In this project, the group develops formal representations of theories of spatiotemporal objects. Such theories deal both with objects in space and time and with spatiotemporal entities, such as space-time points, and they address relations between these entities.

### Research

The project focused on the development and investigation of theories that *directly* fit the ontological and linguistic intuitions we have here: as such, they should not be expressed in set-theoretic terms. But they should be rich and strong enough to do the job for which mathematical theories are usually employed.

Not many theories of this kind are known. The members of the project have used ancient approaches to spatiotemporal entities as a starting point for developing such theories. The project was simultaneously pursuing both interpretive and systematic goals.

##### Results

Karl-Georg Niebergall and Jonathan Beere have worked out a formalization of the first 10 propositions of Euclid’s *Elements* and will extend this formalization to the remainder of Book I and then into the subsequent books. This work has achieved a number of valuable results. For instance, proposition I.4 (often known as “side-angle-side”) has long been seen as problematic. Euclid proves it by a method of superposition, which he prefers to avoid. David Hilbert simply took proposition I.4 as an axiom. While Niebergall and Beere agree with Hilbert in being unable to find an adequate proof of I.4, their attempts to do so have revealed a close connection between the definition of angle and the proposition. Furthermore, they have introduced a distinction between the relations *being-a-part* and *bounding*. Normally, both mathematicians and logicians have treated the relation between, say, the side of a square and the square and the relationship between the square and some regions into which it is divided, as one and the same relation. Euclid seems to distinguish these relations and Niebergall and Beere follow him in this. This gives the theory a structure that is unusual and interesting from the contemporary perspective. On the one hand, the theory is mereological (involving parts and wholes). On the other hand, the theory does not treat the boundaries of objects as parts of those objects.