At the turn of the seventeenth century, Bruno and Cavalieri independently developed two theories, central to which was the concept of the geometrical indivisible. The introduction of indivisibles had significant implications for geometry – especially in the case of Cavalieri, for whom indivisibles provided a forerunner of the calculus. But how did this event occur? What can we learn from the fact that two theories of indivisibles arose at about the same time? These are the questions addressed in this paper. (...) Relying on the methodology of “historical epistemology”, this paper asserts that the similarities and differences between the theories of Bruno and Cavalieri can be explained in terms of “shared knowledge”. The paper shows that the idea – on which both Bruno and Cavalieri build – that geometrical objects are generated by motion was part of the mathematical culture of the time. Tracing this idea back to its Pythagorean origins thus sheds light on the relationship between motion and continuum in mathematics. (shrink)

If anyone still raises doubts about the function of philology and the need for critical editions, this book should definitely put those doubts to rest. It shows that no history of philosophy is possible without intensive philological work with the original texts.

No presente artigo, procuramos mostrar, por intermédio de numerosos exemplos, que a distinção lógica operada por Pierre Duhem entre a física e a metafísica não impossibilita que, do ponto de vista histórico, o autor reconheça a existência de um entrosamento profícuo a cingir as duas áreas do saber. Distinguimos três níveis possíveis de interação entre a física e a metafísica, porque acreditamos que ele aceita (a) que o físico trabalha constantemente movido por ambições metafísicas não autorizadas pela rigorosa lógica, (b) (...) que os sistemas metafísicos contribuíram para muitos dos progressos das teorias físicas, em especial, para o nascimento da ciência moderna e (c) que podemos identificar tradições metafísicas de pesquisa duradouras que se mostraram mais férteis do que algumas tradições positivas. In this article, we aim to show, by means of several examples, that Pierre Duhem's logical distinction operated by Pierre Duhem between physics and metaphysics does not prevent him, from a historical perspective, recognizing that there can be a fruitful interplay between the two fields of knowledge. We distinguish three possible levels of interaction that we believe Duhem accepts between physics and metaphysics: (a) that the physicist works constantly moved by metaphysical ambitions not authorized by rigorous logic; (b) that the metaphysical systems have contributed to the progress of many physical theories and, especially, to the birth of modern science, and; (c) that we can identify long-lasting metaphysical traditions of research which have shown themselves to be more fertile than the positivist traditions. (shrink)

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)