Through the convex Looking Glass: A Helmholtzian lesson for the connection between dynamics and chronogeometry in spacetime theories.

Over the last two decades, the rise of the dynamicist view in the philosophy of spacetime theories has motivated a discussion about the way in which chronogeometric structure and dynamics are connected. Geometricists defend that chronogeometry determines and explains dynamics, whereas dynamicists state that it is the other way around. Both parties assume that the arrow of explanation at issue involves a claim of fundamentality and priority of one of the elements over the other. I challenge this assumption, and I propose a third way to understand the connection. Drawing a lesson from Herman von Helmholtz's and David Hilbert's views on the foundations of geometry, I argue that in spacetime theories chronogeometry and dynamics are inextricably interconnected counterparts, so claims of fundamental explanation and priority, regardless of the direction of the alleged arrow, are misconceptions. The link between chronogeometry and dynamics in spacetime theories is properly understood in terms of a bidirectional arrow, not in terms of a unidirectional arrow of fundamental explanation.