To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory - more precisely, to Hilbert’s (...) main tenet. Ironically, already half a century before the births of these programs the alternative organization has been substantially represented by Lobachevsky's theory on parallel lines. Moreover, although each program’s founder recognised the basic features in a partial way only, all together these programs represented just the four possible foundational approaches. (shrink)

In the last times some scholars tried to characterize Einstein’s distinction between ‘constructive’ – i.e. deductive - theories and ‘principle’ theories, the latter ones being preferred by Einstein. Here this distinction is qualified by an accurate inspection on past physical theories. Some previous theories are surely non-deductive theories. By a mutual comparison of them a set of features - mainly the arguing according to non-classical logic - are extracted. They manifest a new ideal model of organising a theory. Einstein’s paper (...) of 1905 on quanta, qualified by him as a ‘principle theory’, is interpreted according to this model of theory. Some unprecedented characteristic features are manifested. At the beginning of the same paper Einstein declared one more dichotomy about the kind of mathematics in theoretical physics. These two dichotomies are recognised as representing the foundations of theoretical physics. With respect to these dichotomies the choices by Einstein in the paper on quanta are the alternative choices to Newton’s ones. This fact gives reason to the ‘revolutionary’ nature that Einstein attributed to his paper. (shrink)

1. Does Hegel’s The Science of Logic (Hegel 2010) have any relation to or relevance for what is now known as ‘the science of logic’? Here a negative answer is as likely to be endorsed by many conte...

James Prescott Joule’s (1818–1889) bicentenary took place in 2018 and commemorated by the IDTC with a Symposium—‘James Joule’s Bicentenary: Scientific and Pedagogical Issues Concerning Energy Conservation’—at the European Society for the History of Science (ESHS & BSHS), 14th–17th September, 2018, in London. This symposium had three main objectives: It aimed specifically to celebrate James Joule’s achievements considering the most recent historiographical works with a particular focus on the principle of conservation of energy; It served the purpose of discussing the scientific (...) and pedagogical issues related to heat, energy and work and how they are presented in textbooks and worked out in classrooms; It also provided discussions on the present situation of teaching and learning science through the use of History of Science, both in K-12 and college level with an emphasis on energy and related concepts. In the following, the Introduction of this Special Issue on Joule is presented. (shrink)

The aim of this paper is twofold: (1) to show the principal aspects of the way in which Newton conceived his mathematical concepts and methods and applied them to rational mechanics in his Principia; (2) to explain how the editors of the Geneva Edition interpreted, clarified, and made accessible to a broader public Newton’s perfect but often elliptic proofs. Following this line of inquiry, we will explain the successes of Newton’s mechanics, but also the problematic aspects of his perfect geometrical (...) methods, more elegant, but less malleable than analytical procedures, of which Newton himself was one of the inventors. Furthermore, we will also consider the way in which Newtonianism was spread before in England and afterwards on continental Europe. In this respect the Geneva Edition plays a fundamental role because of its complete apparatus of notes, and because it appeared only thirteen years after the publication of the third edition of the Principia (1726). Finally, we will also confront some problems connected to the metaphysics of calculus. Therefore, the case of Newton is one of those in which, starting from mathematics applied to physics, it is possible to connect an impressive series of fundamental arguments such as the role of mathematics in science; the comparison between Newton’s geometrical methods and analytical methods; the way in which Newtonianism was spread as well as the philosophical implications of Newton’s mathematical concepts. (shrink)

Although Hegel is generally not known as a philosopher of mathematics, he maintained a deep interest in the history of mathematics, especially in its transformations between antiquity and the modern age. Charles S. Peirce, who was the son of a distinguished mathematician and was involved in developments in mathematics in the second half of the nineteenth century, was critical of what he perceived as Hegel’s lack of mathematical acumen. Nevertheless, he recognized in Hegel’s Science of Logic structural features of his (...) own mathematically informed philosophy. In this article, I look to Hegel’s discussion of magnitude in The Science of Logic, and especially to his conception of the relation between continuous and discrete magnitudes, in order to articulate a solution he might have offered to difficulties encountered by Peirce in his opposition to Cantor’s set-theoretical analysis of the continuum. It is argued that Hegel’s interest in the ancient Platonic/Pythagorean tradition in mathematics provided him with crucial resources in this regard. (shrink)

In recent decades, the development of sciences and technologies had a significant impact in society. This impact has been object of analysis from several standpoints, i.e., scientific, communication, historical and anthropological. Consequently, serious changes were required by the society. One of these has been the emerging relationship science in society and its foundations of applied sciences. A related foundational challenging is the educational process, which was and still is an unlimited challenge for teachers and professors: i.e., levels of understanding, curricula, (...) activing critical engagement, transfer knowledge—and—skills, management, classroom and subsequently pre-service teachers need special teacher education. Taking into account Joule’s bicentenary commemoration in physics and Nature of Science teaching this paper introduces to a level of inquiring within historical and NoS approaches in the pre-service teachers, typically science, technology, engineering and mathematics. In particular, history and historiography of Joule’s physics are dealt with. (shrink)

Generally speaking, virtual displacement or work concerns to a timely idea according to which a motion of a certain body is not the unique possible motion. The process of reducing this motion to a particular magnitude and concept, eventually minimizing as a hypothesis, can be traced back to the Aristotelian school. In the history and philosophy of science one finds various enunciations of the Principle of Virtual Laws and its virtual displacement or work applications, i.e., from Aristotle to Leibniz’s vis (...) viva, from Maupertuis’ least action to Euler and Lagrange with calculus of variations to Lazare Carnot’s mechanics. In this case study, I will demonstrate that a particular approach used by Lazare Carnot is original by explaining within the historical context of rival approaches such as the development of the Principle of virtual Laws. I will also discuss Carnot’s geometric motion as one of the possible but invertible movements applied to virtual displacement as employed in his theories of machines and collisions. I will then go on to explore how the originality of an invertible motion within his mechanical and, in general, mathematical research program permitted Carnot to introduce a new way of structuring a scientific theory and making mechanics, with respect to the Newtonian paradigm, to scholars and his students of the École polytechnique de Paris. (shrink)

We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, (...) for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. (shrink)

In 1932 Kolmogorov created a calculus of problems. This calculus became known to Deleuze through a 1945 paper by Paulette Destouches-Février. In it, he ultimately recognised a deepening of mathematical intuitionism. However, from the beginning, he proceeded to show its limits through a return to the Leibnizian project of Calculemus taken in its metaphysical stance. In the carrying out of this project, which is illustrated through a paradigm borrowed from Spinoza, the formal parallelism between problems, Leibnizian themes and Peircean rhemes (...) provides the key idea. By relying on this parallelism and by spreading the dialectic defined by Lautman with its ‘logical drama’ onto a Platonic perspective, we will attempt to obtain Deleuze's full calculus of problems. In investigating how the same Idea may be in mathematics and in philosophy, we will proceed to show how this calculus of problems qualifies not only as a paradigm of the Idea but also as the apex of transdiciplinarity. (shrink)