Toward a Metaphysics of Smooth Infinitesimal Analysis as a New Foundation of Physical Geometry

The idea that a circle is a regular polygon with infinitely many sides has a long tradition. This idea was made rigorous by the theory of "smooth infinitesimal analysis" (SIA), alternatively known as "synthetic differential geometry," developed by Lawvere (1980) and others, which features nilpotent infinitesimals (numbers whose squares equal zero). SIA is an intriguing alternative framework for theories of continua, and can better regiment physicists' modes of reasoning with infinitesimals. But to realize this potential, we face a significant obstacle: the axiomatic system of SIA is classically inconsistent, and there are no obvious classical reconstructions for it (Hellman 2006). If this is true, then classical logicians are unable to accept SIA as a realistic theory of space. However, contrary to common belief, I argue that we can interpret SIA as a classically consistent theory of space. To argue for this position, I advance a new understanding of the categorical models for SIA proposed by Moerdijk and Reyes (1991). In particular, those models represent geometric theories that are attractive generalizations of Einstein algebras explored in philosophy of physics