Towards a Philosophy of Real Mathematics
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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. -
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. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration
Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration Bart de Keijzer∗ Tomas B. Klosy Yingqian Zhangz Abstract We study the power index voting game design problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distri- bution. Our goal is to find algorithms that solve this problem exactly. Thereto, we consider various subclasses of simple games, and their asso- ciated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions (also known as shift-maximal losing coalitions) of a game from its list of roof coalitions (also known as shift-minimal winning coalitions), and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time (that is, 2 O(2n · p(n)) for a polynomial p). Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the power index voting game design problem that runs in exponential time. -
Arxiv:1209.4623V1 [Cs.DS] 20 Sep 2012 Who Defined Them in [Ded97]
COUNTING INEQUIVALENT MONOTONE BOOLEAN FUNCTIONS TAMON STEPHEN AND TIMOTHY YUSUN Abstract. Monotone Boolean functions (MBFs) are Boolean functions f : f0; 1gn ! f0; 1g satisfying the monotonicity condition x ≤ y ) f(x) ≤ f(y) for any x; y 2 f0; 1gn. The number of MBFs in n variables is known as the nth Dedekind number. It is a longstanding compu- tational challenge to determine these numbers exactly { these values are only known for n at most 8. Two monotone Boolean functions are inequivalent if one can be obtained from the other by renaming the vari- ables. The number of inequivalent MBFs in n variables was known only for up to n = 6. In this paper we propose a strategy to count inequiva- lent MBF's by breaking the calculation into parts based on the profiles of these functions. As a result we are able to compute the number of inequivalent MBFs in 7 variables. The number obtained is 490013148. 1. Introduction A Boolean function on n variables (BF) is a function f : f0; 1gn ! f0; 1g. A monotone Boolean function (MBF) additionally satisfies the condition n x ≤ y ) f(x) ≤ f(y), for any x; y 2 f0; 1g . We write x ≤ y if xi ≤ yi for all i = 1; 2; : : : ; n, and x < y if x ≤ y and xi < yi for some i. A BF is monotone if and only if it can be written as a combination of conjunctions and disjunctions only. Since each input state in f0; 1gn has two possible output states, there are a total of 22n Boolean functions on n variables. -
The Generation of Hyper-Power Sets
Jean Dezert1, Florentin Smarandache2 1ONERA, 29 Av. de la Division Leclerc 92320, Chatillon, France 2Department of Mathematics University of New Mexico Gallup, NM 8730, U.S.A. The generation of hyper-power sets Published in: Florentin Smarandache & Jean Dezert (Editors) Advances and Applications of DSmT for Information Fusion (Collected works), Vol. I American Research Press, Rehoboth, 2004 ISBN: 1-931233-82-9 Chapter II, pp. 37 - 48 Abstract: The development of DSmT is based on the notion of Dedekind’s lattice, called also hyper-power set in the DSmT framework, on which is defined the general basic belief assignments to be combined. In this chapter, we explain the structure of the hyper-power set, give some examples of hyper-power sets and show how they can be generated from isotone Boolean functions. We also show the interest to work with the hyper-power set rather than the power set of the refined frame of discernment in terms of complexity. 2.1 Introduction ne of the cornerstones of the DSmT is the notion of Dedekind’s lattice, coined as hyper-power set Oby the authors in literature, which will be defined in next section. The starting point is to consider Θ = θ , . , θ as a set of n elements which cannot be precisely defined and separated so that no { 1 n} refinement of Θ in a new larger set Θref of disjoint elementary hypotheses is possible. This corresponds to the free DSm model. This model is justified by the fact that in some fusion problems (mainly those manipulating vague or continuous concepts), the refinement of the frame is just impossible to obtain; nevertheless the fusion still applies when working on Dedekind’s lattice and based on the DSm rule of This chapter is based on a paper [6] presented during the International Conference on Information Fusion, Fusion 2003, Cairns, Australia, in July 2003 and is reproduced here with permission of the International Society of Information Fusion. -
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. -
Download PDF # Integer Sequences
4DHTUDDXIK44 ^ eBook ^ Integer sequences Integer sequences Filesize: 8.43 MB Reviews Extensive information for ebook lovers. It typically is not going to expense too much. I discovered this book from my i and dad recommended this pdf to learn. (Prof. Gerardo Grimes III) DISCLAIMER | DMCA DS4ABSV1JQ8G > Kindle » Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular number, Multiplicative partition, Perrin number, Smooth number, Ulam number, Primorial, Lambek Moser theorem, -
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). -
Dedekind Numbers and Related Sequences
DEDEKIND NUMBERS AND RELATED SEQUENCES by Timothy James Yusun B.Sc. (Hons.), Ateneo de Manila University, 2008 a Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics Faculty of Science c Timothy James Yusun 2011 SIMON FRASER UNIVERSITY Fall 2011 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Timothy James Yusun Degree: Master of Science Title of Thesis: Dedekind Numbers and Related Sequences Examining Committee: Dr. Abraham Punnen, Chair Professor Dr. Tamon Stephen, Senior Supervisor Assistant Professor Dr. Zhaosong Lu, Supervisor Assistant Professor Dr. Michael Monagan, External Examiner Professor, Department of Mathematics Simon Fraser University Date Approved: December 6, 2011 ii Partial Copyright Licence Abstract This thesis considers Dedekind numbers, where the nth Dedekind number D(n) is defined as the number of monotone Boolean functions (MBFs) on n variables, and R(n), which counts non-equivalent MBFs. The values of D(n) and R(n) are known for up to n = 8 and n = 6, respectively; in this thesis we propose a strategy to compute R(7) by considering profile vectors of monotone Boolean functions. The profile of an MBF is a vector that encodes the number of minimal terms it has with 1;2;:::;n elements. The computation is accomplished by first generating all profile vectors of 7-variable MBFs, and then counting the functions that satisfy each one by building up from scratch and taking disjunctions one by one. -
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.