Online research in philosophy

We propose a formal representation of objects , those being mathematical or empirical objects. The powerful framework inside which we represent them in a unique and coherent way is grounded, on the formal side, in a logical approach with a direct mathematical semantics in the well-established field of constructive topology, and, on the philosophical side, in a neo-Kantian perspective emphasizing the knowing subject’s role, which is constructive for the mathematical objects and constitutive for the empirical ones.

In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.

By using some classical reasoning we show that any countably presented formal topology, namely, a formal topology with a countable axiom set, is spatial.

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal (...) topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales. (shrink)