Volume 12, Number 5 May 2018
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ÉCLAT MATHEMATICS JOURNAL Lady Shri Ram College for Women
ECLAT´ MATHEMATICS JOURNAL Lady Shri Ram College For Women Volume X 2018-2019 HYPATIA This journal is dedicated to Hypatia who was a Greek mathematician, astronomer, philosopher and the last great thinker of ancient Alexandria. The Alexandrian scholar authored numerous mathematical treatises, and has been credited with writing commentaries on Diophantuss Arithmetica, on Apolloniuss Conics and on Ptolemys astronomical work. Hypatia is an inspiration, not only as the first famous female mathematician but most importantly as a symbol of learning and science. PREFACE Derived from the French language, Eclat´ means brilliance. Since it’s inception, this journal has aimed at providing a platform for undergraduate students to showcase their understand- ing of diverse mathematical concepts that interest them. As the journal completes a decade this year, it continues to be instrumental in providing an opportunity for students to make valuable contributions to academic inquiry. The work contained in this journal is not original but contains the review research of students. Each of the nine papers included in this year’s edition have been carefully written and compiled to stimulate the thought process of its readers. The four categories discussed in the journal - History of Mathematics, Rigour in Mathematics, Interdisciplinary Aspects of Mathematics and Extension of Course Content - give a comprehensive idea about the evolution of the subject over the years. We would like to express our sincere thanks to the Faculty Advisors of the department, who have guided us at every step leading to the publication of this volume, and to all the authors who have contributed their articles for this volume. -
The Power of Principles: Physics Revealed
The Passions of Logic: Appreciating Analytic Philosophy A CONVERSATION WITH Scott Soames This eBook is based on a conversation between Scott Soames of University of Southern California (USC) and Howard Burton that took place on September 18, 2014. Chapters 4a, 5a, and 7a are not included in the video version. Edited by Howard Burton Open Agenda Publishing © 2015. All rights reserved. Table of Contents Biography 4 Introductory Essay The Utility of Philosophy 6 The Conversation 1. Analytic Sociology 9 2. Mathematical Underpinnings 15 3. What is Logic? 20 4. Creating Modernity 23 4a. Understanding Language 27 5. Stumbling Blocks 30 5a. Re-examining Information 33 6. Legal Applications 38 7. Changing the Culture 44 7a. Gödelian Challenges 47 Questions for Discussion 52 Topics for Further Investigation 55 Ideas Roadshow • Scott Soames • The Passions of Logic Biography Scott Soames is Distinguished Professor of Philosophy and Director of the School of Philosophy at the University of Southern California (USC). Following his BA from Stanford University (1968) and Ph.D. from M.I.T. (1976), Scott held professorships at Yale (1976-1980) and Princeton (1980-204), before moving to USC in 2004. Scott’s numerous awards and fellowships include USC’s Albert S. Raubenheimer Award, a John Simon Guggenheim Memorial Foun- dation Fellowship, Princeton University’s Class of 1936 Bicentennial Preceptorship and a National Endowment for the Humanities Research Fellowship. His visiting positions include University of Washington, City University of New York and the Catholic Pontifical University of Peru. He was elected to the American Academy of Arts and Sciences in 2010. In addition to a wide array of peer-reviewed articles, Scott has authored or co-authored numerous books, including Rethinking Language, Mind and Meaning (Carl G. -
Stephen Cole Kleene
STEPHEN COLE KLEENE American mathematician and logician Stephen Cole Kleene (January 5, 1909 – January 25, 1994) helped lay the foundations for modern computer science. In the 1930s, he and Alonzo Church, developed the lambda calculus, considered to be the mathematical basis for programming language. It was an effort to express all computable functions in a language with a simple syntax and few grammatical restrictions. Kleene was born in Hartford, Connecticut, but his real home was at his paternal grandfather’s farm in Maine. After graduating summa cum laude from Amherst (1930), Kleene went to Princeton University where he received his PhD in 1934. His thesis, A Theory of Positive Integers in Formal Logic, was supervised by Church. Kleene joined the faculty of the University of Wisconsin in 1935. During the next several years he spent time at the Institute for Advanced Study, taught at Amherst College and served in the U.S. Navy, attaining the rank of Lieutenant Commander. In 1946, he returned to Wisconsin, where he stayed until his retirement in 1979. In 1964 he was named the Cyrus C. MacDuffee professor of mathematics. Kleene was one of the pioneers of 20th century mathematics. His research was devoted to the theory of algorithms and recursive functions (that is, functions defined in a finite sequence of combinatorial steps). Together with Church, Kurt Gödel, Alan Turing, and others, Kleene developed the branch of mathematical logic known as recursion theory, which helped provide the foundations of theoretical computer science. The Kleene Star, Kleene recursion theorem and the Ascending Kleene Chain are named for him. -
Russell's Paradox in Appendix B of the Principles of Mathematics
HISTORY AND PHILOSOPHY OF LOGIC, 22 (2001), 13± 28 Russell’s Paradox in Appendix B of the Principles of Mathematics: Was Frege’s response adequate? Ke v i n C. Kl e m e n t Department of Philosophy, University of Massachusetts, Amherst, MA 01003, USA Received March 2000 In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. 1. Introduction Russell’s letter to Frege from June of 1902 in which he related his discovery of the Russell paradox was a monumental event in the history of logic. However, in the ensuing correspondence between Russell and Frege, the Russell paradox was not the only antinomy discussed. -
Computability Computability Computability Turing, Gödel, Church, and Beyond Turing, Gödel, Church, and Beyond Edited by B
computer science/philosophy Computability Computability Computability turing, Gödel, Church, and beyond turing, Gödel, Church, and beyond edited by b. Jack Copeland, Carl J. posy, and oron Shagrir edited by b. Jack Copeland, Carl J. posy, and oron Shagrir Copeland, b. Jack Copeland is professor of philosophy at the ContributorS in the 1930s a series of seminal works published by university of Canterbury, new Zealand, and Director Scott aaronson, Dorit aharonov, b. Jack Copeland, martin Davis, Solomon Feferman, Saul alan turing, Kurt Gödel, alonzo Church, and others of the turing archive for the History of Computing. Kripke, Carl J. posy, Hilary putnam, oron Shagrir, Stewart Shapiro, Wilfried Sieg, robert established the theoretical basis for computability. p Carl J. posy is professor of philosophy and member irving Soare, umesh V. Vazirani editors and Shagrir, osy, this work, advancing precise characterizations of ef- of the Centers for the Study of rationality and for lan- fective, algorithmic computability, was the culmina- guage, logic, and Cognition at the Hebrew university tion of intensive investigations into the foundations of Jerusalem. oron Shagrir is professor of philoso- of mathematics. in the decades since, the theory of phy and Former Chair of the Cognitive Science De- computability has moved to the center of discussions partment at the Hebrew university of Jerusalem. He in philosophy, computer science, and cognitive sci- is currently the vice rector of the Hebrew university. ence. in this volume, distinguished computer scien- tists, mathematicians, logicians, and philosophers consider the conceptual foundations of comput- ability in light of our modern understanding. Some chapters focus on the pioneering work by turing, Gödel, and Church, including the Church- turing thesis and Gödel’s response to Church’s and turing’s proposals. -
Frege and the Logic of Sense and Reference
FREGE AND THE LOGIC OF SENSE AND REFERENCE Kevin C. Klement Routledge New York & London Published in 2002 by Routledge 29 West 35th Street New York, NY 10001 Published in Great Britain by Routledge 11 New Fetter Lane London EC4P 4EE Routledge is an imprint of the Taylor & Francis Group Printed in the United States of America on acid-free paper. Copyright © 2002 by Kevin C. Klement All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any infomration storage or retrieval system, without permission in writing from the publisher. 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Klement, Kevin C., 1974– Frege and the logic of sense and reference / by Kevin Klement. p. cm — (Studies in philosophy) Includes bibliographical references and index ISBN 0-415-93790-6 1. Frege, Gottlob, 1848–1925. 2. Sense (Philosophy) 3. Reference (Philosophy) I. Title II. Studies in philosophy (New York, N. Y.) B3245.F24 K54 2001 12'.68'092—dc21 2001048169 Contents Page Preface ix Abbreviations xiii 1. The Need for a Logical Calculus for the Theory of Sinn and Bedeutung 3 Introduction 3 Frege’s Project: Logicism and the Notion of Begriffsschrift 4 The Theory of Sinn and Bedeutung 8 The Limitations of the Begriffsschrift 14 Filling the Gap 21 2. The Logic of the Grundgesetze 25 Logical Language and the Content of Logic 25 Functionality and Predication 28 Quantifiers and Gothic Letters 32 Roman Letters: An Alternative Notation for Generality 38 Value-Ranges and Extensions of Concepts 42 The Syntactic Rules of the Begriffsschrift 44 The Axiomatization of Frege’s System 49 Responses to the Paradox 56 v vi Contents 3. -
9 Alfred Tarski (1901–1983), Alonzo Church (1903–1995), and Kurt Gödel (1906–1978)
9 Alfred Tarski (1901–1983), Alonzo Church (1903–1995), and Kurt Gödel (1906–1978) C. ANTHONY ANDERSON Alfred Tarski Tarski, born in Poland, received his doctorate at the University of Warsaw under Stanislaw Lesniewski. In 1942, he was given a position with the Department of Mathematics at the University of California at Berkeley, where he taught until 1968. Undoubtedly Tarski’s most important philosophical contribution is his famous “semantical” definition of truth. Traditional attempts to define truth did not use this terminology and it is not easy to give a precise characterization of the idea. The under- lying conception is that semantics concerns meaning as a relation between a linguistic expression and what it expresses, represents, or stands for. Thus “denotes,” “desig- nates,” “names,” and “refers to” are semantical terms, as is “expresses.” The term “sat- isfies” is less familiar but also plausibly belongs in this category. For example, the number 2 is said to satisfy the equation “x2 = 4,” and by analogy we might say that Aristotle satisfies (or satisfied) the formula “x is a student of Plato.” It is not quite obvious that there is a meaning of “true” which makes it a semanti- cal term. If we think of truth as a property of sentences, as distinguished from the more traditional conception of it as a property of beliefs or propositions, it turns out to be closely related to satisfaction. In fact, Tarski found that he could define truth in this sense in terms of satisfaction. The goal which Tarski set himself (Tarski 1944, Woodger 1956) was to find a “mate- rially adequate” and formally correct definition of the concept of truth as it applies to sentences. -
Lambda Calculus and the Decision Problem
Lambda Calculus and the Decision Problem William Gunther November 8, 2010 Abstract In 1928, Alonzo Church formalized the λ-calculus, which was an attempt at providing a foundation for mathematics. At the heart of this system was the idea of function abstraction and application. In this talk, I will outline the early history and motivation for λ-calculus, especially in recursion theory. I will discuss the basic syntax and the primary rule: β-reduction. Then I will discuss how one would represent certain structures (numbers, pairs, boolean values) in this system. This will lead into some theorems about fixed points which will allow us have some fun and (if there is time) to supply a negative answer to Hilbert's Entscheidungsproblem: is there an effective way to give a yes or no answer to any mathematical statement. 1 History In 1928 Alonzo Church began work on his system. There were two goals of this system: to investigate the notion of 'function' and to create a powerful logical system sufficient for the foundation of mathematics. His main logical system was later (1935) found to be inconsistent by his students Stephen Kleene and J.B.Rosser, but the 'pure' system was proven consistent in 1936 with the Church-Rosser Theorem (which more details about will be discussed later). An application of λ-calculus is in the area of computability theory. There are three main notions of com- putability: Hebrand-G¨odel-Kleenerecursive functions, Turing Machines, and λ-definable functions. Church's Thesis (1935) states that λ-definable functions are exactly the functions that can be “effectively computed," in an intuitive way. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
Publications of John Corcoran OCT 04
OCTOBER 2004 PUBLICATIONS OF JOHN CORCORAN I. Articles. MR indicates review in Mathematical Reviews. J indicates available at JSTOR. G indicates available at Google by entering John Corcoran plus the complete title. 1.J Three Logical Theories, Philosophy of Science 36 (1969) 153-77. 2. Logical Consequence in Modal Logic: Natural Deduction in S5 (co-author G. Weaver), Notre Dame Journal of Formal Logic 10 (1969) 370-84. 3. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and the Stratification of Language, Journal of Structural Learning 3 (1971) #1, 55-74. 4. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3 (1971) #2, 1-16. 5. Discourse Grammars and the Structure of Mathematical Reasoning III: Two Theories of Proof, Journal of Structural Learning 3 (1971) #3, 1-24. 6. Notes on a Semantic Analysis of Variable Binding Term Operators (co-author John Herring), Logique et Analyse 55 (1971) 646-57. MR46#6989. 7. Variable Binding Term Operators, (co-authors Wm. Hatcher, John Herring) Zeitschrift fu"r mathematische Logik und Grundlagen der Mathematik 18 (1972) 177-82. MR46#5098. 8.J Conceptual Structure of Classical Logic, Philosophy & Phenomenological Research 33 (1972) 25-47. 9. Logical Consequence in Modal Logic: Some Semantic Systems for S4 (co-author G. Weaver), Notre Dame Journal of Formal Logic 15 (1974) 370-78. MR50#4253. 10, 11, and 12. Revisions of 3, 4 and 5 in Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon and Breach Science Publishers, New York (1976). -
A Truth-Teller's Guide to Defusing Proofs of the Liar Paradox
Open Journal of Philosophy, 2019, 9, 152-171 http://www.scirp.org/journal/ojpp ISSN Online: 2163-9442 ISSN Print: 2163-9434 The Liar Hypodox: A Truth-Teller’s Guide to Defusing Proofs of the Liar Paradox Peter Eldridge-Smith Philosophy, Research School of the Social Sciences, Australian National University, Canberra, Australia How to cite this paper: Eldridge-Smith, P. Abstract (2019). The Liar Hypodox: A Truth-Teller’s Guide to Defusing Proofs of the Liar Para- It seems that the Truth-teller is either true or false, but there is no accepted dox. Open Journal of Philosophy, 9, 152-171. principle determining which it is. From this point of view, the Truth-teller is https://doi.org/10.4236/ojpp.2019.92011 a hypodox. A hypodox is a conundrum like a paradox, but consistent. Some- Received: February 21, 2019 times, accepting an additional principle will convert a hypodox into a para- Accepted: May 5, 2019 dox. Conversely, in some cases, retracting or restricting a principle will con- Published: May 8, 2019 vert a paradox to a hypodox. This last point suggests a new method of avoid- ing inconsistency. This article provides a significant example. The Liar para- Copyright © 2019 by author(s) and Scientific Research Publishing Inc. dox can be defused to a hypodox by relatively minimally restricting three This work is licensed under the Creative principles: the T-schema, substitution of identicals and universal instantia- Commons Attribution International tion. These restrictions are not arbitrary. For each, I identify the source of a License (CC BY 4.0). contradiction given some presumptions. -
Sentence, Proposition, Judgment, Statement, and Fact: Speaking About the Written English Used in Logic John Corcoran
Sentence, Proposition, Judgment, Statement, and Fact: Speaking about the Written English Used in Logic John Corcoran Dedication: for Professor Newton da Costa, friend, collaborator, co-discoverer of the Truth-set Principle, and co-creator of the Classical Logic of Variable-binding Term Operators-on his eightieth birthday. abstract. The five ambiguous words|sentence, proposition, judg- ment, statement, and fact|each have meanings that are vague in the sense of admitting borderline cases. This paper discusses sev- eral senses of these and related words used in logic. It focuses on a constellation of recommended primary senses. A judgment is a pri- vate epistemic act that results in a new belief; a statement is a public pragmatic event involving an utterance. Each is executed by a unique person at a unique time and place. Propositions and sentences are timeless and placeless abstractions. A proposition is an intensional entity; it is a meaning composed of concepts. A sentence is a linguistic entity. A written sentence is a string of characters. A sentence can be used by a person to express meanings, but no sentence is intrinsically meaningful. Only propositions are properly said to be true or to be false|in virtue of facts, which are subsystems of the universe. The fact that two is even is timeless; the fact that Socrates was murdered is semi-eternal; the most general facts of physics|in virtue of which propositions of physics are true or false|are eternal. As suggested by the title, this paper is meant to be read aloud. 1 Introduction The words|sentence, proposition, judgment, statement, and fact|are am- biguous in that logicians use each of them with multiple normal meanings.