# Logic and Philosophy of Mathematics

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Philosophy has always been marked by its self-conscious concern with the activity of reasoning, and one very important facet of the subject has been the development of methods for the systematic study of the ways in which reasons can support conclusions. The development of logic has therefore paralleled the development of philosophy, throughout its history and throughout many of the cultures of the world: work on logic is found among the Ancient Greeks, within Classical Indian philosophy, in Islamic and Western philosophy of the Middle Ages, through to an explosion of activity in modern times.

Research in logic can range from formal work on logical systems to philosophical questions about logic and its applications. Research sometimes focuses on logics which differ from classical logic as typically taught in Elementary Logic classes. Modal logics extend the range of logical concepts to include necessity and possibility and model the rules governing those notions, for instance, while non-classical logics typically employ different rules for basic elements of reasoning and different accounts of their meanings. All of these areas of formal and philosophical work shed light on basic questions about concepts that are fundamental to enormous amounts of human thought.

Mathematics is another area of intellectual activity which pays careful attention to forms of argumentation, because of the role that proofs play within it. (It is a striking fact that some great philosophers have also been major mathematicians, like Descartes and Leibniz.) Some formal work on logic is quite mathematical, but mathematics itself raises many deep philosophical questions. The Philosophy of Mathematics addresses fundamental questions about mathematics itself, our knowledge of mathematics, and the concepts which it involves. It often overlaps with the Philosophy of Logic, but it also deals with metaphysical questions such as whether numbers really exist and, if so, what they are. Discussions of the nature of mathematical entities often form models for discussions of realism and anti-realism about other abstract entities, for example within ethics. It may also interact with questions in Cognitive Science, about our grasp and understanding of mathematical concepts.

These areas of Philosophy have been a strong focus of research in Sheffield for many years. Logic modules are available at all three undergraduate levels, and they can be continued in an MA. PhD students have completed theses on topics such as: higher-order vagueness; the experimental approach to vagueness; intuitionism and revisionism; and applicability, idealisation and mathematisation and have gone on to posts in Kent, Salzburg, the University of East Anglia and the University of Manchester.

Staff in Sheffield work on a range of topics in the area, both separately and collaboratively. For example, Dominic Gregory has written on the metaphysical and semantic implications of the uses of standard model-theoretic techniques in assessing validity, and on modal logic. Rosanna Keefe writes on the logic of vague language and the sorites paradox and she is also interested in other paradoxes and other messy or problematic logical phenomena. She also works on logical pluralism and logical consequence more generally, partly in relation to our everyday reasoning. Dominic and Rosanna are currently working together on the nature of negation, and Yonatan Shemmer is interested in related questions on disagreement and has written about disagreement and dialetheism. Stephen Laurence works on issues concerning the nature and origins of basic numerical representations and our mathematical abilities in both humans and other animals.

Representative publications:

• Gregory D. (2005). “Keeping semantics pure”. Noûs.
• Gregory, D. (2011). “Iterated modalities, meaning and a priori knowledge”. Philosophers’ Imprint.
• Gregory, D. (2017). “Counterfactual reasoning and knowledge of possibilities”. Philosophical Studies.
• Keefe, R. (2000). Theories of Vagueness. Cambridge University Press.
• Keefe, R. (2013). “What logical pluralism cannot be”. Synthese.
• Keefe, R. (2020). “Prefaces, Sorites and guides to teasoning”. In L. Walters & J. Hawthorne (eds), Paradox in Conditionals, Probability, and Paradox: Themes from the Philosophy of Dorothy Edgington. Oxford University Press.
• Margolis E. & Laurence S. (2008). “How to learn the natural numbers: inductive inference and the acquisition of number concepts”. Cognition.
• Bex-Priestley, G. & Shemmer, Y. (2017) “A normative theory of disagreement”. Journal of the American Philosophical Association.