first been published within the preceding thirty years. They were gaining in popularity and by about 1680 would become the [[Theory Acceptance|accepted]] centerpiece of the Cambridge curriculum, as they also would in Paris by 1700.[[CiteRef::Barseghyan (2015)|p. 190]] Newton is known to have been profoundly influenced by Descartes views of space, matter, and God, and by commentaries on Descartes by Henry More (1614-1687). [[CiteRef::Janiak (2014)]]
Descartes saw the ultimate justification of knowledge claims to lie with human reason and the absence of doubt. He relied on classical methods of theorizing and conjectured hypotheses in order to construct scientific propositions.[[CiteRef::Janiak (2016)]] Such a '''rationalist''' approach to knowledge was also championed by Baruch Spinoza (1632-1677), Nicolas Malebranche (1638-1715), and by Gottfried Wilhelm Leibniz.[[CiteRef::Lennon and Dea (2014)]] But, by the early 17th century, experimental researchers like Galileo Galilei and Robert Boyle (1627-1691) had begun to elaborate and practice a very different approach to knowledge based on experimentation and extensive use of mathematics. Following the inductive methodology advocated by [[Francis Bacon]](1561-1626), they maintained that theoretical principles emerged from experimental data by a process of inductive generalization. However, there were also dissenters like Newton's contemporary Christiaan Huygens, who believed that most experimental work involved formulating hypotheses about unobservable entities, which were tested by their observable consequences. This was an early form of hypothetico-deductivism. Newton rejected Cartesian rationalism, and argued that the Cartesians did not sufficiently employ mathematics and experimentation in their work. He rejected the method of hypotheses outright. [[CiteRef::McMullin (2001)]][[CiteRef::Janiak (2016)]] He supported inductivism, and held epistemological views similar to those of his contemporary and friend [[John Locke]](1632-1704), who maintained that all knowledge came from experience.[[CiteRef::Rogers (1982)]]
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=== Newton on Mathematics and Natural Philosophy ===
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.[[CiteRef::Cohen and Smith (Eds.) (2002)|pp. 13-14]] As a mathematical technique, calculus had been the first method that was capable of articulating the quantity of acceleration, unlocking a new world of calculations which geometry as a technique had been incapable of solving.[[CiteRef::Friedman (2002)]] Eventually, 18th century that mathematicians Jacob Hermann and Leonhard Euler expressed Newton’s laws of motion using Newton's own technique of calculus, but in the symbolic expression that Leibniz had developed.[[CiteRef::Smith (2009)|p. 29]] In following 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.[[CiteRef::Smith (2009)]] Although geometry is still taught in schools today, calculus is the primary mathematical technique learned and used in physics and engineering classrooms.
