Leibnizian-Newtonian calculus was a theory that dealt with geometrical objects; the figure continued to play one of the fundamental roles it had played in Greek geometry: it susbstituted a part of reasoning. During the eighteenth century a process of de-geometrization of calculus took place, which consisted in the rejection of the use of diagrams and in considering calculus as an 'intellectual' system where deduction was merely linguistic and mediated. This was achieved by interpreting variables as universal quantities and introducing the (...) notion of function, which replaced the study of curves. However, the emancipation of calculus from its basis in geometry was not comprehensive. In fact, the geometrical properties of curves were attributed de facto to functions and thus eighteenth-century calculus continued implicitly to use principles borrowed from geometry. There was therefore no transition to a purely syntactical theory based on axiomatically introduced terms, a shift which only took place subsequently in modern times. (shrink)

In his writings about hypergeometric functions Gauss succeeded in moving beyond the restricted domain of eighteenth-century functions by changing several basic notions of analysis. He rejected formal methodology and the traditional notions of functions, complex numbers, infinite numbers, integration, and the sum of a series. Indeed, he thought that analysis derived from a few, intuitively given notions by means of other well-defined concepts which were reducible to intuitive ones. Gauss considered functions to be relations between continuous variable quantities while he (...) regarded integration and summation as appropriate operations with limits. He also regarded infinite and infinitesimal numbers as a façon de parler and used inequalities in order to prove the existence of certain limits. He took complex numbers to have the same legitimacy as real quantities. However, Gauss’s continuum was linked to a revised form of the eighteenth-century notion of continuous quantity: it was not reducible to a set of numbers but was immediately given. (shrink)

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's Constraint, (...) that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)

In this paper I illustrate the evolution of series theory from Leibniz and Newton to the first decades of the eighteenth century. Although mathematicians used convergent series to solve geometric problems, they manipulated series by a mere extension of the rules valid for finite series, without considering convergence as a preliminary condition. Further, they conceived of a power series as a result of a process of the expansion of a finite analytical expression and thought that the link between series and (...) analytical expression was not restricted to the interval of convergence. This gave rise to formal aspects that became progressively more evident while the complexity of the theory increased. Asymptotic and recurrence series emerged as the result of the natural evolution of theory, without the difference between these infinite processes with respect to ordinary series being highlighted. The various aspects of dealing with series can be viewed today as derived from different definitions of the sum. However, mathematicians of that time always considered them as different approaches to a single theory that was based on a unique notion of the sum (even though, later in the eighteenth century, they disagree about what this notion was). (shrink)

In this paper I illustrate the evolution of series theory from Leibniz and Newton to the first decades of the eighteenth century. Although mathematicians used convergent series to solve geometric problems, they manipulated series by a mere extension of the rules valid for finite series, without considering convergence as a preliminary condition. Further, they conceived of a power series as a result of a process of the expansion of a finite analytical expression and thought that the link between series and (...) analytical expression was not restricted to the interval of convergence. This gave rise to formal aspects that became progressively more evident while the complexity of the theory increased. Asymptotic and recurrence series emerged as the result of the natural evolution of theory, without the difference between these infinite processes with respect to ordinary series being highlighted. The various aspects of dealing with series can be viewed today as derived from different definitions of the sum. However, mathematicians of that time always considered them as different approaches to a single theory that was based on a unique notion of the sum (even though, later in the eighteenth century, they disagree about what this notion was). (shrink)

Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them.

Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content.

Introduction Première partie – Réflexions sur le développement de la théorie des équations algébriques Section première. Les règles de la méthode Chapitre premier. Le théorème de Lagrange Chapitre II. Le théorème de Gauss Chapitre III. La « méthode générale » d'Abel : preuves « pures » et démonstrations d'impossibilité Chapitre IV. La théorie de Galois Section deuxième – Mathématique universelle Chapitre V. La théorie de Klein Chapitre VI. La théorie de Lie Conclusion. La mathématique universelle Notes Note I. Sur la (...) notion mathématique de l'infini Note II. Sur les constructions géométriques dans les Eléments d'Euclide Note III. Le « principe des relations internes » Bibliographie. (shrink)