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Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. -
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Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online Since 01 October 2014, Connection on 05 November 2020
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 18-3 | 2014 Logic and Philosophy of Science in Nancy (I) Selected Contributed Papers from the 14th International Congress of Logic, Methodology and Philosophy of Science Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.) Electronic version URL: http://journals.openedition.org/philosophiascientiae/957 DOI: 10.4000/philosophiascientiae.957 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 1 October 2014 ISBN: 978-2-84174-689-7 ISSN: 1281-2463 Electronic reference Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.), Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online since 01 October 2014, connection on 05 November 2020. URL : http://journals.openedition.org/ philosophiascientiae/957 ; DOI : https://doi.org/10.4000/philosophiascientiae.957 This text was automatically generated on 5 November 2020. Tous droits réservés 1 This issue collects a selection of contributed papers presented at the 14th International Congress of Logic, Methodology and Philosophy of Science in Nancy, July 2011. These papers were originally presented within three of the main sections of the Congress. They deal with logic, philosophy of mathematics and cognitive science, and philosophy of technology. A second volume of contributed papers, dedicated to general philosophy of science, and other topics in the philosophy of particular sciences, will appear in the next issue of Philosophia Scientiæ (19-1), 2015. Philosophia Scientiæ, 18-3 | 2014 2 TABLE OF CONTENTS Logic and Philosophy of Science in Nancy (I) Preface Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister Copies of Classical Logic in Intuitionistic Logic Jaime Gaspar A Critical Remark on the BHK Interpretation of Implication Wagner de Campos Sanz and Thomas Piecha Gödel’s Incompleteness Phenomenon—Computationally Saeed Salehi Meinong and Husserl on Existence. -
CHAPTER 9 Narrative and the Rationality of Mathematical Practice
August 19, 2011 Time: 12:21pm chapter9.tex CHAPTER 9 Narrative and the Rationality of Mathematical Practice DAVID CORFIELD 1. Introduction How is it to act rationally as a mathematician? For much of the Anglo- American philosophy of mathematics this question is answered in terms of what mathematicians most obviously produce—journal papers. From this perspective, the mathematician’s work is taken to be of interest solely insofar as in consists in deducing the consequences of various axioms and definitions. This view of the discipline, with its strong focus on aspects of mathematics that do not feature largely elsewhere—its use of deductive proof, its supposed capacity to be captured by some formal calculus, the abstractness of the objects it studies—isolates the philosophy of mathematics from philosophical accounts of other forms of enquiry. Against this position, some have refused to class as philosophically insignificant readily observable similarities between mathematics and the natural sciences, such as that each discipline has its own very long history. Mathematics constitutes a continuous intellectual effort stretching back through many centuries, “one of mankind’s longest conversations,” as Barry Mazur beautifully describes it. And, as with the sciences, this is not just any conversation but a series of vigorous, socially embodied arguments as to how the field should progress. Now, one of the few philosophers to make much of these and other similarities with the natural sciences was the philosopher Imre Lakatos. His understanding of what constitutes rational enquiry led him to call on mathematical practitioners not to hide their conceptual thinking behind the formal barrier of journal articles but rather to expose their work in novel ways, telling the stories of the development of their concepts. -
The Importance of Being Externalist About Mathematics – One More Turn?
Philosophica 74 (2004) pp.103-122 THE IMPORTANCE OF BEING EXTERNALIST ABOUT MATHEMATICS – ONE MORE TURN? Bart Van Kerkhove & Hans Comijn1 1. Introduction Over thirty years ago, Stephen Toulmin, in his Human Understanding, argued that a bridging of the gap between scientific practice and its reflective analysis was in order.2 We are convinced that today the argument still stands, above all when it comes to mathematics. Indeed, while opinions might differ over the extent to which bridge-builders have succeeded in their task for science in general, there can be little or no discussion as to the lack of much progress in this respect as far as mathematics is concerned. The thing is that much of the people committed to one of both sides of the divide, respectively mathematicians and meta-reflectors (philosophers, sociologists, and historians), do not themselves think that the gap should be closed, let alone that doing so would be a priority of any sort for any of them. We strongly disagree with this view. Standing in front of the gap, as philosophers cum social scientists, looking hopefully towards the other side, but finding only the tiniest traces of durable crossings, we are prompted to ask if mathematicians have become too unreflective, or if 1 Research for this paper was supported by the Fund for Scientific Research - Flanders (Belgium), for the former author, and the Inter University Attraction Poles program (phase V), granted by the Belgian Federal Science Policy, for the latter. 2 “There are good reasons (…) for re-establishing links between the scientific extension of our knowledge and its reflective analysis, and reconsidering ourselves as knowers in the light of recent extensions to the actual content of our knowledge” (Toulmin [1972], p.2). -
Rationality and Mathematical Narrative David Corfield, School of European Culture and Language, University of Kent
Rationality and Mathematical Narrative David Corfield, School of European Culture and Language, University of Kent We dream in narrative, day-dream in narrative, remember, anticipate, hope, despair, believe, doubt, plan, revise, criticize, construct, gossip, learn, hate and love by narrative. (Barbara Hardy 'Towards a Poetics of Fiction: An Approach Through Narrative', Novel 2, 5-14, 1968: 5) Introduction It is hardly controversial to emphasise the importance of the role of narrative in our lives. Phenomenologists, such as David Carr, have taken this point most seriously. Through the length of his Time, Narrative, and History, Carr argues that we don't just live an unmediated experience to which we later add a narrative gloss, rather our experience is already structured as narrativised. Narrative also provides us with the means to understands each other's actions, as Alasdair MacIntyre remarks: We identify a particular action only by invoking two kinds of context, implicitly if not explicitly. We place the agent's intentions, I have suggested, in causal and temporal order with reference to their role in his or her history; and we also place them with reference to their role in the history of the setting or settings to which they belong. In doing this, in determining what causal efficacy the agent's intentions had in one or more directions, and how his short-term intentions succeeded or failed to be constitutive of long-term intentions, we ourselves write a further part of these histories. Narrative history of a certain kind turns out to be the basic and essential genre for the characterization of human actions. -
Homotopy Type Theory: a Revolution in the Foundations of Mathematics?
Homotopy type theory: A revolution in the foundations of mathematics? David Corfield University of Kent 19 February, 2016 We live in interesting times! A new foundational language for mathematics has just appeared. Why I might have predicted the second coming I came to philosophy on a diet of: I Imre Lakatos I Albert Lautman I Category theory and categorical logic Albert Lautman I Proto-category theorist: thematic similarities everywhere. I Rather than accord logic philosophical priority over other parts of mathematics, we should consider it as any other branch, a place where key ideas recurrently manifest themselves. My Masters thesis - categorical logic Question: What should we make of the two kinds of semantics for intuitionistic logic? I Proof theoretic I Topological Categorical logic went some way to explaining this. Topos theory combines the logical and the spatial. Constructive type theory as `internal language' of a topos. In the intervening 24 years... I However much one speaks of `the rich, lived experience of the mathematician', `mathematics as a tradition of intellectual enquiry', etc., there were always those philosophers content with set theory who claim that nothing goes beyond its bounds. I Many of the `rebels' turned to category theory (Marquis, McLarty, me), and yet categories can be thought of as sets or classes with a certain structure. I However much one protests that set theory has lost contact with the conceptual content of mathematics, still it is possible to maintain mathematics to be the accumulation of timeless truths in the framework of first-order logic + axioms of set theory. Maybe this time I Where Homotopy Type Theory differs is in having logic already contained intrinsically within it. -
Towards a Philosophy of Real Mathematics
TOWARDS A PHILOSOPHY OF REAL MATHEMATICS In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathe- matics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. david corfield holds a Junior Lectureship in Philosophy of Science at the University of Oxford. He is co-editor (with Jon Williamson) of Foundations of Bayesianism (2001), and he has pub- lished articles in journals including Studies in History and Philosophy of Science and Philosophia Mathematica. TOWARDS A PHILOSOPHY OF REAL MATHEMATICS DAVID CORFIELD The Pitt Building, Trumpington Street, Cambridge, United Kingdom The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © David Corfield 2004 First published in printed format 2003 ISBN 0-511-04270-1 eBook (netLibrary) ISBN 0-521-81722-6 hardback From the east to western Ind, No jewel is like Rosalind. -
Proof, Rigour & Informality
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository PROOF, RIGOUR & INFORMALITY: A VIRTUE ACCOUNT OF MATHEMATICAL KNOWLEDGE Fenner Stanley Tanswell A Thesis Submitted for the Degree of PhD at the University of St Andrews 2017 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/10249 This item is protected by original copyright PROOF, RIGOUR & INFORMALITY: A VIRTUE ACCOUNT OF MATHEMATICAL KNOWLEDGE Fenner Stanley Tanswell This thesis is submitted in partial fulfilment for the degree of PhD at the University of St Andrews Date of Submission 31st August 2016 Declaration 1. Candidate's declarations: I, Fenner Stanley Tanswell, hereby certify that this thesis, which is approx- imately 65000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for a higher degree. I was admitted as a research student in September 2013 and as a candi- date for the degree of Doctor of Philosophy in September 2013; the higher study for which this is a record was carried out in the University of St An- drews between 2013 and 2016. Date signature of candidate 2. Supervisor's declaration: I hereby certify that the candidate has fulfilled the conditions of the Resolu- tion and Regulations appropriate for the degree of Doctor of Philosophy in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. -
Real Mathematics
Book Review Towards a Philosophy of Real Mathematics Reviewed by E. Brian Davies Towards a Philosophy of Real Mathematics to the other extreme, letting his book become no David Corfield more than a study of the history of mathematics. Cambridge University Press, 2003 But, being a philosopher, he succeeds in this task. US$105.00, 300 pages The first word of the title of the book is “To- ISBN-13: 9780521817226 wards”, and Corfield admits that he intends to cover a wide range of topics, each of which needs David Corfield starts his interesting book with a to be fleshed out in much more detail. Many of radical rejection of much that has been written his examples come from the fields he knows best, about the philosophy of mathematics. He has but this is inevitable. Its range is wider than the no interest in ontology, epistemology, logicism, expertise of most mathematicians, and Corfield Platonism, Kantianism, nominalism, fictionalism, knows what he is talking about, at least as far as and so forth and does not mention most of them. I could tell. An important merit of the book is its Taking his cue from Nietzsche, Lakatos, and a few large number of quotations. These help to keep others, he likens the traditional approach to the the arguments anchored in the real world—to the examination of a dead body (pp. 3–5). He criticizes extent that mathematicians live in the real world. the attitude that has led many philosophers of Before I continue, a small digression is in order. mathematics to imagine that everything of inter- In most circumstances a reviewer should not read est to their subject occurred between 1880 and other reviews, in order to assure independence. -
Heuristics and Mathematical Discovery: the Case of Bayesian Networks by Donald Gillies, University College London ([email protected])
Heuristics and Mathematical Discovery: the Case of Bayesian Networks by Donald Gillies, University College London ([email protected]) This is the final author version of a paper published in C.Cellucci & D.Gillies (eds.) Mathematical Reasoning and Heuristics, King’s College Publications, 2005, pp. 87-118. Contents 1. Introduction 2. The Development of Artificial Intelligence and the Discovery of Bayesian Networks 3. Heuristics Involved: (a) the Use of Philosophical Ideas 4. Heuristics Involved: (b) New Practical Problems 5. Heuristics Involved: (c) Domain Interaction 6. Heuristics of Mathematical Discovery versus Logic of Mathematical Discovery 1. Introduction I will begin this paper by discussing some ideas to be found in two recent books on the philosophy of mathematics. These are (i) Carlo Cellucci’s Filosofia e matematica, published by La Terza in 2002, and (ii) David Corfield’s Towards a Philosophy of Real Mathematics, published by Oxford University Press in 2003. I will start with Cellucci’s book. In his book, Cellucci is highly critical of the traditional or foundational approach to the philosophy of mathematics, based on the attempt to justify mathematics. Instead he advocates what he calls the heuristic approach to the philosophy of mathematics. As he says (2002, p. viii): ‘According to the dominant point of view the principal problem in the philosophy of mathematics is that of the justification of mathematics. … In this book I maintain instead that the principal problem of reflection on mathematics is that of mathematical discovery. This problem includes the problem of justification …’ I partly agree and partly disagree with this. It is certainly true that traditional philosophy of mathematics focussed exclusively on the problem of the justification of mathematics and neglected the problem of mathematical discovery. -
How Mathematicians May Fail to Be Fully Rational (Version 21 November 2005)
How Mathematicians May Fail To Be Fully Rational (version 21 November 2005) David Corfield Max Planck Institute for Biological Cybernetics, Tübingen, Germany david.corfield AT tuebingen.mpg.de Abstract: 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.