The Other Mathematics: Language and Logic in Egyptian and in General:

397 Pages·2006·5.589 MB·

by  Leo Depuydt

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
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ISBN 978-1-59333-369-0

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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