Newton and Descartes: Mathematics

Newton and Descartes: Mathematics
Isaac Newton studied the works of most, if not all, of the significant mathematicians of his era and of earlier eras. When he began his studies at Cambridge in 1661, Rene Descartes had been dead for just eleven years; the Cartesian influence in mathematics, philosophy, and the natural sciences would have been detectable to the young Newton. Newton read the works of Descartes, studying them carefully; as is often the case in the history of philosophy, the former philosopher will have a twofold effect on the latter: Newton learned and adopted much from Descartes, but would grow to firmly reject other aspects of Cartesian philosophy and mathematics.
Newton's first exposure to Descartes was probably not through a primary text - not through a book written by Descartes himself - but rather a secondary text, a book written by Frans van Schooten, a Dutchman who popularized Cartesian mathematics. One of Newton's most famous phrases - about standing "on the shoulders of giants" - was written in reference to Descartes. Carl Boyer writes:

Early in his first year, however, he bought and studied a copy of Euclid, and shortly thereafter he read Oughtred's Clavis, the Schooten Geometria a Renato Des Cartes, Kepler's Optics, the works of Viete, and, perhaps most important of all, Wallis' Arithmetica infinitorum. Moreover, to this training we must add the lectures that Barrow gave as Lucasian professor, which Newton attended, after 1663. He also became acquainted with the work of Galileo, Fermat, Huygens, and others. It is no wonder that Newton later wrote to Hooke, "If I have seen farther than Descartes, it is because I have stood on the shoulders of giants."
Although Descartes earned lasting fame by securing a central role for algebra, not only in mathematics, but also in "natural philosophy" as the natural sciences and empirical observational sciences were then called, he had not yet progressed to the understanding that an algebraic equation and its curve plotted on a plane were equivalent, i.e., that having either of them amounted to having both, and that given either of them, the other was producible therefrom. Niccolo Guicciardini writes:

The seminal text in Newton’s mathematical formation is a highly abstract essay: Descartes’ Geometrie. He borrowed and annotated the second Latin edition (1659–1661) by Frans van Schooten. Here Descartes had proposed a novel method for the solution — he claimed in the opening sentence — of all the problems of geometry. It was on this text that Newton concentrated his attention. Descartes taught how geometrical problems could be expressed in terms of algebraic equations (this process was termed the resolution or analysis of the problem). He maintained that finding the equation and determining its roots, either by finite formulas or approximations, is not the solution of the problem. It was not a surprise for the contemporaries of Descartes and Newton to read that in order to reach the solution, one had to geometrically construct the required geometrical object. A geometrical problem called for a geometrical construction (a composition or synthesis), not an algebraic result. Traditionally, such constructions were carried out by means of intersecting curves. Thus, Descartes provided prescriptions to construct segments that geometrically represent the roots and are therefore the solution of the problem.
Modern readers will be aware that the Newtonian (and Leibnizian!) calculus enabled mathematicians to think beyond the limits of Cartesian algebra. Descartes was forced to use trial-and-error ('heuristic') methods, making at first only approximations, and then refining them.

By Newton’s day the heuristic method proposed by Descartes was labeled common analysis. It was contrasted with a more powerful new analysis, which tackled problems about tangents and curvature of curves and about the determination of areas and volumes that cannot be reached by the finitist means envisaged by Descartes. Common analysis proceeds by “finite” equations (algebraic equations, we would say) in which the symbols are combined by a finite number of elementary operations. The new analysis instead goes beyond these limitations because it makes use of the infinite and infinitesimal.
As soon as Newton began digesting Cartesian geometry, he began to ponder that algebraic equations might not only yield a curve, but rather also yield information about the curve. It is now commonplace for students in calculus to use the equation to determine 'local minima' and 'local maxima' using Newton's methods.

Newton’s early notes on Descartes’ Geometrie reveal how quick he was in mastering algebra applied to geometry. In 1665 he began to think about how the equation could reveal properties of the curve associated to it via a coordinate system.
The need for information about these curves was both theoretical and practical. Drawing the curves with accuracy, and understanding how to derive information about those curves from their corresponding algebraic equations was simultaneously important both for concrete tasks like lens-grinding and for philosophical tasks in exploring the foundations of mathematics and physics. The reader will recall Newton's interest in telescopes.

Descartes had devised several mechanisms for generating curves. In De Organica Conicarum Sectionum in Plano Descriptione Tractatus (1646), which Newton read in Exercitationum Mathematicarum (1657), van Schooten had presented several mechanisms for generating conic sections. This research field was connected with practical applications, for instance, lens grinding and sundial design, but it was also sanctioned by classical tradition and motivated the highly abstract needs underlined by Descartes. Newton was able to devise a mechanism for generating conics and to extend it to higher-order curves.
Cartesian methods were capable of managing only some types of curves, not all types of curves. Newton's work in physics required that he have more powerful mathematics so that he could work with more types of curves.

In 1665, Newton deployed organic descriptions in order to determine tangents to mechanical lines, that is, plane curves such as the spiral, the cycloid, and the quadratrix that Descartes had banned from his Geometrie. The study of mechanical lines, curves that do not have an algebraic defining equation, was indeed a new, important research field. How to deal with them was unclear. Newton was able to determine the tangent to any curve generated by some tracing mechanism. He decomposed the motion of the tracing point P, which generates the curve, into two components and applied the parallelogram law to the instantaneous component velocities of P.
Descartes had simply been forced to omit the treatment of certain types of curves. The advent of Leibnizian (and Newtonian!) calculus has had a ubiquitous impact on the natural sciences. Carl Boyer writes:

Where Descartes' geometry had once excluded all nonalgebraic curves, the calculus of Newton and Leibniz showed how essential is the role of these in their new analysis.
While the Cartesian approach and its ability to map algebraic equations on a plane still seemed new, Newton was already transcending it, and creating non-algebraic approaches. Niccolo Guicciardini writes:

For instance, the point of intersection of two moving curves will generate a new curve whose tangent Newton was able to determine. Such a method for determining tangents without calculation pleased Newton as much as did his new techniques for the organic description of conics. This was an approach to the study of curves — alternative to the Cartesian algebraic — that Barrow had promoted and that in the 1670s Newton began to couple with ideas in projective geometry. Already in 1665 the master in the common and new algebraic analyses was experimenting with non algebraic approaches to geometrical problems.
Leibniz and Newton broke through the boundaries of Cartesian algebra. Their calculus would fuel the industrial revolution, as more sophisticate steam engines and other devices would be designed with the aid of this newer, higher mathematics. But we might speculate that neither man had such pedestrian interests.

Using Leibnizian jargon, we can say that while differentiation of algebraic functions (accepted by Descartes) leads to algebraic functions, integration can lead to new transcendental functions. Newton referred to what are now called transcendental functions as quantities “which cannot be determined and expressed by any geometrical technique, such as the areas and lengths of curves.” Infinite power series — in some cases fractional power series — were the tool that young Newton deployed in order to deal with these mechanical (transcendental) curves.
Newton's understanding of mathematics was colored by his interests in mystical, theological, and even occult understandings of science. He was convinced that ancient scientists had made marvelous discoveries, and hidden these discoveries in allegories and other cryptic writings. This famed "other side" of Newton enabled him to venture predictions on the exact date of the end of the world, which he envisioned in a fairly orthodox version of a triumphalist second advent - although in other matters, he was anything but orthodox, as his studies of Hebrew and Greek grammar led him to reject Cambridge's Anglican understanding of the Trinity. (Newton's prediction for the end of the word was in the year 2060.) Newton's attempt to reconstruct what he held to be the lost knowledge of the ancients would lead him, he thought, to nearly miraculous or supernatural results.

Lucasian Lectures on Algebra stemmed from a project on which Newton had embarked since the fall of 1669, thanks to the enthusiasm of John Collins: the revision of Mercator’s Latin translation of Gerard Kinckhuysen’s Dutch textbook on algebra. Newton’s involvement in this enterprise was an occasion to rethink the status of common analysis. He began experimenting with what he understood as ancient analysis, a geometrical method of analysis or resolution that, in his opinion, the ancients had kept hidden. In his Lucasian Lectures on Algebra, which he deposited in the University Library of Cambridge in 1684 and from which William Whiston edited the Arithmetica Universalis (1707), Newton extended Cartesian common analysis and arrived at new results in this field. But even in this eminently Cartesian text one can find traces of his fascination with the method of discovery of the ancients. The ancients, rather than using algebraic tools, were supposed to have a geometrical analysis that Newton wished to restore. This was a program shared by many in the seventeenth century. He also made it clear that synthesis, or composition, of geometrical problems had to be carried on — contra Descartes — in terms wholly independent of algebraic considerations. The fascination with ancient analysis and synthesis, a better substitute, he strongly opined, for Cartesian common analysis (algebra) and synthesis (the techniques on the construction of equations prescribed by Descartes), prompted Newton to read the seventh book of Pappus’s Collectio (composed in the fourth Century A.D. and printed alongside a Latin translation in 1588). He became convinced that the lost books of Euclid’s Porisms, described incompletely in Pappus’s synopsis, were the heart of the concealed ancient, analytical but entirely geometrical method of discovery.
While Newton's occult and mystical tendencies may have set him apart from some of his fellow philosophers and mathematicians, they did so by means of degree, not by means of kind. Robert Boyle spent time studying Hebrew grammar; Locke wrote commentaries on Pauline epistles, treating them straightforwardly as divine, inspired, and infallible. Like Newton, Descartes had looked to ancient authors for hints about geometry and algebra.

It seems that Newton did not know that Descartes expressed similar views in the “Responsio ad Secundas Obiectiones” in Meditationes de Prima Philosophia (1641).
While Newton saw mathematics as the structuring principle of the universe, he did not follow Descartes in formulating a rationalistic natural philosophy (in the sense of an a priori natural philosophy). Newton retained an empirical side. A central issue in Newton's thought is the effort to harmonize the a priori with knowledge gained from experience. We can see this struggle in one of Newton's most famous phrases, when he wrote that

propositions collected from observation of phenomena should be viewed as accurate or very nearly true until contradicted by other phenomena.
The phrase "very nearly true" is thought-provoking. In a common notion of truth as a binary opposition to falsehood, a proposition is either true or it is not. To be "very nearly true" is to be false. Yet Newton apparently thought that he was expressing something by writing this way. Perhaps this is the a posteriori at war with the a priori in Newton's mind. Frederick Copleston writes:

That Newton attributed to mathematics an indispensable role in natural philosophy is indicated by the very title of his great work, the Mathematical Principles of Natural Philosophy. The great instrument in the demonstrations of natural philosophy is mathematics. And this may suggest that for Newton mathematical physics, proceeding in a purely deductive manner, gives us the key to reality, and that he stands closer to Galileo and Descartes than to English scientists such as Gilbert, Harvey and Boyle. This, however, would be a misconception. It is doubtless right to stress the importance which Newton attached to Mathematics; but one must also emphasize the empiricist aspect of his thought. Galileo and Descartes believed that the structure of the cosmos is mathematical in the sense that by the use of the mathematical method we can discover its secrets. But Newton was unwilling to make any such presupposition.
Newton morphed from being trans-Cartesian to being anti-Cartesian. He grew to see Cartesian notions as undermining his theological views. Scholars have debated whether Descartes, in his personal life, was a faithful and engaged Roman Catholic, or whether he was more nearly a deist who merely paid lip-service to Roman Catholicism. There is some plausible evidence on both sides. But in his philosophical writings, God is present in a necessary but impersonal way; whatever Descartes may have privately believed, his philosophy is nearer deism than orthodox Roman Catholicism. Newton, by contrast, posited a mystical or spiritualist type of natural philosophy, in which he sought interaction at the intersection between theological questions and his questions in mathematics, physics, and philosophy. Newton rejected Anglicanism; this rejection seems to have been sincerely motivated, because it created otherwise unnecessary obstacles to his work at Cambridge. Instead of Anglicanism, he developed his own belief system as he wrote commentaries on the books of the Old Testament and of the New Testament. Niccolo Guicciardini writes:

Newton intertwined this myth of the ancient geometers with his growing anti-Cartesianism. In the 1670s he elaborated a profoundly anti-Cartesian position, motivated also by theological reasons. He began looking to the ancient past in search for a philosophy that would have been closer to divine revelation. The moderns, he was convinced, were defending a corrupt philosophy, especially those who were under Descartes’ spell. Newton’s opposition to Cartesian mathematics was strengthened by his dislike for Cartesian philosophy. Descartes in the Geometrie had proposed algebra as a tool that could supersede the means at the disposal of Euclid and Apollonius. Newton worked on Pappus’s Collectio in order to prove that Descartes was wrong. He claimed that the geometrical analysis of the ancients was superior to the algebraic of the moderns in terms of elegance and simplicity. In this context, Newton developed many results in projective geometry and concerning the organic description of curves. His great success, achieved in a treatise entitled “Solutio Problematis Veterum de Loco Solido” (late 1670s) on the “restoration of the solid loci of the ancients,” was the solution by purely geometrical means of the Pappus four-lines locus. This result, much more than the new analysis of infinite series and fluxions, pleased Newton because it was in line with his philosophical agenda.
Newton was quite clearly opposed to any mechanistic view of the universe, deistic or atheistic, which saw the universe as running purely on physical principles. To which extent Descartes embraced such a view remains somewhat disputable; while Julien Offray de La Mettrie's book Man a Machine probably goes further in that direction than Descartes himself would have gone, it is nonetheless plausible to see that Le Mettrie took his impetus from Descartes. In any case, Newton posited the existence of a Deity who interacts with the universe in tangible ways. Frederick Copleston writes:

He carried on the work which have been developed by men such as Galileo and Descartes, and by giving to the mechanical interpretation of the material cosmos a comprehensive scientific foundation he exercised a vast influence on succeeding generations. It is not necessary to accept the views of those who rejected Newton's theological ideas and who regarded the world as a self-sustaining mechanism in order to recognize his importance.
The writings of the Dutchman Christiaan Huygens interested Newton: Huygens was applying the latest developments in mathematics in the construction of clocks and various astronomical measuring devices. The Dutchman's mixture of interests was similar to the collection of topics on which Newton was working. Niccolo Guicciardini continues:

One should not forget another factor that determined Newton’s option for geometry in the 1670s: the encounter with Huygens’s Horologium Oscillatorium. In his masterpiece, printed in 1673, Huygens had employed proportion theory and ad absurdum limit arguments (method of exhaustion) and had spurned as far as possible the use of equations and infinitesimals (in his private papers he did employ symbolic infinitesimalist tools, but he avoided them in print). Huygens offered an example to Newton of how modern cutting-edge mathematization of natural philosophy could be presented in a form consonant with ancient exemplars. The Lucasian Professor immediately acknowledged the importance of Huygens's work, and one might surmise that his methodological turn of the 1670s — which in part led him to cool his relationship with Collins and avoid print publication of his youthful algebraic researches — was related not only to a reaction against Cartesianism, but also to an attraction toward Huygens’s mathematical style.
While working in solidly a priori fields like pure mathematics, Newton was nonetheless drawn to the empirical observational sciences like astronomy. Thus he grew disenchanted with much about Cartesianism (if he had ever been enchanted). Descartes, by refining his a priori rules of thought, sought to arrive at an a priori natural philosophy. Carl Boyer writes:
In this he hoped, through systematic doubt, to reach clear and distinct ideas from which it would then be possible to deduce innumerably many valid conclusions. This approach to science led him to assume that everything was explainable in terms of matter (or extension) and motion. The entire universe, he postulated, was made up of matter in ceaseless motion in vortices, and all phenomena were to be explained mechanically in terms of forces exerted by contiguous matter. Cartesian science enjoyed a great popularity for almost a century, but it then necessarily gave way to the mathematical reasoning of Newton. Ironically, it was in large part the mathematics of Descartes that later made possible the defeat of Cartesian science.
The similarities and contrasts between Newton and Descartes did not arise in a vacuum. We have already seen the roles of Huygens and Hooke, Boyle and Galileo, Leibniz and others in the development of the era's thought. A group of thinkers categorized under the heading of the "Cambridge Platonists" played a role in transmitting a Cartesian notion about the nature of space into Newton's thought. Frederick Copleston writes:

For example, in his Enchiridion metaphysicum Henry More argued that the Cartesian geometrical interpretation of nature leads us to the idea of absolute space, indestructable, infinite and eternal. These attributes cannot, however, be the attributes of material things. Absolute space must be, therefore, an intelligible reality which is a kind of shadow or symbol of the divine presence and immensity. More was primarily concerned with arguing that the mathematical interpretation of nature, which separated the corporeal from the spiritual, ought logically to lead to the linking of the one to the other; in other words, he was concerned with developing an argumentum ad hominem against Descartes. But his argument appears to have exercised an influence on the Newtonian conception of space.
Thus one of the most distinctively Newtonian doctrines - Newton's peculiar conceptualization of absolute space as independently existing - may have Cartesian roots.

Although it is probable that no philosopher is entirely consistent or coherent, given that all philosophers are human, it may be argued that Newton presents more internal tensions within his thought than some other philosophers. To which extent those tensions can be resolved, and Newton saved from internal contradiction, is a task for wise scholars. Yet it takes only a little charity to see how Newton viewed mathematics and physics, not as a coldly mechanistic belief system of deists and atheists, but as the handiwork of a present and personal deity who inhabits his universe and whose designs have a teleology to them. Niccolo Guicciardini concludes:

Newton encouraged his acolytes to pursue researches in ancient analysis and never missed the opportunity for praising those, such as Huygens, who resisted the prevailing taste for the symbolism of the moderns, the “bunglers in mathematics.” When the polemic with Leibniz exploded, he could deploy his classicizing and anti-Cartesian theses against the German. Thus, Newton’s last mathematical productions, publications, and (often anonymous) polemical pieces were driven by a philosophical agenda difficult to reconcile with his mathematical practice.
Perhaps the story - with its ambiguous interpretations - of Sweden's Queen Christina is a fitting symbol for the careers of Descartes and Newton. While Descartes could seem rather detached from the Roman Catholicism which professed, Christina converted, upon the death of Descartes, to that faith; Descartes was the only Roman Catholic with whom she'd spent significant amounts of time. Her engagement with that faith seems to have been more passion than that of Descartes; she surrendered her throne for it. Likewise, Newton may have engaged more actively with some aspects of Cartesian thought, while rejecting other aspects of it, than Descartes himself did.