Isaac Newton on Mathematical Certainty and Method

Isaac Newton on Mathematical Certainty and Method

Niccolò Guicciardini
Copyright Date: 2009
Published by: MIT Press
Pages: 448
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  • Book Info
    Isaac Newton on Mathematical Certainty and Method
    Book Description:

    Historians of mathematics have devoted considerable attention to Isaac Newton's work on algebra, series, fluxions, quadratures, and geometry. In Isaac Newton on Mathematical Certainty and Method, Niccolò Guicciardini examines a critical aspect of Newton's work that has not been tightly connected to Newton's actual practice: his philosophy of mathematics. Newton aimed to inject certainty into natural philosophy by deploying mathematical reasoning (titling his main work The Mathematical Principles of Natural Philosophy most probably to highlight a stark contrast to Descartes's Principles of Philosophy). To that end he paid concerted attention to method, particularly in relation to the issue of certainty, participating in contemporary debates on the subject and elaborating his own answers. Guicciardini shows how Newton carefully positioned himself against two giants in the "common" and "new" analysis, Descartes and Leibniz. Although his work was in many ways disconnected from the traditions of Greek geometry, Newton portrayed himself as antiquity's legitimate heir, thereby distancing himself from the moderns. Guicciardini reconstructs Newton's own method by extracting it from his concrete practice and not solely by examining his broader statements about such matters. He examines the full range of Newton's works, from his early treatises on series and fluxions to the late writings, which were produced in direct opposition to Leibniz. The complex interactions between Newton's understanding of method and his mathematical work then reveal themselves through Guicciardini's careful analysis of selected examples. Isaac Newton on Mathematical Certainty and Method uncovers what mathematics was for Newton, and what being a mathematician meant to him.

    eISBN: 978-0-262-25886-9
    Subjects: Mathematics, History of Science & Technology

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-xii)
  3. Preface
    (pp. xiii-xx)
  4. Abbreviations and Conventions
    (pp. xxi-xxvi)
  5. I Preliminaries
    • 1 Newton on Mathematical Method: A Survey
      (pp. 3-18)

      Mathematics played a prominent role in Newton’s intellectual career. This was not, of course, his only concern. A polymath and polyhistor, Newton devoted years of intense research to the reading of the Books of Nature and Scripture, deploying the tools of the accomplished “chymist” (at the furnace and at the desk), instrument maker (he made his own instruments, among them the first reflecting telescope), experimentalist, astronomer, biblical interpreter, and chronologist. In all these fields mathematics entered as one of the most powerful and reliable tools for prediction and problem solving, and as the language that guaranteed accuracy and certainty of...

    • 2 Newton on Certainty in Optical Lectures
      (pp. 19-30)

      The appointment to the Lucasian Chair (1669) led Newton to ponder over his role as a mathematician. Now he was not just a young creative protégé of Barrow but rather a professor who would soon address the Royal Society with new theories concerning the nature of light. That Newton began investigating the role of mathematics in natural philosophy is evident in his first set of lectures on optics, which he supposedly delivered between January 1670 and the end of Michaelmas term in 1672. As Lucasian Professor, Newton had to deliver one lecture each week during term and to deposit at...

    • 3 Descartes on Method and Certainty in the Géométrie
      (pp. 31-58)

      Few works in the history of mathematics have been more influential than Descartes’Géométrie(1637). The canon defined in this revolutionary essay was to dominate the scene for many generations, and its influence on the young Newton cannot be overestimated. Indeed, most of Newton’s mathematical work can be understood as a development of and a response to theGéométrie, which he came to know in its Latin translation (1659–1661) due to Frans van Schooten. It is thus essential to recall some salient aspects of this pivotal text.¹

      The difficulties that one encounters in historically evaluating theGéométrieare, in...

  6. II Against Cartesian Analysis and Synthesis
    • [II Introduction]
      (pp. 59-60)

      Just after the creative burst of hisanni mirabiles, when the tumultuous experience of discovery left very little energy for considerations on exactness and rigor, Newton began posing questions concerning mathematical method, and he did so very seriously. Part II considers a time span from the late 1660s to the early 1680s. This period was a turning point in Newton’s intellectual career. His new status as Lucasian Professor (1669) required an approach different from the unsystematic forays into new territory permitted the young protégé of Barrow. The encounter in 1673 with Huygens’sHorologium Oscillatoriumwas probably another factor that induced...

    • 4 Against Descartes on Determinate Problems
      (pp. 61-78)

      Sometime between the autumn of 1683 and early winter of 1684, Newton, according to the statutes of the Lucasian Chair, deposited hisLucasian Lectures on Algebra.¹ The lectures bear dates from 1673 to 1683, but these were added in retrospect, and it is highly unlikely that they were ever delivered to Cambridge students. From several points of view, and notwithstanding Newton’s professed anti-Cartesianism, these lectures can be described as a fulfillment of Descartes’ program because algebra is here extensively presented as the tool to be used in the analysis (or resolution) of problems. Indeed, the lectures were printed in 1707...

    • 5 Against Descartes on Indeterminate Problems
      (pp. 79-108)

      This chapter considers Newton’s criticisms of the Cartesian approach to indeterminate problems. These are problems that, when reduced to algebraic symbolism, typically lead to a polynomial equation in two unknownsxandy. Their solution is a curve whose points with coordinates (x,y) in a Cartesian coordinate system satisfy the equation. Of course, equations in more than two unknowns can also occur. Recall that the process that leads from the statement of the problem to the equation is the analysis (or resolution) of the problem, and the process that leads from the equation to the construction of the solution...

    • 6 Beyond the Cartesian Canon: The Enumeration of Cubics
      (pp. 109-136)

      Newton’s first attempts to enumerate cubic curves date from the late 1660s or early 1670s.¹ In these early works he was able to reduce, via a change of coordinate axes, the general form of a third-degree polynomial to four cases.² He reconsidered the classification of cubic curves in the late 1670s,³ reaching most of the results that he later, in 1695, systematized in a slim treatise that was to appear in 1704 as an appendix to theOpticksunder the titleEnumeratio Linearum Tertii Ordinis.⁴

      It is interesting first to consider the text of the printedEnumeratioand then step...

  7. III New Analysis and the Synthetic Method
    • 7 The Method of Series
      (pp. 139-168)

      One of the key elements of Newton’s new analysis is the use of infinite series, or infinite equations. The binomial theorem that Newton formulated in the winter of 1664–1665 is only the first step in the new analytical method that allowed him to handle mechanical curves and calculate the areas of curvilinear figures. These techniques are based on the kind of “inductive” generalizations that Newton found in the work of Wallis, most notably inArithmetica Infinitorum(1656).¹ I turn to some salient features of this work before dealing with the method of series that Newton codified inDe Analysi...

    • 8 The Analytical Method of Fluxions
      (pp. 169-212)

      Newton’s first attempts to codify the method of fluxions date from the October 1666 tract.¹ This chapter, however, deals with the analytical version that Newton fully developed inDe Methodis(1671) and inDe Quadratura(1691–1692).² The method is divided into a direct and an inverse part. Newton considered the techniques of the direct method as having been brought to perfection in his 1671 treatise. After 1671 he sought both to improve the algorithm of the inverse method and reach a better conceptual foundation for the direct method. Newton kept on working on these issues until the early 1690s,...

    • 9 The Synthetic Method of Fluxions
      (pp. 213-232)

      The procedures considered in chapter 8 are analytical. But recall that, from Newton’s point of view, analysis (resolution) did not provide asolution. After the resolutive, analytical stage, a geometrical construction or synthetic stage must follow. Thus, Newton developed a synthetic method of fluxions.

      At the beginning ofDe Methodis, Newton somewhat parenthetically mentioned the need to provide a synthetic demonstration “from proper foundations” for the inverse method of fluxions. In a section devoted to the inverse problem (Problem 2), he wrote, We have at last done with the problem [Problem 2] but its demonstration still remains. Not (in such...

  8. IV Natural Philosophy
    • 10 The Principia
      (pp. 235-258)

      Newton’s earliest studies on the laws of motions and gravity took place in late 1664 and early 1665.¹ As in the case of mathematics, his starting point was Descartes. Newton commented upon Part 2 of Descartes’Principia Philosophiae(1644) with particular insight. It is believed that the title of Newton’smagnum opuswas conceived as a criticism of the French philosopher, whose work, Newton thought, lacked adequate mathematical principles. From Descartes, Newton learned about the law of inertia, which was to become the first axiom or law of motion of thePrincipia: a body is at rest or moves in...

    • 11 Hidden Common Analysis
      (pp. 259-266)

      This chapter considers how Newton employed algebraic equations in thePrincipia. He did employ algebra, which for several purposes proved to be a useful analytical tool. But he did not print the Cartesian (or common) analysis but rather the geometrical synthesis, that is, thecompositiobut not theresolutio. As explained in chapter 3, the Cartesian canon of problem solving required that, after having reduced a problem to an algebraic equation, one had to produce a geometrical construction followed by a geometrical demonstration that the construction solved the problem.

      The use of algebraic equations was well known to even modestly...

    • 12 Hidden New Analysis
      (pp. 267-290)

      Proposition 30, Book 1, tackles a rather elementary problem, one that can be resolved via common analysis. Is there any trace of the use of new analysis in thePrincipia? Indeed, there are some occurrences of application of the method of series and fluxions.¹ In this chapter I give two examples from which it will be clear that Newton deployed rather advanced quadrature techniques. Corollary 3, Proposition 41, Book 1, and Corollary 2, Proposition 91, Book 1, are analyzed in some detail.

      As mentioned in section 10.2.7, point 3a, there are a number of propositions that begin with the statement...

  9. V Ancients and Moderns
    • 13 Geometry and Mechanics
      (pp. 293-308)

      The first edition of Newton’sPrincipiaopens with a “Praefatio ad Lectorem.”¹ The first lines of thisPrefacehave received scant attention from historians, even though they contain the very first words addressed to the reader of one of the greatest classics of science. Instead, it is the second half of thePrefaceto which historians have often referred in connection with their treatments of Newton’s scientific methodology.

      Roughly in the middle of thePreface, Newton defined the purpose of philosophy as a twofold task: to investigate the forces of the phenomena of nature and, having established the forces, to...

    • 14 Analysis and Synthesis
      (pp. 309-328)

      In the 1690s, Newton researched extensively on geometry. He pursued projects of restoration of the analysis of the ancients that had already occupied him in the 1670s in his work on the Pappus problem. As we mentioned in chapter 5, Newton became convinced that the ancients’ method of analysis was related to the organic construction of curves and to the study of projective transmutations between curves. In fact, in his manuscripts on geometrical method he often moved from a Pappian characterization of the method of analysis to a treatment of projective transformations.¹

      Newton’s writings on geometry in the 1690s culminated...

  10. VI Against Leibniz
    • [VI Introduction]
      (pp. 329-330)

      When Newton confronted Leibniz in the dispute over priority, he was concerned with building up forensic and historical documentation, most notablyCommercium Epistolicum(1713) and its anonymous “Account” (1715), whose purpose was to prove Leibniz’s plagiarism. Many historians have amassed evidence about Newton’s obsessive approach to the priority dispute, his lack of fairness, and his egotism. It is also clear that Leibniz had to be opposed for a series of very solid reasons having to do with philosophical and even political issues. The German, who was employed by the Hanover family, after the accession of George I to the throne...

    • 15 The Quarrel with Leibniz: A Brief Overview
      (pp. 331-338)

      The controversy between Newton and Leibniz has been studied in detail, most notably by Rupert Hall, to whose bookPhilosophers at WarI am deeply indebted. In this chapter I give an overview of the main stages of the controversy.

      Newton formulated his method of series and fluxions between 1664 and 1666. He continued working and refining the method, obtaining new results and new versions until the 1690s. He also let it circulate in manuscript form, since he was proud of the results he had achieved, but at he first rejected the idea of printing it. The reasons that lay...

    • 16 Scribal Publication, 1672–1699
      (pp. 339-364)

      It is well known that Newton, prior to the printing of his mathematical works in the eighteenth century, disseminated knowledge about his mathematical discoveries through correspondence with other mathematicians or via intermediaries such as John Collins and Henry Oldenburg. Correspondence was one of the main vehicles of publication for seventeenth-century mathematicians. However, in his correspondence (§16.3) Newton disclosed only a fraction of his mathematical output; many important results, especially details about proof methods, remained buried in his manuscripts. This aspect of Newton’s policy of publication has been extensively researched, especially in studies focused on the priority dispute with Leibniz. Many...

    • 17 Fluxions in Print, 1700–1715
      (pp. 365-384)

      Newton’s policy of controlled scribal publication was no longer tenable at the turn of the seventeenth century. In 1699, Fatio had bluntly posed the question of priority in hisLineae Brevissimi Descensus Investigatio Geometrica, noting that on the Continent the calculus was unjustly attributed to Leibniz. As a privileged acolyte of Newton he informed his readers that his great mentor had devised an equivalent algorithm well before the publication of Leibniz’s “Nova Methodus.” In 1695, Wallis had complained to Newton,

      [Y] our Notions (ofFluxions) pass there with great applause, by the name ofLeibniz’s Calculus Differentialis. . ....

  11. Conclusion
    (pp. 385-388)

    After this detailed analysis of Newton’s writings on mathematical method it is time to briefly evaluate what has been achieved. Does Newton emerge as a creative and innovative philosopher of mathematics? Clearly, the answer is no. From this viewpoint, he does not compare with Descartes and Leibniz. His methodological views were framed in terms that are commonplace in the history of seventeenth-century mathematics. Newton’s conceptions on analysis and synthesis, or on the merits of geometry over algebra, were shared by many of his contemporaries, including Hobbes, Fermat, Huygens, and Barrow. Descartes and Leibniz, would have subscribed to many of Newton’s...

  12. A Brief Chronology of Newton’s Mathematical Work
    (pp. 389-390)
  13. References
    (pp. 391-412)
  14. Index
    (pp. 413-422)