Newton versus Lagrange: Two Rival Interpretations of Spacetime

The conflict between the Newtonian and the Leibnizian picture of spacetime is well-known and widely discussed in the literature. However, two competing metaphysical pictures of spacetime have not been addressed so far: The Newtonian and the Lagrangian pictures. According to the Newtonian picture, spacetime can be considered as the time evolution of a spatial hypersurface. According to the Lagrangian picture, spacetime is a unified whole that can be examined in an all-at-once manner, and its (temporal) evolution is not meaningful, except locally or quasi-locally. Although the Newtonian picture has been the dominant metaphysical picture since Newton's time, our explicit familiarity with these two competing pictures is recent. It is rooted in Smolin's criticism of modern cosmology and Wharton's research on the foundations of quantum mechanics. Here, we describe the views of Smolin and Wharton, then we will argue that two famous formulations of general relativity, namely the Hamiltonian formulation and the Lagrangian formulation, require the Newtonian and the Lagrangian picture, respectively. Also, we will argue that: (1) the Lagrangian picture is more compatible with the philosophy of relativity theory than the Newtonian picture, while the Newtonian picture is only acceptable for practical and computational purposes. And, (2) the Lagrangian picture is universal and applicable to any spacetime, while the Newtonian picture can only be used for certain spacetimes. Therefore, the Lagrangian picture is a more plausible metaphysical picture than the Newtonian.