The article investigates the role of symbolic means of knowledge representation in concept development using the historical example of medieval diagrams of change employed in early modern work on the motion of fall. The parallel cases of Galileo Galilei, Thomas Harriot, and René Descartes and Isaac Beeckman are discussed. It is argued that the similarities concerning the achievements as well as the shortcomings of their respective work on the motion of fall can to a large extent be attributed to their (...) shared use of means of knowledge representation handed down from antiquity and the Middle Ages. While the interpretation of medieval diagrams was unproblematic in the scholastic context from which they arose, in the early modern context, which was characterized by the confluence of natural philosophy and practical mathematics, it became ambiguous. It was the early modern mathematicians’ work within this contradictory framework that brought about a new conceptualization of motion which, in particular, eventually led to an infinitesimal concept of velocity. In this process, the diagrams themselves remained largely unchanged and thus functioned as a catalyst for concept development. (shrink)

As I have read the scholium, it divides into three main parts, not including the introductory paragraph. The first consists of paragraphs one to four in which Newton sets out his characterizations of absolute and relative time, space, place, and motion. Although some justificatory material is included here, notably in paragraph three, the second part is reserved for the business of justifying the characterizations he has presented. The main object is to adduce grounds for believing that the absolute quantities are (...) indeed distinct from their relative measures and are not reducible to them. Paragraph five takes this up for the case of time. Paragraphs eight to twelve endeavor to do this for rest and motion by appealing to their properties, causes and effects. In arguing that absolute motion is not reducible to any particular form of the relative motion of bodies with respect to one another, and thus, as is directly argued in the third argument, must be understood in terms of motionless places, Newton thereby constructs an indirect case that absolute space is indeed something distinct from any relative space. Paragraph thirteen functions as conclusion to this line of inquiry and comments on how, in the light of this, the names of these quantities are to be interpreted in the scriptures. The third and final part consists of paragraph fourteen alone, and addresses the question: given that true motion is motion with respect to absolute space, but the parts of the latter are not perceivable, is it possible for us to know the true motions of individual bodies? Newton illustrates how this may be done from the evidence provided by their apparent motions and the forces which are the causes and effects of true motion. This forms a bridge to the body of the work insofar as the purpose of the Principia, according to Newton, is to show how this, and the converse problem, of inferring true and apparent motions from the forces, can be dealt with.Part II of this paper will appear in the next issue of Studies in History and Philosophy of Science. (shrink)

Abstract.As an undergraduate at Cambridge, Newton entered into his ‘Waste Book’ an assumption that we have named the Equivalence Assumption (The Younger): ‘‘ If a body move progressively in some crooked line [about a center of motion]..., [then this] crooked line may bee conceived to consist of an infinite number of streight lines. Or else in any point of the croked line the motion may bee conceived to be on in the tangent.’’ In this assumption, Newton somewhat imprecisely describes two (...) mathematical models, a ‘‘polygonal limit model’’ and a ‘‘tangent deflected model,’’ for ‘‘one-body motion,’’ that is, for the motion of a ‘‘body in orbit about a fixed center,’’ and then claims that these two models are equivalent. In the first part of this paper, we study the Principia to determine how the elder Newton would more carefully describe the polygonal limit and tangent deflected models. From these more careful descriptions, we then create Equivalence Assumption (The Elder), a precise interpretation of Equivalence Assumption (The Younger) as it might have been restated by Newton, after say 1687. We then review certain portions of the Waste Book and the Principia to make the case that, although Newton never restates nor even alludes to the Equivalence Assumption after his youthful Waste Book entry, still the polygonal limit and tangent deflected models, as well as an unspoken belief in their equivalence, infuse Newton’s work on orbital motion. In particular, we show that the persuasiveness of the argument for the Area Property in Proposition 1 of the Principia depends crucially on the validity of Equivalence Assumption (The Elder). After this case is made, we present the mathematical analysis required to establish the validity of the Equivalence Assumption (The Elder). Finally, to illustrate the fundamental nature of the resulting theorem, the Equivalence Theorem as we call it, we present three significant applications: we use the Equivalence Theorem first to clarify and resolve questions related to Leibniz’s ‘‘polygonal model’’ of one-body motion; then to repair Newton’s argument for the Area Property in Proposition 1; and finally to clarify and resolve questions related to the transition from impulsive to continuous forces in ‘‘De motu’’ and the Principia. (shrink)

The first proposition of the Principia records two fundamental properties of an orbital motion: the Fixed Plane Property (that the orbit lies in a fixed plane) and the Area Property (that the radius sweeps out equal areas in equal times). Taking at the start the traditional view, that by an orbital motion Newton means a centripetal motion – this is a motion ``continually deflected from the tangent toward a fixed center'' – we describe two serious flaws in the Principia's argument (...) for Proposition 1, an argument based on a polygonal impulse approximation. First, the persuasiveness of the argument depends crucially on the validity of the Impulse Assumption: that every centripetal motion can be represented as a limit of polygonal impulse motions. Yet Newton tacitly takes the Impulse Assumption for granted. The resulting gap in the argument for Proposition 1 is serious, for only a nontrivial analysis, involving the careful estimation of accumulating local errors, verifies the Impulse Assumption. Second, Newton's polygonal approximation scheme has an inherent and ultimately fatal disability: it does not establish nor can it be adapted to establish the Fixed Plane Property. Taking then a different view of what Newton means by an orbital motion – namely that an orbital motion is by definition a limit of polygonal impulse motions – we show in this case that polygonal approximation can be used to establish both the fixed plane and area properties without too much trouble, but that Newton's own argument still has flaws. Moreover, a crucial question, haunted by error accumulation and planarity problems, now arises: How plentiful are these differently defined orbital motions? Returning to the traditional view, that Newton's orbital motions are by definition centripetal motions, we go on to give three proofs of the Area Property which Newton ``could have given'' – two using polygonal approximation and a third using curvature – as well as a proof of the Fixed Plane Property which he ``almost could have given.''. (shrink)

A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

François Viète is often regarded as the first modern mathematician on the grounds that he was the first to develop the literal notation, that is, the use of two sorts of letters, one for the unknown and the other for the known parameters of a problem. The fact that he achieved neither a modern conception of quantity nor a modern understanding of curves, both of which are explicit in Descartes’ Geometry, is to be explained on this view “by an incomplete (...) symbolization rather than by any obstacle intrinsic in the system.” Descartes’ Geometry provides only a “clearer expression” of themes already sounded in Viète’s work, one that perfects Viète’s literal calculus and gives it “its modern form”; it merely continues the “‘new’ and ‘pure’ algebra which Viète first established as the ‘general analytic art’.” It can seem, furthermore, that this must be right, that had there been some obstacle intrinsic to Viète’s system that barred the way to a modern conception of quantity and a modern understanding of curves, then Descartes’ Geometry would have had to have taken a very different form than it did. As it was, Descartes had only to improve Viète’s symbolism, free himself of the last vestiges of the ancient view of geometrical and arithmetical objects, and apply the new symbolism to the study of curves in order to achieve what Viète did not but could have. But what, really, is the status of this “could have”; what would it actually have taken for Viète to achieve Descartes’ results in the Geometry? The interest of the question lies in its potential to better our understanding of the nature of modern mathematics. (shrink)

Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and (...) 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods. (shrink)

The first edition of Newton’s Principia opens with a “Praefatio ad Lectorem.” The first lines of this Preface have 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 the Preface that historians have often referred to in connection with their treatments of Newton’s scientific methodology. Roughly in the middle of the Preface, Newton defines the purpose of philosophy (...) as a twofold task: to investigate the forces of the phenomena of nature and, once having established the forces, to demonstrate the remaining phenomena. Newton then introduces a distinction between the first two books, which deal with general propositions, and the third, where the propositions are applied to particular instances of celestial phenomena. From these phenomena, Newton claims, the force of gravity, thanks to which bodies tend toward the Sun, is derived. By assuming this force in mathematical propositions, other motions are deduced: the motions of the planets, the comets, the moon, and the sea. Newton then declares his hope that phenomena relative to small particles will also be explained, as per the celestial ones, thanks to the understanding of attractive forces hitherto unknown to philosophers. The Preface ends with a well-deserved laudatio of Edmond Halley and an apologetic passage concerning certain imperfections in the presentation of advanced subjects, such as the motions of the moon. It is these lines to which historians have given most attention in their various studies. (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

Summary A sketch is given of a way of looking at science. Research is viewed as a complex of cognitive processes with a theoretical and experimental sides. A distinction is made between context of discovery and context of presentation. In the latter paragons of science come into play. From this platform the theory of research of Christian Huygens is examined, in its contemporary situation between Baconian empiricism and Cartesian rationalism, and in connection with Galileo's outlook on method. Huygens' attitude on (...) legitimating the results of his research production is also examined. The paper employs a method of case study, which is also discusses. (shrink)

A sketch is given of a way of looking at science. Research is viewed as a complex of cognitive processes with a theoretical and experimental sides. A distinction is made between context of discovery and context of presentation. In the latter “paragons of science” come into play. From this platform the “theory of research” of Christian Huygens is examined, in its contemporary situation between Baconian empiricism and Cartesian rationalism, and in connection with Galileo's outlook on method. Huygens' attitude on legitimating (...) the results of his research production is also examined. The paper employs a method of case study, which is also discusses. (shrink)

In the preface to the Principia (1687) Newton famously states that “geometry is founded on mechanical practice.” Several commentators have taken this and similar remarks as an indication that Newton was firmly situated in the constructivist tradition of geometry that was prevalent in the seventeenth century. By drawing on a selection of Newton's unpublished texts, I hope to show the faults of such an interpretation. In these texts, Newton not only rejects the constructivism that took its birth in Descartes's Géométrie (...) (1637); he also presents the science of geometry as being more powerful than his Principia remarks may lead us to believe. (shrink)

(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...

Sir Isaac Newton was one of the greatest scientists of all time, a thinker of extraordinary range and creativity who has left enduring legacies in mathematics and the natural sciences. In this volume a team of distinguished contributors examine all the main aspects of Newton's thought, including not only his approach to space, time, mechanics, and universal gravity in his Principia, his research in optics, and his contributions to mathematics, but also his more clandestine investigations into alchemy, theology, and prophecy, (...) which have sometimes been overshadowed by his mathematical and scientific interests. (shrink)

Hobbes and Wallis's "battle of the books" illuminates the intimate relationship between science and crucial seventeenth-century debates over the limits of sovereign power and the existence of God.

Presents Newton's unifying idea of gravitation and explains how he converted physics from a science of explanation into a general mathematical system.

Newton's philosophical views are unique and uniquely difficult to categorise. In the course of a long career from the early 1670s until his death in 1727, he articulated profound responses to Cartesian natural philosophy and to the prevailing mechanical philosophy of his day. Newton as Philosopher presents Newton as an original and sophisticated contributor to natural philosophy, one who engaged with the principal ideas of his most important predecessor, René Descartes, and of his most influential critic, G. W. Leibniz. Unlike (...) Descartes and Leibniz, Newton was systematic and philosophical without presenting a philosophical system, but over the course of his life, he developed a novel picture of nature, our place within it, and its relation to the creator. This rich treatment of his philosophical ideas will be of wide interest to historians of philosophy, science, and ideas. (shrink)

This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)