In the correspondence with Clarke, Leibniz proposes to construe physical theory in terms of physical (spatio-temporal) relations between physical objects, thus avoiding incorporation of infinite totalities of abstract entities (such as Newtonian space) in physical ontology. It has generally been felt that this proposal cannot be carried out. I demonstrate an equivalence between formulations postulating space-time as an infinite totality and formulations allowing only possible spatio-temporal relations of physical (point-) objects. The resulting rigorous formulations of physical theory may be seen (...) to follow Leibniz' suggestion quite closely. On the other hand, physical assumptions implicit in the postulation of space-time totalities are made explicit in the reconstruction of the space-time versions from the physical-relation versions. (shrink)

An infinitary characterisation of the first-order sentences true in all substructures of a structure M is used to obtain partial reduction of the decision problem for such sentences to that for Th(M). For the relational structure $\langle\mathbf{R}, \leq, +\rangle$ this gives a decision procedure for the ∃ x∀ y-part of the theory of all substructures, yet we show that the ∃ x 1x 2 ∀ y-part, and the entire theory, is Π 1 1 -complete. The theory of all ordered subsemigroups (...) of $\langle\mathbf{R}, \leq, +\rangle$ is also shown Π 1 1 -complete. Applications in the philosophy of science are mentioned. (shrink)

We suggest that there can be epistemologically significant reasons why certain mathematical structures - such as the Real numbers - are more important than others. We explore several contexts in which considerations bearing on the choice of a fundamental numerical domain might arise. 1) Set theory. 2) Historical cases of extension of mathematical domains - why were negative numbers resisted, and why should we accept them as part of our fundamental numerical domain? 3) Using fewer reals in physics, without really (...) noticing. (shrink)