Mechanics, mathematical physics and foundations of mathematics in the 18th and 19th Centuries

7 July 2008 - 12 July 2008

Aims

After the invention of infinitesimal calculus, rational mechanics is the grand affair of eighteenth-century mathematics. As Jean Bernoulli said, armed with the differential equations of the new calculus mathematicians ventured to the explore ‘an unknown continent’. They approached and solved very difficult problems such as those of the isochrone and the brachistochrone, which are among the most well-known examples. Not only rigid body mechanics, but fluid dynamics as well were subjected to the laws of the new calculus. According to D’Alembert, the successes justified the ‘faith’ in the calculus, but at same time they posed new problems. The integration of the wave equation (that is, the vibrating string problem) raised ticklish questions that divided the opinions of mathematicians for a long time: What are the functions? How are they represented analytically? On the other hand, the calculus runs up against insurmountable problems. How is three-body motion described with respect to the universal law of gravity? Neither Euler and Lagrange went beyond the particular solutions. At the end of the century, analytical mechanics and celestial mechanics were given a systematic exposition in important treatises of Lagrange and Laplace. What meaning and the significance does the Newtonian model have for the explication of natural phenomena? How can the actions of numerous ‘fluids’ which concern the physical explanation of electrostatic or magnetic phenomena and of the propagation of heat be considered? How can they be described by the instrument of analysis? The new century opens with Fourier’s research on heat; those concerning the potential culminate in the works of Gauss and Dirichlet. What are the relationships between the new physical theories and the contemporaneous invention of new tools of geometry and real and complex analysis? Are successful descriptions of natural phenomena such as mechanics, thermodynamics and electromagnetism which characterise the mathematical physics of the nineteenth century a guarantee of solidity of the foundations of analysis or, more generally, of mathematics? Or is it precisely because of those successes that the problem of establishing stable and rigorous foundations for mathematics is once again examined with vitality? Of course, the workshop does not aim at providing complete and exhaustive answers to the many questions which the development of mechanics and of mathematical physics raise in the period under consideration. Its aim is to give to scholars from Italy and abroad an occasion to discuss and to compare their opinions and research. In this way we hope to contribute to the clarification of some aspects of the theme from a mathematical, historical and epistemological point of view.