THE OTHER MATHEMATICS The Other Mathematics Language and Logic in Egyptian and in General LEO DEPUYDT GORGIAS PRESS 2008 First Gorgias Press Edition, 2008 Copyright © 2008 by Gorgias Press LLC All rights reserved under International and Pan-American Copyright Conventions. No part of this publication may be reproduced, stored in a re- trieval system or transmitted in any form or by any means, electronic, mechani- cal, photocopying, recording, scanning or otherwise without the prior written permission of Gorgias Press LLC. Published in the United States of America by Gorgias Press LLC, New Jersey ISBN 978-1-59333-369-0 GORGIAS PRESS 180 Centennial Ave., Piscataway, NJ 08854 USA www.gorgiaspress.com Library of Congress Cataloging-in-Publication Data Depuydt, Leo. The other mathematics : language and logic in Egyptian and in general / Leo Depuydt. -- 1st Gorgias Press ed. p. cm. Includes bibliographical references and index. ISBN 978-1-59333-369-0 (alk. paper) 1. Egyptian language--Sentences. 2. Egyptian language--Conditionals. 3. Lan- guage and logic. 4. Boole, George, 1815-1864. I. Title. PJ1201.D47 2008 493'.1--dc22 2008044028 The paper used in this publication meets the minimum requirements of the American National Standards. Printed in the United States of America Our bodies are given life from the midst of nothingness. Existing where there is nothing is the meaning of the phrase, “Form is emptiness.” That all things are provided by nothingness is the meaning of the phrase, “Emptiness is form.” One should not think that these are two separate things. Tsunetomo Yamamoto (d. ca. 1700), Hagakure: The Book of the Samurai (1716), chap. 2 (trans. William Scott Wilson) When x and y are regarded as classes we cannot but observe that not-x and not-y are themselves just as much classes as those of which they are the contradictories. John Venn, Symbolic Logic (1894), 307 ô’ ... ášô’ Rìá ›ðÜñ÷åéí ôå êár ìx ›ðÜñ÷åéí Päýíáôïí ô² ášô² êár êáôN ô’ ášôü ... áœôç äx ðáó§í dóôr âåâáéïôÜôç ô§í Pñ÷§í ... Päýíáôïí ... ¿íôéíï™í ôášô’í ›ðïëáìâÜíåéí åqíáé êár ìx åqíáé ... Pîéï™óé äx êár ôï™ôï Pðïäåéêíýíáé ôéícò äéE Pðáéäåõóßáí · hóôé ãNñ Pðáéäåõóßá ô’ ìx ãéãíþóêåéí ôßíùí äås æçôåsí Pðüäåéîéí êár ôßíùí ïš äås · ”ëùò ìcí ãNñ QðÜíôùí Päýíáôïí Pðüäåéîéí åqíáé · åkò Tðåéñïí ãNñ Uí âáäßæïé ªóôå ìçäE ïœôùò åqíáé Pðüäåéîéí · A single thing cannot at the same time possess and not possess the same attribute all else being the same. … That is the firmest of all the axioms. … Assuming that whosoever can both be and not be the same thing is impossible.… Some ask for proof, but only because they lack education. For not knowing of what one needs to seek proof and of what not shows lack of education. Proving everything is definitely impossible. One would just recede into infinity (in trying to prove everything by something else) and the final step would still be without proof. Aristotle, Metaphysics, 4.3.9–10, 4.4.2 (L.D.’s translation) (From the Preface, adapted) Hiding under the human skull is the most complex structure in the universe, the human brain, the seat of thought. No concept has inspired the present investigation more than that thought is subject to absolute limitations. Yet, thought is perhaps more readily conceived as limitless. Just think of the human imagination in its various forms: literary, religious, visual, and so on. Anything seems possible when it comes to thought. Then again, the brain is a material structure that is not infinite. It therefore seems eminently reasonable to suppose that what the brain does is not infinite either. The larger aim of the present investigation is to achieve a better sense of the absolute limitations of thought and of the precise and distinct levels of thought that reach up to this final border beyond which thought is not possible. Humility is a common concept in the realms of religion and morality. But rational thought has its own kind of humil- ity, namely the acute awareness of its own absolute limitations. What are these limita- tions? How smart are we really? Max Planck recommended the study of philosophy only when conjoined to the study of more specific subjects. In this spirit, the larger concept outlined above is studied here in relation to a narrower domain. This narrower domain is the Egyptian language, whose history is the longest attested of any language. The focus is specifically on cer- tain striking phenomena of Egyptian, along with their parallels in other languages. These phenomena lay bare some of the fundamental fiber of human thought. Since the mid-nineteenth century, Aristotelian and scholastic logic has been fully superseded by modern scientific logic. The pioneer is George Boole (1815–1864). In the late 1930s, an M.I.T. graduate student named Claude Shannon adapted Boolean algebra for electronic circuits and the computer age began. In the present investiga- tion, several facets of Egyptian are treated in detail in light of modern scientific logic. But no prior knowledge of logic is presupposed. A brief history of logic is provided. All that is needed from logic is defined here internally in fully explicit terms. Topics pertaining to Egyptian treated in the present work include: sharp and simple definitions of condition and premise, of the difference between condition and premise, and of how one gets from condition to premise and back; the balanced sen- tence or Wechselsatz; the conditio sine qua non and the language; and the intriguing ques- tion of whether we moderns are smarter or more sophisticated than the ancient Egyptians or than ancient peoples in general. The treatments of these distinct but also interconnected topics ultimately all have these general purposes: to expose ever more clearly the basic articulation of thought into three levels and to suggest the apparent inability of thought to break out of this tripartite pattern as its absolute limitation. The three-level model is provisionally able to absorb and incorporate in complete transpar- ency a number of abstract and much discussed terms such as “causality,” “condition,” “result,” “consequence,” “premise,” “thought,” “truth,” “certainty,” “right and wrong,” and many others. Everyone senses more or less what these terms mean. But defining them precisely is another matter. An engineering application is added at the end in support of the notion that, if a func- tional physical mechanism for how we draw inferences that lead us to act can exist in an electronic circuit, one of analogous structure must exist in the brain. C ONTENTS Acknowledgments..........................................................................xvii Abbreviations...................................................................................xxi Symbols...............................................................................................xxi Introduction: The Other Mathematics..............................................1 0.1 How Smart Are We?............................................................................1 0.2 What Is a Condition?............................................................................2 0.3 Two Main Lines of Research..............................................................6 0.4 Earlier Research on Conditional Clauses..........................................6 0.5 Earlier Research on Logic...................................................................9 0.6 Outline of the Present Investigation................................................11 0.7 The Other Mathematics: Logic and Mathematics Proper as Two Members of Deductive Thought........................................14 0.7.1 The separate histories of logic and mathematics.........................14 0.7.2 Deduction...........................................................................................15 0.7.3 Axiomatic observations of reality...................................................15 0.7.4 Quantity..............................................................................................16 0.7.5 Attribute.............................................................................................16 0.7.6 Rules valid with attributes but not with quantities.......................17 0.7.7 Combination classes.........................................................................19 0.7.8 Other axiomatic observations about reality..................................20 0.7.9 Condition and deduction.................................................................20 0.8 On Logic and Its Proper Domain, and on Truth, Certainty, Right and Wrong, and Knowledge Itself........................................21 0.8.1 Logic in relation to truth and thought...........................................21 0.8.2 The domain of logic..........................................................................22 0.8.3 What are truth and certainty?..........................................................23 0.8.4 Right and wrong................................................................................23 0.8.5 What is knowledge?..........................................................................24 0.9 Boole’s Algebra and the Boolean Algebra of Computing............24 0.10 Boole and the Theory of Probabilities............................................30 0.11 “If” and “When” in English.............................................................31 0.12 On Style and on Coptic Transliterations.........................................32 vii viii THE OTHER MATHEMATICS 1 Two Conditional Sentence Types...............................................35 2 Basic Concepts Of Logic............................................................39 2.1 On the History of Logic....................................................................39 2.2 Two Levels of Making Statements and the Parallelism between the Two.....................................................43 2.3 Types of Statement: Boole’s Three Types and Five Types Discussed by Venn.........................................................45 3 Conditional Sentence with jrand the Balanced Sentence..........49 4 Logical Properties of Types 1 and 2.............................................51 4.0 Brief Comparison of Types 1 and 2.................................................51 4.1 Reversibility of the Equation (Type 1 Only)..................................51 4.2 Balanced Negation (Type 1 Only)....................................................53 4.3 Restriction on the Combination of Classes (Type 1 Only)..........53 4.4 Expressing the Relation between Type 1 and Type 2...................57 5 Balanced Sentences and Logical Types 1 and 2.........................59 5.0 Four Properties in Relation to Types 1 and 2................................59 5.1 Reversibility as a Fact of Balanced Sentences................................59 5.1.1 Examples of reversibility..................................................................59 5.1.2 The meaning of paired balanced sentences: Equation (from which reversibility naturally follows)..................66 5.1.3 Association of the balanced sentences with the logical statement of Type 1 (and of the sentence with jr with the logical statement of Type 2)................................66 5.1.4 Translating balanced sentences........................................................67 5.1.5 The absence of inversion with sentences with jr..........................67 5.1.6 A single thought expressed both as a balanced sentence and as a sentence with jr, confirming the postulated difference between the two...............68 5.2 Balanced Negation as a Fact of Balanced Sentences.....................69 5.3 Restriction to Certain Combinations as a Fact of Balanced Sentences.......................................................70 5.4 Two Sentences with jr Correspond to One Balanced Sentence.75 6 From Condition to Premise and Back........................................79 6.1 Condition and Premise......................................................................79 6.1.1 Definition............................................................................................79 6.1.2 Definite and indefinite processes....................................................80 6.1.3 The problem: Is deriving conditions from premises a definite process?...........81 6.2 From Condition to Premise..............................................................82 6.2.1 Two events correspond to eight sequences of protasis and apodosis...................................................................82 CONTENTS ix 6.2.2 Eight sequences correspond to four denials of a combination’s existence, making four pairs...........................83 6.2.3 Conclusive and inconclusive derivations of premises from conditions...........................................................85 6.2.4 Sixteen derivations of premises from conditions.........................87 6.2.5 The eight conclusive derivations of premises from conditions...........................................................87 6.2.6 Conclusion: Derivation of premises from conditions is a definite process...........................................................................89 6.3 From Premise to Condition..............................................................89 6.3.1 Two events correspond to eight sequences of premise and result or consequence............................................89 6.3.2 Derivation of premises from conditions: A definite process.....90 6.4 From Condition to Premise and Back: Derivations Channeled through Four Pairs....................................90 6.5 Concrete Application:The Difference between Mark 9:47 and Matthew 18:9.............................................91 7 The (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:6)(cid:3)(cid:8)(cid:9)(cid:6)(cid:4)(cid:10)(cid:8)(cid:11)(cid:12)(cid:13)(cid:8)(cid:4)(cid:3)(cid:4).....................................................................93 8 The Balanced Sentence: Collection of Examples.......................97 8.0 Purpose of the Collection and Criteria of Classification of the Examples......................................97 8.0.1 Exclusive writings and distinctive writings of verb forms..........97 8.0.2 %Dmm.f is always (1) passive, (2) future, and (3) substantival.....98 8.0.3 Balanced sentences with the second versus the first verb form written distinctively as substantival............................130 8.0.4 A notable absence in balanced sentences as formulas..............131 8.1 Both Verb Forms Exclusively Written as Substantival...........................135 8.2 Both Verb Forms Distinctively Written as Substantival.........................137 8.2.1 Book of the Dead, chapter 90, Papyrus of Nu (Budge 1898: 192, lines 10–12).....................................................137 8.2.2 CT 5.326g–h, Coffin B2L..............................................................139 8.2.3 Pyr. §412b.........................................................................................139 8.2.4 Pyr. §696d TN..................................................................................141 8.2.5 CT 3.61f–k, Coffins B1C, B2L.....................................................143 8.2.6 CT 6.302m–p, Coffin B1Bo..........................................................143 8.2.7 CT 3.24a–25b, Coffins S1C, S2C..................................................143 8.2.8 CT 3.115e–h, Papyrus Gardiner II...............................................144 8.2.9 Temple at Deir el-Bahri (Naville 1901: plate CXIV, line 18 from the right).....................145 8.2.10 A Special Case: Pyr. §193c Nt.......................................................145 8.2.11 Urkunden 4.305,8..............................................................................146