Jonathan Fay

Faced with the mathematical possibility of non-Euclidean geometries, 19th Century geometers were tasked with the problem of determining which among the possible geometries corresponds to that of our space. In this context, the contribution of the Belgian philosopher-mathematician, Joseph Delboeuf, has been unduly neglected. The aim of this essay is to situate Delboeuf’s ideas within the context of the philosophies of geometry of his contemporaries, such as Helmholtz, Russell and Poincaré. We elucidate the central thesis, according to which Euclidean geometry is given special status on the basis of the relativity of magnitudes, we uncover its hidden history and show that it is defensible within the context of the philosophies of geometry of the epoch. Through this discussion, we also develop various ideas that have some relevance to present-day methods in gravitational physics and cosmology.

We argue that the fundamental assertion underlying Mach's critique of Newton's first law is that inertial motion is not motion in the absence of causes; rather, it is motion whose cause lies in some homogeneous aspect of the environment. We distinguish this formal requirement (Mach's principle) from two hypotheses which Mach considers concerning the origin of inertia: that the distant stars play (1) a merely “collateral” or (2) a “fundamental” role in the causal determination of inertial motion. In his later writings, Mach deliberately avoids referring to the concept of causation, and indeed, this has made the interpretation of Mach's principle a subject of widespread controversy. However, in his earlier writings, the substance of Mach's critique is less ambiguously expressed. Therefore, close attention is given to Mach's early writings and the evolution of his thought. Various accounts in the secondary literature on Mach's principle, in particular those of Norton and DiSalle, are assessed on this basis. We end with a defence of the Machian status and legitimacy of the early Einstein's research program.