Department of Philosophy & Moral Science

In this paper I challenge Paolo Palmieri's reading of the Mach-Vailati debate on Archimedes's proof of the law of the lever. I argue that the actual import of the debate concerns the possible epistemic (as opposed to merely pragmatic) role of mathematical arguments in empirical physics, and that construed in this light Vailati carries the upper hand. This claim is defended by showing that Archimedes's proof of the law of the lever is not a way of appealing to a non-empirical source of information, but a way of explicating the mathematical structure that can represent the empirical information at our disposal in the most general way.

But surpassing all stupendous inventions, what sublimity of mind was his who dreamed of finding means to communicate his deepest thoughts to any other person, though distant by mighty intervals of space and time! Of talking with those who are in India; of speaking to those who are not yet born and will not be born for a thousand or ten thousand years; and with what facility, by the different arrangements of twenty characters upon a page!" (Galileo 2001, pp. 120-1)

kin to the mathematical recreations, John Wilkins' Mathematicall Magick (1648) elaborates the pleasant, useful and wondrous part of practical mathematics, dealing in particular with its material culture of machines and instruments. We contextualize the Mathematicall Magick by studying its institutional setting and its place within changing conceptions of art, nature, religion and mathematics. We devote special attention to the way Wilkins inscribes mechanical innovations within a discourse of wonder. Instead of treating ‘wonder’ as a monolithic category, we present a typology, showing that wonders were not only recreative, but were meant to inspire Wilkins' readers to new mathematical inventions.

In his "Galileo's machines, his mathematics, and his experiments" , Peter Machamer proposed to call Galileo's use of the simple machines "models of intelligibility". These machines function as models of intelligibility by "directing attention to what is important in a problem and by exhibiting what relations exist among those important elements" [14, p.72]. As models they made it possible for Galileo to search in a directed way for the mathematical structures that could characterize phenomena of motion, such as the free fall or projection of bodies. The simple machines, among which the Archimedean balance occupied a privileged place, were ideally suited to this role: they are (1) material objects, (2) for which there existed successful mathematical treatments of the relations between their basic properties, and (3) that could be easily manipulated in a controlled way. They showed how to see ideal mathematical structures within concrete and material phenomena, a way of looking that was informed by the use of specific mathematical techniques (proportional geometry and Euclidean style proofs), representational conventions (allowing the coordination of the phenomenon with the mathematical structure), and a practice in which the objects treated were actively manipulated and dealt with to achieve certain goals.