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Six Septembers: Mathematics for the Humanist Patrick Juola Duquesne University, Juola@Mathcs.Duq.Edu University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Zea E-Books Zea E-Books 4-6-2017 Six Septembers: Mathematics for the Humanist Patrick Juola Duquesne University, [email protected] Stephen Ramsay University of Nebraska - Lincoln, [email protected] Follow this and additional works at: http://digitalcommons.unl.edu/zeabook Part of the Algebra Commons, Applied Mathematics Commons, Digital Humanities Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Juola, Patrick and Ramsay, Stephen, "Six Septembers: Mathematics for the Humanist" (2017). Zea E-Books. 55. http://digitalcommons.unl.edu/zeabook/55 This Book is brought to you for free and open access by the Zea E-Books at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Zea E-Books by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. SIX SEPTEMBE R S Mathematics for the Humanist Patrick Juola and Stephen Ramsay Scholars of all stripes are turning their attention to mate- rials that represent enormous opportunities for the future of humanistic inquiry. The purpose of this book is to im- part the concepts that underlie the mathematics they are likely to encounter and to unfold the notation in a way that removes that particular barrier completely. This book is a primer for developing the skills to enable humanist scholars to address complicated technical material with confidence. Patrick Juola is Professor of Computer Science and Di- rector of the Evaluating Variations in Language Laboratory at Duquesne University in Pittsburgh, PA. He is an inter- nationally noted expert in text analysis, security, forensics, and stylometry. He has taught at the University of Colorado and Oxford University, and conducted research at Johns Hopkins University, Carnegie Mellon University, MIT, PGP Inc., and AT&T Bell Laboratories. He is the author of Prin- ciples of Computer Organization and Assembly Language (2007) and Authorship Attribution (2008). Stephen Ramsay is Susan J. Rosowski Associate Profes- sor of English at the University of Nebraska–Lincoln. He is a specialist in digital humanities, theory of new media, and theater history. He is the author of Reading Machines: Toward an Algorithmic Criticism (2011). Cover: William Blake, “Newton” (c. 1804) Zea Books Lincoln, Nebraska ISBN 978-1-60962-111-7 doi: 10.13014/K2D21VHX Six Septembers: Mathematics for the Humanist Patrick Juola and Stephen Ramsay Zea Books Lincoln, Nebraska 2017 Version March 27, 2017 Copyright © 2017 Patrick Juola and Stephen Ramsay. This is an open access work licensed under a Creative Commons Attribution 4.0 International license. ISBN 978-1-60962-111-7 https://doi.org/10.13014/K2D21VHX Zea Books are published by the University of Nebraska–Lincoln Libraries Electronic (pdf) edition available online at http://digitalcommons.unl.edu/zeabook/ Print edition available from http://www.lulu.com/spotlight/unlib UNL does not discriminate based upon any protected status. Please go to unl.edu/nondiscrimination Contents Introduction 1 Preface 9 1 Logic and Proof 13 1.1FormalLogic..................... 16 1.1.1 Deduction and Induction . ........ 17 1.1.2 Validity, Fallacy, and Paradox ........ 18 1.2TypesofDeductiveLogic............... 26 1.2.1 Syllogistic Logic . ............ 27 1.2.2 Propositional Logic . ............ 31 1.2.3 Predicate Logic ................ 49 1.2.4 Possible Other Logics ............ 55 1.3TheSignificanceofLogic............... 58 1.3.1 Paradoxes and the Logical Program . .... 58 1.3.2 Soundness, Completeness, and Expressivity . 61 1.3.3 The Greatest Hits of Kurt Gödel . .... 64 2 Discrete Mathematics 71 2.1SetsandSetTheory.................. 74 2.1.1 Set Theory, Informally ............ 74 iii Contents 2.1.2 Set Theory, Notation . ............ 77 2.1.3 Extensions of Sets . ............ 81 2.2RelationsandFunctions................ 83 2.2.1 Cartesian Products . ............ 83 2.2.2 Relations . ................ 84 2.2.3 Functions . ................ 86 2.2.4 Equivalence Relations and Partial/Total Orders 88 2.3GraphTheory..................... 92 2.3.1 Some Definitions . ............ 94 2.3.2 Planar Graphs . ................ 96 2.3.3 Chromatic Graphs . ............ 97 2.3.4 Bipartite Graphs . ............101 2.3.5 Eulerian Graphs . ............104 2.3.6 Hamiltonian Graphs . ............108 2.3.7 Trees . ....................110 2.4 Induction and Recursion . ............111 2.4.1 Induction on Numbers ............112 2.4.2 Induction on Graphs . ............114 2.4.3 Induction in Reverse . ............116 2.5Combinatorics.....................123 3 Algebra 133 3.1StructureandRelationship..............135 3.2 Groups, Rings, and Fields . ............140 3.2.1 Operations . ................140 3.2.2 Groups ....................143 3.2.3 Rings . ....................147 3.2.4 Fields . ....................150 iv Contents 3.3FurtherExamples...................155 3.3.1 Classical Examples . ............156 3.3.2 Vectors ....................158 3.3.3 Matrices ....................162 3.3.4 Permutations . ................169 3.3.5 Topological Transformations . ........170 3.3.6 Cryptology . ................180 3.4Answerstochapterpuzzles..............188 3.4.1 Sudoku ....................188 3.4.2 Cryptograms . ................189 4 Probability and Statistics 191 4.1 Probability . ....................195 4.1.1 Manipulating Probabilities . ........199 4.1.2 Conditional Probability ............203 4.1.3 Bayesian and Frequentist Probabilities ....206 4.2 Probability and Statistics . ............216 4.2.1 Normal Distribution . ............218 4.2.2 Other Distributions . ............227 4.2.3 Central Tendency . ............234 4.2.4 Dispersion . ................236 4.2.5 Association . ................241 4.3 Hypothesis Testing . ................245 4.3.1 Null and Experimental Hypotheses . ....246 4.3.2 Alpha and Beta Cutoffs ............249 4.3.3 Parametric Tests . ............253 4.3.4 Non-parametric Tests . ............260 4.3.5 Interpreting Test Results . ........264 v Contents 4.4SomeHumanisticExamples.............267 4.4.1 Hemingway’s Sentences . ........267 4.4.2 The Federalist Papers.............269 4.5Lying:aHow-ToGuide................272 5 Analysis and Calculus 277 5.1TheMathematicsofChange.............279 5.1.1 Exhaustion . ................280 5.1.2 Paradoxes of Infinity . ............282 5.2Limits.........................292 5.2.1 Limits and Distance . ............293 5.2.2 Formal Definitions of Limits . ........294 5.2.3 Limits at a Point: The Epsilon-Delta Defini- tion......................302 5.2.4 Continuity . ................304 5.2.5 Motionless Arrows and the Definition of Speed309 5.2.6 Instantaneous Speed as a Limit Process . 314 5.2.7 The Derivative ................315 5.2.8 Calculating Derivatives ............319 5.2.9 Applications of the Derivative ........324 5.3Integration.......................333 5.3.1 The Area Problem . ............333 5.3.2 Integrals ....................335 5.4 Fundamental Theorem of Calculus . ........337 5.5 Multidimensional Scaling . ............339 5.6CalculusMadeBackwards..............343 vi Contents 6 Differential Equations 347 6.1Definitions.......................349 6.2TypesofDifferentialEquations............353 6.2.1 Order of a Differential Equation . ....354 6.2.2 Linear vs. Non-Linear Equations . ....355 6.2.3 Existence and Uniqueness . ........355 6.2.4 Some Preliminary Formalities ........356 6.3First-OrderLinearDifferentialEquations......358 6.3.1 Solutions via Black Magic . ........359 6.3.2 A Simple Example . ............360 6.4Applications......................368 6.4.1 Naive Population Growth . ........368 6.4.2 Radioactive Decay . ............370 6.4.3 Investment Planning . ............371 6.4.4 Temperature Change . ............374 6.4.5 More Sophisticated Population Modeling . 377 6.4.6 Mixing Problems and Pollution . ....380 6.4.7 Electrical Equipment . ............381 6.5SethiAdvertisingModel...............384 6.6 Computational Methods of Solution . ........387 6.7ChaosTheory.....................389 6.7.1 Bifurcation . ................390 6.7.2 Basins of Attraction and Strange Attractors . 393 6.7.3 Chaos and the Lorenz Equations . ....394 6.8FurtherReading....................397 A Quick Review of Secondary-School Algebra 403 1 Arithmetic Expressions ................404 vii Contents 2 Algebraic expressions . ................407 3 Equations.......................408 4 Inequalities . ....................411 5 StoryProblemsandMathematicalModels......412 6 FurtherReading....................418 viii Introduction Look at mathematics: it’s not a science, it’s a monster slang, it’s nomadic. Deleuze and Guattari A Thousand Plateaus This is a book on advanced mathematics intended for humanists. We could defend ourselves in the usual way; everyone can benefit from knowing something about mathematics in the same way that everyone can benefit from knowing something about Russian liter- ature, or biology, or agronomy. In the current university climate, with its deep attachment to “interdisciplinarity” (however vaguely defined), such arguments might even seem settled. But we have in mind a much more practical situation that now arises with great regularity. Increasingly, humanist scholars of all stripes are turning their attention toward materials that represent enormous opportunities for the future of humanistic inquiry. We are speaking, of course, of the vast array of digital materials that now present themselves as objects of study: Google Books
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