The Tangled Origins of the Leibnizian Calculus

The following sections are included:

• Preface

• Acknowledgments

• Contents

There is a conventional interpretation of Leibniz's mathematical achievements, specifically his discovery of the calculus, found in histories of mathematics like Boyer (1959), Burton (1985), or the more technical monographs by Baron (1969), Edwards (1979), or Hofmann (1974a). It goes roughly as follows: Leibniz's contribution was a predictable synthesis–mainly through improvements in notation–of earlier results of many seventeenth-century mathematicians such as Fermat (1601–1665), Cavalieri, Descartes, Barrow, Sluse (1625–1685), and others. Calculus in the form of contemporary investigation of tangent or quadrature problems was "in the air":…

The previous description of mathematical change may be a caricature, but much of it is undoubtedly true. Whether we are Platonists, believing that mathematics discovers a timeless realm of super-sensible abstract truth, or think of mathematics as some sort of "social construction" conforming to the standards of a community of mathematicians, who are themselves affected by external cultural factors, the actual theorems of mathematics seem more usefully characterized as "discoveries" than "inventions" or "constructions," since they are initially unknown and follow (often only after a great deal of hard work and ingenuity) from a more general conceptual structure. We may or may not "invent" this structure, but much mathematical activity consists of the discovery of its consequences. For example, if we cast aside the problem of the ontological status of the system of positive integers, it is a fact that both the Lagrange Four Square Theorem stating that every positive integer can be expressed as the sum of at most four (not necessarily different) square integers¹ and the theorem that every even number is the sum of at most 300,000 primes² are true consequences of that system. Before their discovery we were just as unaware of them as medieval Europeans were of the New World. On a more elementary level once we have discovered (or invented) the concept of the derivative y′ of a function y, then the quotient rule

follows necessarily. In many cases we may also conjecture that certain mathematical statements are true without having discovered whether or not they actually are. In these circumstances our position is analogous to that of Columbus who believed, but did not know, that he might be close to China or that of a scientist conjecturing yet unverified properties of an unfamiliar virus. For instance, we believe that there is an infinity of twin primes (like, (3,5), (5,7), (11, 13), (17, 19), etc.) or that the Goldbach conjecture that every even number greater than or equal to 4 is the sum of only two primes (much better than 300,000!) is true,³ but we could be wrong and in the same position as Columbus concerning the location of the New World. In all these cases and in countless others human wishes, ideology, or social circumstances seem irrelevant. In this sense the actual content of mathematics is certainly "internal," developing out of its past substance, and representing cumulative knowledge. We know far more mathematics today than, say, a century ago, and the same was true in 1911 as compared to 1811…

To get a more intimate familiarity with the geometric values we have been describing and to appreciate their technical impact on infinitesimal problems of interest prior to Leibniz, let us look in some detail at Isaac Barrow, Newton's mentor at Cambridge, who is a perfect mathematical representative of them and in many ways an antithesis to Leibniz. Barrow is not the only exemplar of the classical paradigm. We could have picked several of the mathematicians previously mentioned, including Leibniz's mentor Christiaan Huygens. But Barrow is a particularly illuminating example, both because of his chronological closeness to and purported influence upon Leibniz…

It is a familiar cliche that mathematical talent is exhibited early in life, in most cases evident in childhood or adolescence. If so, Leibniz is an exception. There were few clues in his early life that he would become a mathematician, let alone a mathematical revolutionary. Although his prodigious talents were evident at an early age, they seemed at first to lie in the direction of philosophy, theology, and law rather than mathematics and science…

The opportunity that changed the history of western mathematics was a consequence of a memorandum, the Consilium Aegyptiacum (1671–72) drawn up by Leibniz recommending that the European states undertake a new crusade against the Turks. Anticipating Napoleon, he proposes that France should take the lead by an invasion of Egypt. Leibniz's proposal was part of an effort to divert Louis XIV's destructive ambitions from Holland (and potentially Germany). Boineburg had already vaguely communicated the plan in correspondence with the French foreign minister, Simon Arnauld de Pomponne (1618–1699). Since there seemed to be some interest Leibniz was sent to Paris in March 1672 by Boineburg with the mission of formally presenting the Egyptian project at the French Court. A secondary goal was to secure a considerable amount of money in the form of property rents and pensions which the French owed Boineburg.¹ Evidently Leibniz's memorandum was not taken seriously, since Louis allied with Charles II of England, Sweden, and the Bishopric of Münster had declared war on April 6 and duely launched an invasion of on Holland in May, barely two months after Leibniz's arrival in Paris (as late as September Leibniz was even unable to meet Pomponne).² A further effort is made in November 1672 when Leibniz is joined by Boineburg's son-in-law and nephew of the Archbishop-Elector Melchior Friedrich von Schönborn (1644–1717) who was carrying a proposal for a Peace Conference in Cologne. This too was ignored by the French and the war continued until 1678.³ But at least these diplomatic disappointments allowed Leibniz to spend the next four years in Paris, then the culturally most developed city in Europe, with relatively few demands put on his time by the authorities in Mainz.⁴…

Leibniz himself in a 1703 letter to James Bernoulli and in his unpublished Historia et origo calculi differentialis, written in 1713 as an answer to the Commercium Epistolicum report issued by the Royal Society accusing him of plagiarizing Newton, dates his discovery of the calculus to the 1673-4 period when he first extended Pascal's method of computing the surface area of a sphere to general solids of revolution and then arrived at his method of transmutation. He also mentions work in this period using differences as a tool to sum numerical series as well as a new notation "invented by him at a later date" in which ʃ x and dx respectively represent the partial sums of a series (whose general term is x) and the difference of successive terms.¹ Still later (no precise date is given), he says that he realized that dx and dy represent infinitesimal differences of the abscissa and ordinate, since "the infinitely small lines occurring in diagrams were nothing else but the momentaneous differences of the variable lines."² The difference notation, originally applied to series, could then be used to denote the horizontal and vertical sides of the "characteristic triangle" which Leibniz says he discovered and named in 1673.³…

Besides his work in infinitesimal analysis, Leibniz made repeated investigations in logic, beginning in the Mainz period and continuing throughout his life. Leibniz's approaches to logic and mathematics were quite similar. In both fields he wanted to create a "calculus," centering on the construction of a new notation which by its manipulation would make reasoning and the discovery of new truths nearly automatic. And as we will see in Chapter 8, he considered most of his results in these areas to be mere examples of a vast scheme to find a way of mechanizing all thought, which would both serve as an aid to discovery and be capable of settling religious, political, and scientific controversies. Both Leibniz's mathematics and his logic also share another characteristic. They are simultaneously revolutionary and are still in many respects embedded in a traditional paradigm. His calculus of tangents and quadratures, as we have shown, was not our familiar functional and algebraic calculus which uses his notation. On the contrary, like Descartes he was still heavily influenced by the dominant geometrical orientation of his time; his calculus consequently was one of geometric objects: subtangents, subnormals, and the like, not one of functions. In a similar fashion, his many attempts to construct a "calculus" of logic are mixtures of both revolutionary insight and of Aristotelian traditionalism. There are aspects to Leibniz's logic which were not surpassed until the work of George Boole, William S. Jevons, Gottlob Frege, Schröder, (1841–1902) and Bertrand Russell (1872–1970) in the mid-nineteenth and early twentieth centuries. Indeed, he was so far ahead of his time that some of his work could be mistaken for arguments in a contemporary axiomatic modern algebra text, while others (such as the "alphabet of human thoughts") have not yet been implemented. At the same time he was even more firmly rooted in Aristotelian traditionalism than he was in what we have called the "geometric paradigm." In some ways, in fact, he was less advanced logically than the Stoics or later scholastics such as William of Ockham. This conservatism was probably a consequence of both his educational experience and the philosophy he developed. Leibniz was a superbly trained seventeenth century Central European university graduate. He had thoroughly mastered the university curriculum and its Aristotelian values were deeply imprinted in him. He very much respected Aristotle's logic, and some of his own core logical ideas were shaped by his own metaphysics which had a significant inheritance from Aristotle. Unfortunately, unlike his contributions to mathematics, Leibniz's work in logic had no influence either during his life time or later. He published nothing in the field, and almost all of his writings (as is the case as well with many of his other interests) are rough drafts or fragments and contain various imperfections and inconsistencies. After his death in 1716 his manuscripts were more or less forgotten. Nineteenth century historians such as Karl von Prantl (1820–1888) misinterpreted and denigrated his logical ideas which they did not understand.¹ And by the time his worth was discovered in the late nineteenth century, the bulk of his work been surpassed by contemporary logicians such as Frege and Schröoder.²…

Even in his mature period one has the impression that Leibniz's knowledge of Scholastic and Renaissance philosophy was deeper than his detailed knowledge of either ancient or contemporary mathematics. This was undoubtedly true up to his second visit to London in 1676 and perhaps well beyond, and is certainly understandable. He had read almost every major work in philosophy beginning well before his teens and had also a rigorous academic training in it; by age thirteen or so he could read, he says, the scholastic philosophers with "no ordinary delight."¹ On the other hand, he was twenty six before he began in 1672 to study mathematics seriously under the tutelage of Christiaan Huygens. During the Paris years he rapidly lost his earlier near total ignorance of mathematics. By 1674 or 1675 he certainly had, as he testified, mastered the geometry of Descartes and had additionally read under Huygens direction Pascal's Lettre de A. Dettonville (1659), Gregory St. Vincent's Opus geometricum (1647), James Gregory's Geometriae pars universalis (1668), and Wallis' Arithmetica Infinitorum (1656). Leibniz also speaks of an acquaintance with Cavalieri work and a "close study of Slusius."² All this was enough to allow him to reach the research frontier in infinitesimal analysis. But to repeat some earlier judgements, it is doubtful that he ever had the kind of deep mastery of the classical geometric tradition characteristic of Huygens, Barrow or Newton, all of whom could clinch an argument by a citation from one of the ancient masters. Leibniz was certainly very gifted mathematically and his early discoveries are impressive. But considered only in isolation they are no more impressive than results found by others. We have already pointed out, for example, that his arithmetical quadrature of a circle or conic sections was just one of a family of infinite series/infinite product representations found earlier by mathematicians of the calibre of Gregory, Wallis, or Newton. His quadrature of the cycloid was similar to what had been done by Roberval or Huygens. And we certainly can find in Barrow or Huygens results at least as ingenious as his various transmutation theorems, some of which he may, in fact, have "borrowed" from Barrow.³ And in respect to problem solving or the ability to handle or synthesize involved mathematical arguments, he was probably equal to the more gifted of his contemporaries, but ranked no higher. Mathematically, he is perhaps broader than Newton since he made discoveries in several unrelated areas (some of which we discuss below), but in pure geometric power and mathematical depth he is distinctly below Newton who in the seventeenth century was in a class by himself…

Where did the ideas discussed in the previous chapter come from and what motivated them? The search for perfect philosophical languages, an Ars combinatoria, etc. which would serve as a universal and infallible means of communication and obtaining knowledge did not spring up ex nihilo in the seventeenth century. The political and religious situation of the period was almost certainly a potent psychological factor in stimulating it. Thinkers like Wilkins, Dalgarno, or Leibniz still embodied the medieval heritage of rationalism. They were completely untouched both by the modern awareness of the historical mutability of human thought or by the postmodern sense of relativism which it eventually entailed. They felt that human reason properly directed ought to be able to settle any disputed question whether in it was in religion, philosophy, or politics, and to which there was only one correct answer. Everywhere, however, in the mid-seventeenth century there was intellectual strife, especially in religion. Men and women could die horribly (by fire) over esoteric points of religious dogma such as predestination or the nature of the Eucharist. Europe had just ended the Thirty Years War, the most destructive conflict (even in comparison with the Napoleonic wars) until the World Wars of the twentieth century, and Britain its bloody Civil War. Yet there were sincere intellectuals arguing every side of these conflicts…

We have seen that Leibniz was unable to obtain a permanent position in Paris after the death of his employer the Elector of Mainz Johann Philipp von Schönborn, and so for want of other options had to enter the service of the Duke of Hanover Johann Friedrich of Brunswick-Lünberg and to accept the Duke's request that he reside in Hanover. Until his death in 1716 he was the ducal librarian employed in an extraordinary number scientific, genealogical, political, and historical researches, most of which have long been forgotten. At the same time he was able to pursue a vast correspondence with practically every major (and minor) intellectual figure in Europe, refine his calculus, formulate dynamical theories from a point of view quite different from Newton's, and develop the intricate philosophy of monads for which he is primarily remembered today. Although he was soon recognized as a European philosopher and mathematician of the first rank, we shall see that this was accompanied by a gradual loss of status and increasing neglect at home…

One of our major theses is that Leibniz's calculus combined with Descartes analytic geometry represented the beginnings of a mathematical analog of a Kuhnian "paradigm shift," a phrase which has become a cliché not only in the history of science but in journalism, corporate management theory, or automobile advertisements (one wonders what its inventor, a rigorously trained Harvard physicist, would have made of it all). But as we have argued in Chapters 1 and 2 the analogy with scientific change goes only so far. Rejected scientific theory is viewed as false. Its claims are no longer seen to correspond to the way the world is. Despite the assertions by sociological proponents of the "Strong Programme" or various "postmodern" theorists of science, this situation has been rarely the case in mathematics. The conclusions of Viète or Apollonius are as valid today as they were when they were made. On the other hand, mathematics can certainly abandon a previously dominant "thought style" or subject matter and go off in new directions in such a profound way that the previous work is essentially forgotten or even made inaccessible, and this was the fate of seventeenth century geometry…

The following sections are included:

• Appendix A A Transmutation Theorem of Leibniz

• Appendix B Leibniz's Series Quadrature of a Conic

• Appendix C Syllogistic Logic

• Appendix D The Vis Viva Dispute

• Appendix E Some Applications of Conics, Mechanical Curves and Neusis in Greek Geometry

• Appendix F Infinitesimals

• A Note on the Author

• Bibliography

• Index