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Science in Axiomatic Perspective Msc in Logic

PREFACE The axiomatic method counts two thousand and three hundred years circa. Suppes [61] has proposed the category of Euclidean-Archimedean tradition to refer to the axiomatic theories that have been developed before the inven-tion/discovery of the non-Euclidean geometries. Among these theories the first axiomatic system that we know is Euclid's Elements [16], a mathematical tractate consisting of thirteen books in which three centuries of Greek mathematical knowledge were given an order and were presented as a unified theory. 1 Euclid produced another axiomatic theory, the Optics [15]. This represents a theory of vision in Euclidean perspective rather than a tractate on physical optics. It is interesting that Archimedes's Treatise [12], probably the first book on mathematical physics, is an axiomatic theory. The axiomatic method in the Euclidean-Aristotelian tradition was transmitted during the medieval age and scholarship in history of science has established the use of the axiomatic method in scientific tractates through all periods from antiquity up to the sixteenth–seventeenth-century Scientific Revolution [5]. In the context of the Scientific Revolution an important ax-iomatic theory is Newton's Principia [41, 61]. The axiomatic method covers a too big period of history and philosophy of science and we cannot deal with it in this thesis. So we skip the analysis of the axiomatic method in the Euclean-Archimedean tradition and begin our analysis in the nineteenth century when the axiomatic method entered in the modern phase. As Suppes puts it [61, p. 225]: " The historical source of the modern viewpoint toward the axiomatic method was the intense scrutiny of the foundations of geometry in the nineteenth century. Undoubtedly the most important driving force behind this effort was the discovery and development of non-Euclidean geometry at the beginning of the nineteenth century and IV is contained plane geometry, in Book II are contained the rudiments of geometrical algebra, Books V–X develop number theory (Note that number theory is developed without stating the axioms. An axiomatic treatment of number theory will appear only at the end of the nineteenth century, as we will see.) and Books X–XIII develop solid geometry.