Conceptualism and contextualism in the recent historiography of Newton's \textit{Principia}.
Observing that recently the \textit{Principia} has been the object of renewed interest among mathematicians and physicists, and referring to a lot of such publications, the author notices that several interesting and fruitful questions have emerged concerning Newton's mathematization of motion. The philosophical views on the nature of mathematics orient historical research along different lines, which are called by the author \textit{conceptualism} and \textit{contextualism}. The former can lead to cumulative, linear history, and the latter to a history where ramifications and perhaps even revolutionary changes are possible. The author would like to advocate an approach to the study of the mathematical methods of Newton's \textit{Principia} in which both these aspects are taken into consideration. Newton was a thinker who did not draw a clear-cut distinction between natural philosophy, theology, mathematics, and alchemy. In the publication policy he followed the dualism between public and private, mostly in theology, alchemy, and optics, but also in the case of mathematics. His mathematical discoveries in the late 1660s and early 1670s were printed decades later, especially those concerning infinite series and the calculus, which were rendered available to the mathematical community through letters and manuscript circulation. He found it convenient to avoid print publication of his fluxional method that appeared to him not well-founded from a logical point of view and distant from the rigor attained by the ancient geometrical synthesis. In the case of \textit{Principia} Newton affirmed to have made use of the modern analysis as a heuristic tool and to have retranslated a pristine analytical text into geometrical form in order to conform his work to the style of the ancient geometers. He viewed himself as a heir of an ancient mathematical tradition. The interrelation between a mechanically based geometry and natural motion allows Newton to defend the superiority of his method compared to Leibniz's: the latter can be useful in the art of discovery, but is of no use in the science of demonstrations. Specially are considered the recent studies by \textit{M. Nauenberg} and \textit{C. Wilson} [in \textit{J. Z. Buchwald} (ed.) et al., Isaac Newton's natural philosophy, Cambridge, MA: MIT Press; 139--188, 189--224 (2000; Zbl 0985.01003)] of Newton's approach to the study of the Moon's motion, now looking for intersections between conceptualism and contextualism. It is noted that Newton was actually reinterpreting the classical tradition into a completely new area: the geometrization of motion. There is the plausible existence of hidden parts behind the printed text of the \textit{Principia}. When we read the manuscripts that circulated in the Newtonian circle we find that Newton discussed the algorithmic methods necessary to fill the gaps in the printed text, but they were not considered as a ``calculus''. Thus, according to Newton, a problem is solved by the geometrical construction (or synthesis), while the resolution (the analytical step) can be avoided in publication; the integration technique can be kept hidden.
scientific article; zbMATH DE number 2037200
scientific article