In the seventeenth and eighteenth centuries, mathematicians and physical philosophers managed to study, via mathematics, various physical systems of the sublunar world through idealized and simplified models of these systems, constructed with the help of geometry. By analyzing these models, they were able to formulate new concepts, laws and theories of physics and then through models again, to apply these concepts and theories to new physical phenomena and check the results by means of experiment. Students’ difficulties with the mathematics of (...) high school physics are well known. Science education research attributes them to inadequately deep understanding of mathematics and mainly to inadequate understanding of the meaning of symbolic mathematical expressions. There seem to be, however, more causes of these difficulties. One of them, not independent from the previous ones, is the complex meaning of the algebraic concepts used in school physics, as well as the complexities added by physics itself. Another source of difficulties is that the theories and laws of physics are often applied, via mathematics, to simplified, and idealized physical models of the world and not to the world itself. This concerns not only the applications of basic theories but also all authentic end-of-the-chapter problems. Hence, students have to understand and participate in a complex interplay between physics concepts and theories, physical and mathematical models, and the real world, often without being aware that they are working with models and not directly with the real world. (shrink)

Modern philosophy is often seen as characterized by a shift of focus from the things themselves to our knowledge of them, i.e., by a turn to the subject and subjectivity. The philosophy of Immanuel Kant is seen as the site of the emergence of the idea of a subject that constitutes the object of knowledge, and thus plays a central role in this narrative. This study examines Kant’s theory of knowledge at the intersection between the history of science and the (...) history of the modern subject, on the one hand, and in the tension between modern experimental and mathematical science and more traditional Aristotelian conceptions of epistemic perfection, on the other. The dissertation consists of four chapters. In the first chapter, I examine Kant’s concept of experience, and its relation both to Early Modern experimentalism and to the Wolffian tradition. In the second chapter, I argue that Kant adheres to a broadly Aristotelian conception of epistemic perfection – the ideal of understanding – but transforms this ideal into the self-understanding of reason, where reason can only have insight into the products of its own activity. In the third chapter, I use Kant’s conception of space and time to exemplify such products of reason, and argue that, for Kant, space and time are constructively generated representations that function as principles for ordering empirical knowledge. In the fourth and final chapter, I examine Kant’s conception of the subject, and situate it in relation to both the long history of the modern subject and German Enlightenment philosophy. Whereas the modern philosophical conception of the subject is usually taken to combine an ‘I’ functioning as the subject to which mental acts are attributed and an ‘I’ that has the ability to immediately perceive itself as the subject of these acts, I argue that Kant reconceives this relation between the ‘I’ and its acts as a purely intellectual self-relation. The unity of the ‘I’ is not a perceived unity, but a unity brought about by the intellect. (shrink)

An oft-cited truism about the emergence of a new, ‘mechanistic’ approach to natural philosophy in the seventeenth century is that it was inspired by analogy to the workings of clockwork. In Authori...

The mathematical nature of modern science is an outcome of a contingent historical process, whose most critical stages occurred in the seventeenth century. ‘The mathematization of nature’ (Koyré 1957 , From the closed world to the infinite universe , 5) is commonly hailed as the great achievement of the ‘scientific revolution’, but for the agents affecting this development it was not a clear insight into the structure of the universe or into the proper way of studying it. Rather, it was (...) a deliberate project of great intellectual promise, but fraught with excruciating technical challenges and unsettling epistemological conundrums. These required a radical change in the relations between mathematics, order and physical phenomena and the development of new practices of tracing and analyzing motion. This essay presents a series of discrete moments in this process. For mediaeval and Renaissance philosophers, mathematicians and painters, physical motion was the paradigm of change, hence of disorder, and ipso facto available to mathematical analysis only as idealized abstraction. Kepler and Galileo boldly reverted the traditional presumptions: for them, mathematical harmonies were embedded in creation; motion was the carrier of order; and the objects of mathematics were mathematical curves drawn by nature itself. Mathematics could thus be assigned an explanatory role in natural philosophy, capturing a new metaphysical entity: pure motion. Successive generations of natural philosophers from Descartes to Huygens and Hooke gradually relegated the need to legitimize the application of mathematics to natural phenomena and the blurring of natural and artificial this application relied on. Newton finally erased the distinction between nature’s and artificial mathematics altogether, equating all of geometry with mechanical practice. (shrink)

We introduce the question whether there are specific kinds of writing modalities and practices that facilitated the development of modern science and mathematics. We point out the importance and uniqueness of symbolic writing, which allowed early modern thinkers to formulate a new kind of questions about mathematical structure, rather than to merely exploit this structure for solving particular problems. In a very similar vein, the novel focus on abstract structural relations allowed for creative conceptual extensions in natural philosophy during the (...) scientific revolution. These preliminary reflections are meant to set the stage for the following contributions in this volume. (shrink)