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**1 - 6**of**6**### Free crossed resolutions of groups and presentations of modules of identities among relations

, 2008

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### GRAPHS OF MORPHISMS OF GRAPHS Bangor Mathematics Preprint 06.04

, 2006

"... This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an ap-plication, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion ..."

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This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an ap-plication, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton’s (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equiv-

### 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 tradition-constituted 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.

### david.corfield AT tuebingen.mpg.de

, 2005

"... 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 ..."

Abstract
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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 tradition-constituted 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. To the extent that mathematicians do not contribute to this writing, from the MacIntyrean perspective they are failing to act fully rationally.

### Constructing Orders By Means of Inductive Definitions

"... We present a class of algebraic theories that are enriched over a novel Symmetrical Monoidal Closed structure on the category of graphs, whose free models are posets that are equipped with an induction principle, which is easily formalized in type theory. We give examples. The development of comput ..."

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We present a class of algebraic theories that are enriched over a novel Symmetrical Monoidal Closed structure on the category of graphs, whose free models are posets that are equipped with an induction principle, which is easily formalized in type theory. We give examples. The development of computer science has given a new impulse to the theory of inductive denitions. It was classically based on set theory a la Zermelo (for a survey see [Acz77]), but the needs of the theory of data types, and those of type theory, has compelled people to look towards universal algebra and category theory for inspiration and paradigms. In particular it has been known for a long time that the notion of free structure is closely related to that of induction principle, at the very least since Lawvere's categorical axiomatization of natural numbers [Law64]. But a lot of mathematical structures, be they algebraic or topological, admit a free model, and it is also known that those that can be said to den...