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A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
Narrative and the Rationality of Mathematical Practice
, 2008
"... The possibility that narrative might play a crucial role in the practice of mathematics has been paid little attention by philosophers. The majority of Anglophone philosophers of mathematics have followed those working in the logical empiricist tradition of the philosophy of science by carving apart ..."
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The possibility that narrative might play a crucial role in the practice of mathematics has been paid little attention by philosophers. The majority of Anglophone philosophers of mathematics have followed those working in the logical empiricist tradition of the philosophy of science by carving apart rational enquiry into a ‘context of discovery ’ and a ‘context of justification’. In so doing, they have aligned the justification component with the analysis of timeless standards of logical correctness, and the discovery component with the historical study of the contingent, the psychological, and the sociological. The failings of this strategy are by now plain. In this debating arena there can be no discussion of the adequacy of current conceptions of notions such as space, dimension, quantity or symmetry. Such matters become questions purely internal to the practice of mathematics, and no interest is shown in the justificatory narratives mathematicians give for their points of view. In this talk, I would like to outline the views of the moral philosopher, Alasdair MacIntyre, whose descriptions of traditionconstituted forms of enquiry are highly pertinent to the ways in which mathematics can best be conducted, and allow us to discern the rationality of debates concerning, say, the mathematical understanding of space. An essential component of a thriving research tradition is a narrative account of its history, the internal obstacles it has overcome, and its responses to the objections of rival traditions.
Kleisli enriched
"... For a monad S on a category whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules ( = profunctors, distributors), for some quantale V. The assignment V is shown to belong to a global adjunc ..."
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For a monad S on a category whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V(bi)modules ( = profunctors, distributors), for some quantale V. The assignment V is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative, unital) quantale V the prototypical example of a modular monad, namely the presheaf monad on V Cat, the category of (small) Vcategories. We discuss in particular the question whether the Hausdorff monad on V Cat is modular.