Abstract: Many mathematicians have cited depth as an important value in their research. However, there is at present no analysis of mathematical depth that is generally admitted. In this talk I will try to make some progress on this question. I will begin with a discussion of Szemerédi’s theorem, that every sufficiently “dense” subset of N contains an arbitrarily long arithmetic progression. This theorem has been judged deep by many mathematicians. Using this theorem as a case study, I will continue by presenting and discussing several different analyses of mathematical depth. In particular I will attend to the objectivity of depth judgments under each analysis.

Abstract: In this talk, I will give a Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. I will discuss Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s Four-Valued Logic. I will underline how non-classical logics require different verification games and how different logics require different game theoretical conditions in such games.

Abstract: The dominant view in philosophy was and is that mathematics requires fully axiomatised set theoretic foundations (almost invariably Zermelo-Fraenkel set theory including the axiom of choice). In mathematical practice sets are extensively used but the representation of simple set properties is a sufficient basis for very many proofs. These set properties formalise natural intuitions about sets or capture mathematical ideas of a non-set theoretic kind. Philosophical interest in set theoretic foundations is motivated by concerns about ontology and epistemology rather than by an interest in understanding contemporary mathematical practice. Given this the foundational significance of category theory deserves much more philosophical attention than it has had.

Abstract: One of the announced goals of the Reverse Mathematics program is to calibrate which set existence principles are required to prove classical theorems whose statements are not overtly set theoretic in character (e.g. those of analysis, algebra, or combinatorics). The axiom system based on Weak König's Lemma [WKL] -- i.e. every infinite subtree of the full binary tree has an infinite path -- may appear to be an outlier in this regard: not only does WKL fail to have the form of an unqualified assertion of set existence, it is also not initially clear how it might demarcate between the commitments of different programs in the foundations of mathematics. One goal of this talk will be to put these observations into historical context by considering how WKL came to be isolated as a combinatorial principle in its own right. This story has much to do with the Gödel (1929) completeness theorem for first-order logic and its subsequent arithmetization by Hilbert and Bernays (1939). I will argue on this basis that both WKL and the completeness theorem play a role in how we should understand statements of mathematical existence.

School of Mathematics

University of Birmingham

http://web.mat.bham.ac.uk/R.W.Kaye/

Dr Walter Dean

Department of Philosophy

University of Warwick

http://go.warwick.ac.uk/whdean