"HISTORY OF THE CALCULUS, THE"   Term Paper ID:18921 Click to get this paper

Essay Subject: (Carl Boyer). Reviews work on evolution & philosophy of this mathematical discipline.... More...

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Paper Abstract: (Carl Boyer). Reviews work on evolution & philosophy of this mathematical discipline. Paper Introduction: The purpose of this research is to examine The History of Calculus by Carl B. Boyer. The plan of the research will be to set forth the general ideas in the book, and then to explore details contained in the book that summarize the concepts of calculus that Boyer wants to emphasize. Boyer's The History of the Calculus is put forward as one of the few histories of how the discipline of calculus evolved, apart from an explanation of how to use it mathematically. It is by positioning the ideas of mathematics and philosophy that influenced calculus that Boyer gradually moves toward an explanation of how calculus can actually be used and applied. The background of Boyer's approach appears to be the idea that a richer understanding of how calculus came to "be" in the world of ideas can lead to an understanding of how it can be employed in Text of the Paper: The entire text of the paper is shown below. However, the text is somewhat scrambled. We want to give you as much information as we possibly can about our papers and essays, but we cannot give them away for free. In the text below you will find that while disordered, many of the phrases are essentially intact. From this text you will be able to get a solid sense of the writing style, the concepts addressed, and the sources used in the research paper. What Newton and Leibniz each did in his way was to "organize theviews, methods, and discoveries involved in the infinitesimal analysis intoa new subject characterized by a distinctive method of procedure" (187).This is important because the greatest discovery of calculus seems to havebeen that it represented a process for understanding rather than a once-and-for-all single explanation of every problem and paradox in the universe.The idea of change and integration, and the idea that they are inverse, aretwo key points of insight that these two mathematicians introduce into thehistory of calculus. In particular, the doctrine of latitudeof forms was to have an impact on the development of algebra, geometry, andlater calculus. That is, mathematics determines what conclusions will follow logically from given premises. In the final analysis, the Continental method ofnotation prevailed, in part because it was finally recognized thatinfinity, by its nature, could not be manipulated arithmetically, and thatthe idea or doctrine of limits resurfaced decisively and permanently in theformulation of a specific method of deriving proofs. . Meanwhile, Johann Kepler did work on mathematics as pure abstraction,but more in line with arithmetic as an aspect of mysticism than of adiscipline unto itself. In developing what Boyer calls "the rigorous formulation," the notionof inverseness and ratio appears, just as it had in the work of Newton andLeibniz. The point is, Kepler sought to explain theuniverse in mathematical terms, and to more or less work out the problemsthat would result in explaining everything: "The belief that the universewas an ordered mathematical harmony, so strongly shown in Kepler'sMysterium cosmographicum, was combined with Platonic and Scholasticspeculations on the nature of the infinite, giving him a modification ofArchimedes' mensurational work which was to be a powerful influence inshaping the development of the calculus" (1 7). It remained for mathematicians to discover the nature of thatreality. . Meanwhile,however, Leibniz was seeking a method for finding sums and differences ofinfinitesimals, ultimately finding an inverse proof of Newton'sdifferential operation, leading to the discovery of the fundamental inverseproperty of calculus. Had their work been more closely associated with the geometrical procedures of Archimedes and less bound up with the philosophy of Aristotle, it might have been more fruitful (87). Throughout the period, there is evidence of an increasingappreciation of ratios and relationships that apply to many cases, asopposed to a concern to calculate and get the "right" answer to a specificproblem. . In this regard, Simon Stevin dealt withsuch calculus concepts as infinity and limit in terms of geometric proofsrather than abstract mathematical proofs. This hasimplications for all kinds of study and scholarly discipline. The background of Boyer's approach appears to be the ideathat a richer understanding of how calculus came to "be" in the world ofideas can lead to an understanding of how it can be employed in the realworld of applied and theoretical mathematics. . Although Newton apparently preferred to link his method of fluxions with the idea of a limiting ratio, he so often used infinitesimals for dispatch that we shall find many of his successors later interpreting the fluxions themselves as infinitely small quantities, confusing them with moments (2 -2 1). Arguments with respect to the infinite do not proceed,therefore, as do those concerning finite quantities" (7 ). Boyersums up the significance of work in the period: The almost simultaneous appearance of such rules and formulas indicates that shortly after the middle of the seventeenth century infinitesimal considerations were so widely employed and had developed to such a point that, given a suitable notation, a unifying analytic algorithm was almost bound to follow. Newton first had in mind infinitely small quantities [not spatial measurements!] which are not finite nor yet precisely zero. Around the same period, Galileo proceeded even further than Kepler,concluding that the infinite, infinitesimal, and the continuum wereparadoxes, or more exactly mysteries of the quantified universe. The various experts were trying to findthe most efficient method of recording, or "notating," the differentials.Meanwhile, of course, they were also trying to determine exactly whatLeibniz or Newton had "meant" when they had been discussing infinitesimals,limits, and infinity. Back of any discovery or invention there is invariably to be found an evolutionary development of ideas making its geniture possible. In particular, themeasurement of change or variation in the "natural" order of things becameimportant. . Having defined the derivative in terms of limits, he then expressed the differential in terms of the derivative (275). It allows the thinker to test the thought bymeans of a procedure or process that works out quantitative relationships.The results of the test (or proof) that are derived have to be consistent,but this does not necessarily detract from their creativity. It is made the central concept of the differential calculus, and the expression "differential" is then defined in terms of the derivative. Calculus, of course, is far more complex than these "fundamental"properties imply, but both Newton and Leibniz realized that ratios and notindividual calculations were the important elements of abstraction towardthe infinite or infinitesimals. The purpose of this research is to examine The History of Calculus byCarl B. But as Boyer also points out about this period, the results were found either through verbal arguments or geometrically from the graphical representation of the form, rather than by means of arithmetic considerations based upon the limit concept. This is the basis upon which Boyer asserts, "Howstartlingly apropos, with respect to the development of the calculus, isthe Pythagorean dictum: All is number!" (298; emphasis in original). The agent of transition from theconcrete (computational) or metaphysical contemplation of mathematic to theabstract was that of geometry. Cavalieri and Roberval took theseconcepts one step further in considering infinitesimals and indivisibles.Pascal, Fermat, Huygens, and Descartes had different approaches to thecalculus, but each anticipated the later work of Newton and Leibniz.Pascal connected geometry and pure numbers, and he considered hisconnection of algebra and geometry an "elaboration" of those twodisciplines (152). The entities of mathematics had an ontologicalreality, independent of common sense, and the postulates were discovered byreason alone. The concept of the infinitesimal as putforth by Cauchy was even more subtle, and it was extremely useful becauseit employed a concept whose discovery had been important to the advance ofmathematics in general, that of zero. On the Continent themetaphysical rationalism of Leibniz was neglected by his followers,whofreely attempted to interpret the differentials as actual infinitesimals oreven as zeros,and who criticized Leibniz for his hesitancy in this respect"(224). Rarely--perhaps never--is a single mathematician or scientist entitled to receive the full credit for an "innovation," nor does any one age deserve to be called the "renaissance" of an aspect of culture. Boyer says that Newton's theory sometimes confused latercommentators, but he also suggests that Newtonian theory was purelyabstract in arithmetic instead of geometric terms, which represented amajor advance. Just as important, ultimate definitions of the infinitesimal,limit, and infinite appear, and very much in terms that can be associatedwith ratio and proportion. New York: Dover, 1949.----------------------- 16 In this connection, the concept of fluxions was vital, andit was this concept that really led to differential calculus. "Ghosts of departed quantities" they were fittingly called by the critics of the method in the following century. This is the reasonfor Boyer's discussion of "the period of indecision," in which proponentsof Newton or Leibniz tried to rationalize the rules for calculus method:"In England Newton's lack of clarity and his inconsistency in notation wasfollowed by a confusion of fluxions with moments. Where Newton and Leibniz differed in theirmethod of working out proofs (notation), however, became a point ofdeparture and controversy for subsequent commentators. Boyer cites Aristotle's declaration of twoinfinities--potential and actual--and says that some medievalists, notablyWilliam of Occam, agreed, while others, notably Gregory of Rimini,(correctly) disagreed. Boyer gives a clue asto why none of them actually reached differential and integral calculuswhen he says that they anticipated "portions" of these disciplines (185).Because of the particular method or predisposition of a philosopher, hisconcept of mathematics and the universe developed in a certain way. In other words, in the medieval period, themain philosophical focus was on spiritual and mystical matters rather thanon mathematical abstractions as such. Another aspect that Boyer emphasizes in his conclusion of his studyof the history of calculus is how much it owes to classical thought, eventhough the proofs tried by subsequent mathematicians deviated from puremathematical abstraction. . What is important about this secondreference to the Hindu mathematical system is that it points up thecultural differences between West and East in the approach to abstractproblem-solving. What began in classical timesas a concern for the paradoxes that arose because of the seeming indefiniteaspect of ratios (rather than firm calculations) thus proceeded, in theformal designation of the differential, toward an embrace of paradox ormore exactly a recasting of what is possible to be proved, inferred, ordeduced in mathematics. Other critics, who focused almost exclusively onNewton, took different viewpoints, with the result that in England thedevelopment of differential calculus was retarded. . From this definition, Cauchy could proceed to adefinition of infinity, as a kind of limitless variable, which can increase"beyond any given number" (275). What becomes clear in Boyer'sdiscussion of the contributions of individual commentators in the 18thcentury is that there was progress toward refining the method of proof, nowthat most mathematicians agreed that what was being proved or demonstratedby the calculus was itself valid. This is not to say thatgeometric concepts were not important, but that the mathematicians had notseen beyond geometry into pure abstraction as yet. It might follow from this that mathematics in general and thecalculus in particular are elements of the "art" of the possible. But thesemetaphysical speculations are less interesting mathematically than aninsight by Richard Suiseth (the "Calculator"), who disposed of themetaphysical disputes implied by a spiritualistic approach to mathematics,by saying that "a finite part can have no ratio to an infinite whole. Although this made mathematics independent of naturalscience, it did not give it the postulational freedom it enjoys today"(3 4). Boyer. That is, even when the Hindus encountered a concept thatwould prove useful to the emergence of calculus, such as the concept of theabsolute negative and the irrational roots of quadratic constructions (61),they did not develop the concept because their way of looking atmathematics was more oriented toward using it to do computational tricksthan toward looking to see what mathematical abstractions might revealabout the nature of the universe. What the formal description of the calculus has achieved, accordingto Boyer, is an ability to disregard virtually any element of arelationship that seems irrelevant to the logic of abstract relationships!Neither moralistic philosophy nor a philosophy that seeks for understandingsolely on evidence that can be observed, such as in the physical sciences.This may be the reason that Boyer concludes his study with a discussion ofwhat the calculus and mathematics in general are not, only to say that itis a way of reaching for what is possible in all other areas of philosophyor science: Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. A.L. But Boyer is careful to note that the deduction of theformal calculus was not really the work of Cauchy only, and moreover, thatCauchy, like others, built on the work of commentators who had worked inprevious periods. ButGalileo's insight was to depart from the idea of infinity as an aspect ofmagnitude and "focused attention, as had Plato, upon the infinite asmultiplicity or aggregation" (115). Meanwhile, Boyer notes that the Aristotelian idea of reality, asbeing dependent on the natural world experience, was consistent with themathematics of geometry. The history of calculus, therefore, is a history of a way of thinkingabout quantities, measurements, and ratios that is logical and structured.It is also the history of creative thought that can arise based on (but nottied to) that structure. The fact that they could combine all previousinsights into their own theories represented a great leap forward for thediscipline. In other words, the creative speculations of mathematics that aremade possible by the calculus were implied, but not permanently set inplace by, the idea that abstract number had a reality independent ofobservation. In this regard, Boyer notes the Platonic worldview of realities as important for the formulation of the calculus as"pure" abstraction: "Under this view, the conceptions of the infinite andthe infinitesimal were not excluded, inasmuch as reason was not subject tothe world of sensation. Thisconclusion, he said, would be conceded by the imagination, for the contrarywould imply that any part, when added to the whole, would not change it inmagnitude. The History of the Calculus and Its Conceptual Development. The "anticipation" was for theRenaissance and subsequent periods. It was Rimini who suggested there was no realitycontradiction implicit in the notion of the "actual" infinity. One critic, Berkeley, attacked the logical system of Newton as animperfect method of proof, saying that it was too inductive (creative?) andnot deductive enough. Such doctrines as the latitude of forms, measurement ofintensity, velocity, degree, infinite series, and rates of change came infor analysis during this period. It continued itsdevelopment on the Continent, however. This explains Boyer'shistorical approach, which is really an approach to the history of abstractthinking in the Western world. Because numbers lendthemselves to abstraction, there was a certain tendency toward equatingmathematics with mystical experience. Boyer cites Cauchy's "formal precision" of definition (276), which isvital in understanding the analytic method to be employed in the use of thecalculus. The reason theissue arose was that in mathematics, as in religion, there was a concern todeal with infinity, the continuum, and the divisibility of theinfinitesimal and points. His formulation was precisely that given by Bolzano: Let the function be y = f(x); to the variable x give an increment x i; and form the ratio _ y f(x + i) - f(x) ------ = --------------------- _ x i The limit of this ratio ("when it exists") as i approaches zero he represented by f (x), and he called this the derivative of y with respect to x . The plan of the research will be to set forth the generalideas in the book, and then to explore details contained in the book thatsummarize the concepts of calculus that Boyer wants to emphasize. Thus, forexample, despite his inclination to use mathematics as a way of explaininguniversal truths, Nicholas of Cusa was able to derive a "characteristicquadrature of the circle" (91), which in fact was the,quadrature of allcircles of like kind. Cauchy's work was most important in thisregard. In other words, the infinitesimal could be used as a constant. These offer too great difficulty of conception, so Newton next focused attention on their ratio, which in general is a finite number. Works CitedBoyer, Carl B. The idea of expressing the function of a number (value) in termsof its relationship between variables also refined the conception ofabstract number as an expression of ratios. Instead, he consideredthe concept as an aspect of value: "When the successive values attributedto a variable approach indefinitely a fixed value so as to end by differentfrom it by as little as one wishes, this last is called the limit of allthe others" (272; emphasis added). It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. Why the nature of mathematical abstraction is important tounderstanding the medieval contributions toward calculus becomes clear whenBoyer explains that "Scholastic [medieval] speculations invariably centeredupon the metaphysical question of the reality of indivisibles, rather thanupon the search for a representation which should be consistent with thepremises of mathematics" (67). In such insights, Kepleranticipated the calculus, although his concept of irrational numbers, theinfinite, and the infinitesimal was incomplete. What is most important about the mathematician-philosophers of the 17th century, however, is that their attempts toexplain paradoxes became increasingly sophisticated. . With regard to the concept of limit, Cauchy not only gave a clear-cut definition of the term but also marked off the calculus from classicalgeometry and toward arithmetic. . He follows this with a review ofthe medieval "contributions" to the development of calculus, again withbrief reference to the Hindus. [I]t remained for . The history of the calculus furnishes a remarkably apt illustration of this fact (299). This is how Boyer explains it,quoting Cauchy: "One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero." An infinitesimal was consequently not different from other variables, except in the understanding that it is to take on values converging toward zero as a limit (273). Boyer's The History of the Calculus is put forward as one of the fewhistories of how the discipline of calculus evolved, apart from anexplanation of how to use it mathematically. And from this, there proceeded the ideaof derivative, upon which could be based the notion of the differential. It was during this period, too, thatthere were attempts to explain the natural world in terms of spiritualityrather than "pure" mathematics without spiritual content. The importance of the medieval mathematicians in thehistory of calculus, then, was to set the stage (after the age of "belief")for the application of mathematics to an understanding of the natural,instead of spiritual, universe. . It was during the medieval period that other concepts emerged thatsought to explain what seemed to be mysterious. In the period following the medieval period, which Boyer refers to asthe "century of anticipation," the shift from spiritual to materialconsiderations of mathematics occurred. Leibniz had considered differentials as the fundamental concepts, the differential quotient being defined in terms of these; but Cauchy reversed this relationship. This was true, for example, of thePythagorean adherents. "Differentiation," says Boyer, "is in general thefundamental operation, integration being regarded simply as the inverse ofthis" (2 6). The differential thus represents simply a convenient auxiliary notion permitting the application of the suggestive notation of Leibniz without the confusion between increments and differentials which this symbolism had engendered. In particular, he notes that another greattradition, the Hindu, also used a number system as the basis forphilosophical thought, but that the underlying way of thinking about theuniverse differed markedly from that of the Western of Hellenic traditionfrom which calculus eventually derived. It is by positioning theideas of mathematics and philosophy that influenced calculus that Boyergradually moves toward an explanation of how calculus can actually be usedand applied. The concept of limit, for example, Cauchymore or less considered as irrelevant for geometry. This is in spite of the fact that it wasAristotle who also noted the "possible" or "potential" infinity. From this it follows that the notion of infinity can be embraced onone hand, while specific calculations along a continuum toward infinity canalso be made. Itdid not matter whether the infinitesimal was rational or irrational, or howsmall it was in fact. After an introductory overview of the notion of numbers as an"ultimate" category of human thought, Boyer discusses the earliest knownuses of advanced mathematics in Babylonia, Egypt, and Greece, which formedthe theoretical basis of what would become calculus. The conjunction of mathematics and philosophy, or of mathematics and science, is frequently of great service in suggesting new problems and points of view (3 8). Knowing this ratio, one may now substitute for the infinitesimal quantities forming it any other easily conceived finite magnitudes having the same ratio, such as quantities which are thought of as the velocities or fluxions of those entering into the equation. Newton and Leibniz, to bind all of this work into what represents probably the most effective instrument for scientific investigation that mathematics has ever produced (186). If this paper is not what you are looking for, you can search again: Search for: or Click here to request an essay written just for you.