Peirce e dedekind: La definizione di insieme finito

Starting from Peirce's repeated claims of priority with respect to Dedekind's definition of finite set [R. Dedekin, Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888), Definizione 64], this paper traces the history of Peirce's definition and its role in his research on the foundations of arithmetic. This brings to light some remarkable and neglected achievements of Peirce in this field. It also shows that his priority claims are unjustified, although understandable in terms of his desire for acknowledgment of his pioneering work on the foundations of arithmetic.     

The Method of Scientific Discovery in Peirce’s Philosophy: Deduction, Induction, and Abduction

In this paper we will show Peirce’s distinction between deduction, induction and abduction. The aim of the paper is to show how Peirce changed his views on the subject, from an understanding of deduction, induction and hypotheses as types of reasoning to understanding them as stages of inquiry very tightly connected. In order to get a better understanding of Peirce’s originality on this, we show Peirce’s distinctions between qualitative and quantitative induction and between theorematical and corollarial deduction, passing then to the distinction between mathematics and logic. In the end, we propose a sketch of a comparison between Peirce and Whitehead concerning the two thinkers’ view of mathematics, hoping that this could point to further inquiries.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

Mathematical methodology in the thought of charles S,Peirce

It is fitting that the celebration of Peirce's New Elements of Mathematics should be taking place in New York City, where Peirce was often to be found attending mathematical meetings at Columbia University and where he consulted the resources of the old Astor Library for the production of many of his writings. This paper considers Peirce—a lifelong student of logic—as he examined scientific and mathematical methodology on all levels, in ages past as well as in the then-contemporary literature. Peirce hoped to create an exact philosophy by applying the ideas of modern mathematical exactitude. He developed a semiotic pattern of mathematical procedure with which to test validity in all areas of investigation.     

Carolyn Eisele's place in Peirce studies

Carolyn Eisele's unique, ongoing career as a scholar is sketched, and the importance of her contributions to Peirce Studies and other fields is emphasized. The essay concludes with a series of suggestions about how to interpret Peirce's works based on themes related to the pioneering efforts of Dr. Eisele.     

Peirce's place in mathematics

This article compares treatments of the infinite, of continuity and definitions of real numbers produced by the German mathematician Georg Cantor and Richard Dedekind in the late 19th century with similar interests developed at virtually the same time by the American mathematician/philosopher C. S. Peirce. Peirce was led, not by the internal concerns of mathematics which had motivated Cantor and Dedekind, but by research he undertook in logic, to investigate orders of infinite sets (multitudes, in his terminology), and to introduce the related concept of infinitesimals. His arguments in support of the mathematical and logical validity of infinitesimals (which were rejected by such eminent mathematicians as Cantor, Peano, and Russell at the turn of the century) are considered. Attention is also given to the connections between Peirce's mathematics, his philosophy, and especially his interest in continuity as it was related to his Pragmatism.     

Johann Heinrich Lambert,mathematician and scientist, 1728 – 1777

1977 is the two hundredth anniversary of the death of Johann Heinrich Lambert, a little known but nonetheless intriguing figure in 18th century science. In the general histories of science and mathematics Lambert's contributions are often described piecemeal, with each discovery and invention usually divorced both from the method by which he arrived at it and from the totality of his intellectual endeavour. To the student of optics he is remembered for his cosine law in photometry, to the astronomer for his work on comets, to the meteorologist for his design of a gut hygrometer, and to the mathematician for his work on non-Euclidean geometry and his demonstration of the irrationality of π and e. There is no doubt that each of these contributions had a definite importance of its own; but it is not the aim of the present article to enumerate in this way the high points of Lambert's scientific and mathematical work, rather to describe it for once as a unified whole, and to relate it to the contemporary intellectual outlook.     

Boscovich's geometrical principle of continuity,and the “mysteries of the infinity”

In this paper we give a detailed account of Boscovich's geometrical principle of continuity. We also compare his ideas with those of his forerunners and successors, in order to cast some light on his possible sources of inspiration and to underline the elements of novelty in his approach to the subject.     

Descartes and the birth of analytic geometry

The traditional thesis that analytic geometry evolved from the concepts of axes of reference, co-ordinates, and loci, is rejected. The origins of this science are re-defined in terms of Egyptian, Greek, Babylonian, and Arabic influences merging in Vieta's Isagoge in artem analyticam (1591) and culminating in a work of his pupil Ghetaldi published posthumously in 1630. Descartes' Vera mathesis, conceived over a decade earlier, served to revive and strengthen the important link with logic and thereby to extend the field of application of this analytic method to the corporeal and moral worlds.     

A two-stage stochastic model for labour wastage

A stochastic model for labour wastage is presented which accounts for the employee's variability on the propensity to leave through dependence on both his length of service and his tenure-in-current-state. The basic assumption underlying the model is that the employee's personal characteristics, the job characteristics and the external labour market conditions are stochastic over time, therefore affecting his decision to stay or leave the company. It is shown that the model provides a good fit to a variety of observed leaving patterns for several companies reported in the literature, explains the relationships among a number of important occupational variables, and is useful for planning purposes in predicting future developments.     

Neo-Kantian foundations of geometry in the German Romantic period

While mathematics received relatively little attention in the idealistic systems of most of the German Romantics, it served as the foundation in the thought of the Neo-Kantian philosopher/mathematician Jakob Friedrich Fries (1773–1843). It fell to Fries to work out in detail the implications of Kant's declaration that all mathematical knowledge was synthetic a priori. In the process Fries called for a new science of the philosophy of mathematics, which he worked out in greatest detail in his Mathematische Naturphilosophie of 1822. In this work he analyzed the foundations of geometry with an eye to clearing up the historical controversy over Euclid's theory of parallels. Contrary to what might be expected, Fries' Kantian perspective provoked rather than inhibited a reexamination of Euclid's axioms. Fries' attempt to make explicit through axioms what was being implicitly assumed by Euclid while at the same time wishing to eliminate unnecessary axioms belies the claim that there was no concern to improve Euclid prior to the discovery of non-Euclidean geometry. Fries' work therefore serves as an important historical example of the difficulties facing those who wanted to provide geometry with a logically secure foundation in the era prior to the published work of Gauss, Bolyai, and others.     

Cauchy and the infinitely small

We consider Cauchy's use of the infinitely small in his textbooks. He never examined fully his concept of variables with limit zero, and he sometimes argued as if he were using actual infinitesimals. Occasionally he adopted an epsilon-delta approach. The author argues that historical evaluations of mathematical analysis may and should be made in the light of both standard and non-standard analysis. From this point of view, Cauchy's move toward founding analysis entirely on the standard real number system does not seem to have been inevitable. Some historical observations by the founder of non-standard analysis, Abraham Robinson, are extended, and in one case contested. It is shown that some of Cauchy's alleged errors are explained if he is admitted to have been thinking of actual infinitesimals and infinitely large integers. Cauchy's definitions of differential in his textbooks are examined, and the author shows that the earlier of his two definitions of total differential works well, but the later does not.     

A remark on He’s variational iteration technique for solving nonlinear equations

The variational iteration method (VIM) can be usefully applied for solving many linear and nonlinear scientific and engineering problems. In this note we show that He’s approach for solving nonlinear equations, arising from the VIM, is, actually, Schröder’s method presented in his classical work from 1870.     

Cartan,Schouten and the search for connection

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by Élie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925–1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions.     

E.W. Chittenden and the early history of general topology

E.W. Chittenden's work and its influence on the early history of general topology are examined. Particular attention is given to his work in metrization theory and its role in the background of the Aleksandrov-Uryson Metrization Theorem. A recounting of Professor Chittenden's career, a list of his students and his publications and a chronology in the early history of General Topology are also included.     

The astronomical work of Carl Friedrich Gauss (1777–1855)

Gauss's interest in astronomy dates from his student-days in Göttingen, and was stimulated by his reading of Franz Xavier von Zach's Monatliche Correspondenz… where he first read about Giuseppe Piazzi's discovery of the minor planet Ceres on 1 January 1801. He quickly produced a theory of orbital motion which enabled that faint star-like object to be rediscovered by von Zach and others after it emerged from the rays of the Sun. Von Zach continued to supply him with the observations of contemporary European astronomers from which he was able to improve his theory to such an extent that he could detect the effects of planetary perturbations in distorting the orbit from an elliptical form. To cope with the complexities which these introduced into the calculations of Ceres and more especially the other minor planet Pallas, discovered by Wilhelm Olbers in 1802, Gauss developed a new and more rigorous numerical approach by making use of his mathematical theory of interpolation and his method of least-squares analysis, which was embodied in his famous Theoria motus of 1809. His laborious researches on the theory of Pallas's motion, in which he enlisted the help of several former students, provided the framework of a new mathematical formulation of the problem whose solution can now be easily effected thanks to modern computational techniques.Up to the time of his appointment as Director of the Göttingen Observatory in 1807, Gauss had little opportunity for engaging himself in practical astronomical work. His first systematic observations were concerned with re-establishing the latitude of of that observatory, which had been well-determined by Tobias Mayer more than fifty years earlier. However, he found a small but not negligible discrepancy between results obtained independently from stellar and solar observations, as well as irregularities among later measurements of polar altitudes (made at the new observatory completed in 1816), which he was never able to explain, despite repeated attempts to do so using different instruments and observational techniques. Similar anomalies were also detected by a number of other astronomers at around this time. These may have been associated--at any rate, partially--with the phenomenon identified later in the century as a “variation of latitude” due to minor periodic fluctuations in the Earth's axis of rotation produced by meteorological and geological factors.