Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

Many Valued Generalised Quantifiers for Natural Language in the DisCoCat Model

DisCoCat refers to the Categorical compositional distributional model of natural language, which combines the statistical vector space models of words with the compositional logic-based models of grammar. It is fair to say that despite existing work on incorporating notions of entailment, quantification, and coordination in this setting, a uniform modelling of logical operations is still an open problem. In this report, we take a step towards an answer. We show how one can generalise our previous DisCoCat model of generalised quantifiers from category of sets and relations to category of sets and many valued rations. As a result, we get a fuzzy version of these quantifiers. Our aim is to extend this model to all other logical connectives and develop a fuzzy logic for DisCoCat. The main contributions are showing that category of many valued relations is compact closed, defining appropriate bialgebra structures over it, and demonstrating how one can compute within this setting many valued meanings for quantified sentences.EPSRC Career Acceleration Fellowship EP/J002607/

Incorporating Perspectival Elements in a Discrete Mathematics Course

Discrete mathematics is a vast field that can be explored along many different paths. Opening with a unit on logic and proof and then taking up some additional core topics (induction, set theory, combinatorics, relations, Boolean algebra, graph theory) allows one to bring in a wealth of relevant material on history, philosophy, axiomatics, and abstraction in very natural ways. This talk looks at how my 2019 textbook on discrete mathematics, focused in this way, came to be, and it highlights the various perspectival elements the book includes