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The historical and epistemological reflection on the applications of mathematical techniques to the Sciences of Nature - physics, biology, chemistry, and geology - today generates attention and interest because of the increasing use of mathematical models in all sciences and their high level of sophistication. The goal of the meeting and the papers collected in this proceedings volume is to give physicists, biologists, mathematicians, and historians of science the opportunity to share information on their work and reflect on the and mathematical models are used in the natural sciences today and in way mathematics the past. The program of the workshop combines the experience of those working on current scientific research in many different fields with the historical analysis of previous results. We hope that some novel interdisciplinary, philosophical, and epistemological considerations will follow from the two aspects of the workshop, the historical and the scientific? This proceedings includes papers presented at the meeting and some of the results of the discussions that took place during the workshop. We wish to express our gratitude to Sergio Monteiro for all his work, which has been essential for the successful publication of these proceedings. We also want to thank the editors of Kluwer AcademidPlenum Publishers for their patience and constant help, and in particular Beth Kuhne and Roberta Klarreich. Our thanks to the fallowing institutions: -Amministrazione Comunale di Arcidosso -Comunita Montana del Monte Amiata ?Center for the History of Physics, UCLA -Centre F.