THE NATURE 0? DEFINITION IN HIGH-SCHOOL GEOMETRY

A CRITIQUE OF CURRENT PRACTICES

DISSERTATION Presented In Partial Fulfillment of the Requirements For the Degree Doctor of Philosophy in the Graduate School of The Ohio State university

By SHELDON STEPHEN lyîYERS, B.8., B.E.» M, Ed,

******

The Ohio State University 1955

Approved by:

Adviser Department of Education AOKNOWLEDCaÆHNTS

The writer would like to express his sinoere eppreoiation to Professor Nathan Lazar for making unstint- Ingly available his Intellectual resources and wise counsel during the completion of this study. The writer would also like to express his appreciation to Professor Harold P. Fawcett who first stimulated his Interest In the teach­ ing of geometry.

11 t able o f C0NTMT3

CHAPTER page I. INTRODUCTION...... 1 Statement of the problem...... 1 Importance of the p r o b l e m ...... 3 Scope and limitations of the study...... 4 Definitions of terms ...... 5 Preview of the study...... , 6 II. ARISTOTELIAN ORIGINS OF SOME DEFINING PRACTICES IN PLANE GE0L5ETRY TETTBOOKS . . . 8 Definitions should be in the form of "genus at differentia,"...... 10 Definitions should not use obscure language and should use terms which are prior and more intelligible ...... 18 Prior and more intelligible terms .... 18 Obscure language ...... S7 Definitions through opposites, circular definitions, and definition by coordi­ nate species ...... SO Circular defiiiitions...... 31 Sequences of ùufinition ...... 33

undefined teriLs...... 36 S u m m a r y ...... 51

iii CHAPTER PACE Definitions should not oontain redundant or superfluous information •••••••* 55 Definitions must be reversible...... 58 There is only one definition of a thing • • 74 Definitions state what a thing is but not the foot that the thing exists...... 85 Purposes of definitions given by textbooks . 99 Chapter s u m m a r y ...... * ...... 101 III. TREATMENT OF DEFINITION IN I.iATHEMATIOS METHODS BOOKS ...... 103 Per genus et differentiam method of defining 103 The question of whether there is only one, or more than one, definition of a term . . 106 Use of simpler or previously defined terms in definition...... # 110 Undefined t e r m s ...... 112 The relation of meaning to definition • , * 118 The problem of existanoe and definition. • . 121 The question of reversibility and definition...... 124 Redundancy or overloading in definition • • 124 Other pedagogical sources on definition in geometry ...... 128

iv CHAPTER PAGE Chapter Summary •••.«•*•«••••• 13£ 17. TREATMENT BY LOGIC AND MATHIWATICS BOOKS OF CERTAIN DIFFICULTIES CONCERNING DEFINITIONS FOUND IN CONTEMPORARY GEOMETRY BOOKS .... 136 Dlffloultlea arising out of the Aristotelian distinotion between essenoe and property . 136 Dlffioulties arising out of the Aristotelian notion that Glasses and rank of intelli­ gibility are absolute ...... 145 We do not know the meaning of a term until we define it...... 151 Definitions can be true or f a l s e ...... 153 Definitions establish exlstenoe ...... 155 Dlffioulties relating to the reversibility of definitions...... 158 Other theoretical points about definition. . 161 Real and nominal d e f i n i t i o n ...... 16£ Stipulative and lexical definition .... 164 Analytic and synthetic definition .... 166 Causal or genetic definition ...... 167 Operational definition ...... 167

Explicit definition and definition in use. 169

▼ CHAPTER PAGE SxtenslTe or donotatlv# definition and oonnotative definition • • ...... 170 Oateneive definition ...... 171 Implioatire definition ...... ••••• 171 Coordinating definition ...... 173 Descriptive definition...... • . • • 173 Definitions in mathematical systems • • • 174 Chapter s u m m a r y ...... 176 V. smiMARY, CONCLUSIONS, AND RECOLîMENûATIONS . . 178 Recommendations...... 181 Recommendations for other s t u d i e s .... 185 BIBLIOGRAPHY...... 188 Part I. High School Geometry Textbooks Examined. 189 Part II. Methods manuals in mathematics and related w o r k s ...... 193 Part III. Logic and philosophy-of-mathematics b o o k s ...... 199 Part IV. Periodical l i t e r a t u r e...... 200

▼i LIST OF TABLES TABLE PAGE I. Number of Plane Geometry Textbooks^ by PubXloatlon Date, yfhloh Do or Do Not Mention Undefined Terms ...... 47

vii CHAPTER I

INTRODUCTION

Statement of the Problem

Almost universally students in plane geometry are told to define a figure by placing it in its next larger class and by giving characteristics which will distinguish that figure from all others in the class* Thus, they will see that the textbook definition, "A triangle is a polygon with three sides,*' comfortably meets the requirements which they have been taught, Since no suggestion is made that there can be another way to define, they do not question the method. Later, in the same textbook, the follcwing definition may be encountered, "'./hen one straight line meets another straight line so as to form equal adjacent angles, the lines are said to be perpendicular." Usually there is no hint that this definition does not follow the rule prescribed earlier. Later on, when a student attempts to define a parallel­ ogram as a quadrilateral with the opposite sides equal, he is rebuked and informed that "equality of sides" is a deducible property and that the essential, or defining property, of a parallelogram must be "opposite sides are parallel," Yet, were he to glance furtively into several £ other textbooksr he might find in one the definition, *▲ rectangle is an equiangular qualdrilateral," and in another, "A rectangle is a parallelogram with one right angle." But not having other books available, usually, which would enable him to mcounter alternate definitions, he is thereby given the impression that every figure has an essential property that must be used for the definition. As a result of this, he is given the additional impression that there is only one correct definition for each figure. Early in the course, the student is often informed that if he wants to understand a term, he must memorize its definition carefully. After memorizing the definition of adjacent angles, he later finds that he is unable to iden­ tify them and has a very confused idea of what they are. Contrary to the widely-held opinion that the function of definitions is to convey the meaning of terms, they did not do so in the case just cited. In the light of the three, foregoing illustrations, the question might well be raised as to what is the basis of these difficulties and what can be done to correct them. One suspects further that the above difficulties are symptomatic of underlying causes which may have produced other problems with regard to definition in geometry. The purpose of this study will therefore be to investi­ gate the treatment accorded definitions in contemporary, hlgh-BOhool, geometry teztbooke» to determine the natare and extent of tmeound defining jo^aetioee, to determine the origin of theme praotioea, and to reoomaend improvement in aoeordanoe with m o d e m pedagogical, logical, and mathe­ matical theory*

Probl*»

An examination of contemporary geometry textbooks reveals that there is firmly imbedded in them a tradition about definition which gives rise to a number of unsound practices* This can have serious conséquences for the future intellectual growth of the student, for definitions occupy a key role in all organized knowledge systems, such as physics, economics, philosophy and law* Geometry has long been accepted as an important training ground for the development of critical and logical thinking* Therefore, it would seem extremely important in plane geometry, where the student first studies a logical system, that he utilize the soundest defining practices* This study should serve to point out the source and nature of defining difficulties in plane geometry. It should also focus the attention of teachers, textbook writers, and mathematics educators, on the most approved theories and methods of defining today with particular reference to geometry* This should make it possible for 4 teaohers to work out eroatlvaly in tha olaasrooa thalr own mathodologr of definition# within the framawork of sound theory*

8eOP* SB&. T.imltmtion. Of $6* Sj-SOX

In particular, this study will undertake to show the extent to which defining practices in contemporary geometry textbooks and in books and other souroes related to the teaching of mathematics originate In the logical works of

Aristotle* Certain of these practices will then be criti­ cized by means of documented comments from the current mathematical and logical literature* This literature will then be used to present the various acceptable methods of defining* The relation of these methods to geometry is pointed out* The various types of information in this study will be gathered from the following sources: (1) Contemporary practices in geometry— high-school geometry textbooks, contemporary mathematics methods books, and articles from Journals (2) Origin of contemporary practices— the logical works of Aristotle and commentaries on these works (3) Mathematical and logical theory of definition— contenq>orary books and periodicals dealing with logic and mathematics The study will confine itself largely to textbooks and methods books used in the United States and to a few in Canada and Great Britain* Most of the mathematical and 5 logloal lltaratura used waa from th# Uhitad Stataa with a f#w important itam# from Germany, France, and Italy. The study was confined to demonstrat iv e plane geometry, con­ sidered as a science of space*

Definitions of Texas

The following terms are used so frequently In this study that It seems desirable to define them in this con­ venient place: ition. This refers to the method of defining developed §y Aristotle and illustrated at the beginning of this chapter. It involves stating the larger class of a thing and then its distinguishing character­ istic or property* The entire definition, according to Aristotle, gave the essence of a thing* Genus. This refers to the immediately larger class of a thing in the Aristotelian definition* Thus, "polygon** was thought to be the genus of "triangle* ** Differentia (pi* differentiae)* This refers to the distinguishing characteristic or essential property in the Aristotelian definition* Thus, "having three sides" was the differentia of "triangle." Aristotelian definition is referred to as definition "per genua et differentiam* " Definiens* This refers to the defining phrase* riAfThis refers to the term being defined* Denotation, extension. These terms will be used interchangeably to refer to the class of examples of a term being defined* intension. These terms will be used intexônangeably to refer to the property or set of proper­ ties which may be used for defining purposes* Arbitrary. With regard to definition, this word means "complete freedom of choice*" Arbitrary definitions are sometimes called stipulative definitions* Oonventlonal. With regard to definition, this word means "aooording to usage." Conventional definitions are laxieal definitions. NsBlnaJ^. This word with respect to definition, means "completely verbal.** In definition, it means a term is related by definition to a longer set of tezns. A nominal definition consists entirely of words and is about words. Real. In traditional logic a real definition is one where the existence of the term being defined has been established. Such definitions were presumably about things, rather than words. Postulational system. This means an organized, related body of knowledgeconsisting at the beginning of a set of undefined terms, postulates, axioms, and definitions with a sequence of theorems derived from these by deductive means. Meaning. This is a very aofciguous word with any one of a number of interpretations. One form of meaning of a word is when the examples to which the word refers are known. This is sometimes called the "denotative or extensional meaning." Another form of meaning of a word is when the properties of the thing or class of things to which the word refers are known. This is sometimes called the "connotstive or intensional meaning." Sometimes the meaning of a word like "water** is given by saying it is the substance formed when hydrogen and oxygen combine chemically. This would be called a genetic or causal meaning, but carries no denota­ tive or oonnotative meaning. A very important form of mean­ ing is conveyed operationally as follows: "Measuring a dis­ tance" means "to count unitsof length as they are laid down successively end-to-end." This paragraph does not exhaust the various interpretations of the word "mesning."

Preview of the Study

In Chapter II the following six practices will be traced to the works of Aristotle, mostly through Euclid: (1) Definitions should be in the form of genus et differen­ tiae, (£} Definitions should not use obscure language and should use terms which are prior and more intelligible. (3) Definitions should not oontain redundant or superfluous information, (4) Definitions must be reversible, (5) There is only one definition of a thing, (Ô) Definitions state what a thing is but not the fact that the thing exists* The manner and extent to which each praotice ooours in con-* temporary geometry textbooks will be shown* Not all of these practices are bad, and some become objectionable by virtue of the way they are handled. Chapter II will con­ clude with a survey end clasaificatlon of purposes of definition found stated in contemporary geometry books* Chapter III will present the material on definition to be found in each of twenty-four books dealing with the teaching of mathematics. Other related sources will also be surveyed. Chapter 17 will refute— with the aid of m o d e m mathe­ matical and logical literature— certain traditional and wide-spread defining practices to be found in contemporary geometry textbooks and methods books* In addition, using the same literature, Chapter 17 will present and discuss various modern ways of defining and show which of these ways can occur in geometry. Chapter 7 will consist of a summary and a list of logical and pedagogical recommendations* CHAPTER II

ARISTOTELIAN ORIGINS OF SOME DEFINING PRACTICES IN PLANE GEOMETRY TEXTBOOKS

Most of the prevailing praotioea oonoeming definitions in plane geometry textbooks have their origin in the works of Aristotle, a Greek philosopher and biologist who lived from 364 B.C. to 321 B.C. The first oomplete rendering of the elements of geometry as a postulational system was by Suolid while a professor of mathematios at the University of Alexandria, Egypt, o. 500 B.C. to o. 275 B.C. Little is known of Euclid's life prior to his Alexandrian period and one can only speculate whether he studied at the Athenian Lyceum, the school founded by Aristotle, or at the older Academy founded by Plato. Heath seems inclined toward the latter view: It is most probable that Euclid received his mathematical training in Athens from the pupils of Plato; for most of the geometers who could have taught him were of that school, and it was in Athens that the older writers of elements (Eudoxus, etc.), and the other mathematicians of whose works Euclid's Elements depend, had lived and taught.1

^Sir Thomas Heath, Euclid. The Elements, p. 2.

8 9 Uhtll he was about thirty years old, Aristotle also studied under Plato about a generation before Euclid, Aristotle's logical work influenced both schools and later prevailed over the entire western world. According to Heath, the logical structure of Euclid's Elements was com­ pletely Aristotelian down to minute details of the defining process,^ Aristotelian definition has been briefly defined in the first chapter and will shortly be elaborated. Down through the years, in spite of oLodifi oat ions by Legendre and the development of contemporary American textbooks in geometry, the principles of defining as well as the logical structure followed the pattern fixed by Euclid and remained Aristotelian, Criticism in this study will be directed toward an apparently blind obeisance to a tradition which has the following consequences: 1, An implicit denial of equally sound alternate methods of defining. 2, The assumption that figures have an essential property which must be used for defining while all other properties are deducible, 3, Assumes that figures belong to absolute classes rigidly ranked on the basis of simplicity, 4, Implies that definitions are true or false according to whether or not they state the essence of a figure.

2 Ibid,. p, 146, 10 Th# remainder of thie ohepter will present by direot quotation the defining practices advocated by Aristotle and shorn the nature and extent of these practices in contempo­ rary geometry textbooks# The following six practices found in contemporary geometry textbooks will be shown to occur directly in the works of Aristotle: (1) Definitions should be in the form of "genus et differentia" (class and difference)# (£) Definitions should not use obscure language and should use texms which are prior and more intelligible# (3) Definitions should not contain redundant or superfluous information# (4) Definitions must be reversible# (5) There is only one definition of a thing# (6) Definitions state what a thing is but not the fact that the thing exists# A seventh topic will be included in this chapter deal­ ing with purposes of definitions given by textbooks# One should note that Aristotle applied the above char­ acteristics to scientific definitions as opposed, for example, to dictionary definitions. 1. gl^aitio^i k# the fojrm "gen^ff et

According to Aristotle, a term being defined is the name of a class of things called a "species#** In a defini­ tion the species of a term must first be placed in the next higher class, which includes among its species the species of the term, and then the essential characteristics 11 (differentiae) whloh distinguish the partloular speoles from other speoles within the larger olass (genus) must be stated. Aristotle's adoption of this method was probably stimulated by his actlrltles as a biologist. Aooordlng to some historians, his former pupil, Alexander the Great of uaoedon, sent to Aristotle at the Lyoeum In Athens large numbers of specimens of flora and fauna oolleoted In his Asiatic campaigns. These provided Aristotle with an exper­ imental basis for developing a theory of classification and definition. This led Aristotle to the notion of absolute hierarchies of classes and the concept of the prior and posterior. In Aristotelian definitions, genus and species were any two successive classes where the former Immediately Included the latter. Membership by specific figures In the successive classes of genus end species was considered by Aristotle to be fixed and unalterable. Today, we know, for example, that the rectangle, rhombus, square, parallelo­ gram, and trapezia can be defined to be equal species of the genus **quadrilateral” (found In Euclid) or the rectangle, rhombus, and square can be defined so as to be species of parallelogram (many contemporary books). In general text­ books do not convey the Impression that there are different possible ways to classify figures and therefore are Instru­ mental In continuing this outmoded tradition. xz

In Suolid the "genus et differentia" method of defining is illustrated hj the definition of triangle, whioh Suolid oalled "trilateral figure": "Reotilinear figures are those oohtained by straight lines, trilateral figures being those oontained by three • • • straight lines#In this case, the speoies "trilateral figure" is placed in the genus "reotilinear figures" by means of the relative pronoun "those," and the differentia is designated by the phrase "oontained by three straight lines." The fact that the figure is closed is part of Suolid*s definition of figure: "A figure is that whioh is oontained by any boundary or boundaries."^ Today praotioally every textbook presents this definition in a slightly altered, but Aristotelian form: "A triangle is a polygon having three sides." The fact that a triangle is a closed figure and consists of straight lines is included in the definition of "polygon." While this definition serves in geometry, it by no means is the last word on the definition of "triangle." The following two alternatives are possible: 1. Aristotelian, but different : "A triangle is a polygon whose angles sum to half a turn." From this definition all the other properties of a triangle oould be deduced.

^Ibld.. p. 187.

*Ibid.. p. 182. 13 £• Non-Arlstotellan and dlf farent : **Whan thraa non-oolinaar points ars Joined by straight lines in space, the resulting figure is called a tri­ angle* ** This type is later discussed in Chapter 17 as a genetic definition* a one of the problems confronting Aristotle as he developed his method of definition can be seen in the fol­ lowing lengthy passage dealing with the subject "man": We must first inquire about definitions arising out of divisions* There is nothing in the defini­ tion ezoept the first-named genus and the differ­ entia* The other genera are the first genus and along with this the differentiae that are taken with it. e*g* the first may be *animal,' the next ^animal which is two-footed,' and again 'animal whioh is two- footed and featherless,' and similarly if the defini­ tion includes more terms* And in general it makes no difference whether it Includes many or few terms,— nor, therefore, whether it includes few or simply two; and of the two one is differentia and the other genus, e*g* in 'two-footed animal* 'animal* is genus, and the other is differentia* If then genus abso­ lutely does not exist apart from the species whioh it as genus includes, or if it exists but exists as matter (for the voioe is genus and matter, but its differentiae make the kinds, i*e* the letters out of it), clearly the definition is the formula whioh comprises the differentiae* But it is also neoessary in division to take the differentia of the differentia; e*g* 'endowed with feet* is a differentia of 'animal'; again we must know the differentia of 'animal endowed with feet' qua endowed with feet* Therefore we must not say, if we are to speak rightly, that of that whioh is endowed with feet one part has feathers and one is featherless; if we say this we say it through inca­ pacity; we must divide it qua cloven-footed or not- oloven; for these are differentiae in the foot; cloven-footedness is a form of footedness* And we always want to go on so till we come to the speoies that contain no differences. And then there will be as many kinds of foot as there are differentiae, and the kinds of animals endowed with feet will be equal in number to the differentiae * If then this is so, clearly the last differentia will be the substance 14 of the thing and its definition, since it is not right to state the same things more than once in oar definitions; for it is saperfluous. And this does happen; for when we say * animal whioh is endowed with feet, and two-footed* we have said nothing other than *animal having feet, having two feet*; and we divide this by the proper divi­ sion, we shall be saying the same thing many times— as many times as there are differentiae. If then a differentia of a differentia be taken at each step, one differentia— the last- wili be the form and the substemoe; but if we divide according to accidental qualities, e.g. if we were to divide that which is endowed with feet into the white and the black, there will be as many differentiae as there are processes of divi­ sion. Therefore it is plain that the definition is the formula which contains the differentiae, or, according to the right method, the last of these.° The previous passage conveys a notion which is typical of Aristotle— that of an analytical regress from that which is posterior to that whioh is more prior, ultimately arriv­ ing at the substance or essential nature of a thing. Heath in his introduction to his translation of Euclid makes the following comments about "prior" and "genus and differ­ entiae": Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things defined. . . . But the terms "prior" and "better known" are, as usual suscepti­ ble of two meanings; they may mean (1) absolutely or logically prior and better known, or (S) better known relatively to us. . . . It follows therefore that a thing should be defined by means of the absolutely

^Aristotle, Itotaphvslca. Pp. 1037^ - 1038^ 30. (See bibliography for translation used*) 15 prior and not the relatively prior. In order that there may be one sole and Immutable definition* This Is further enforoed by referenoe to the requirement that a good definition must state the genus and the differentiae, for these are among the things whioh are. In the absolute sense, better known than, and prior to, the speoles.^ It can be seen from the above quotation that the genus and differentiae method of defining Is a means of following Aristotle*s rule that definitions should be In terms of what is prior and more intelligible* Furthermore, this method of defining Is olosely related to the oonoept of essenoe and substanoe which Is fundamental to the whole Aristotelian theory of definition* By substance, Aristotle meant: Substanoe, in the truest and primary and most definite sense of the word, Is that which Is neither predloable of a subject nor present In a subject; for Instance the Individual man or horse* But In a secondary sense those things are called substances within which, as species, the primary substances are Included; also those which, as genera, include the species* For Instance, the Individual man Is Included In the species *man,* and the genus to which the species belongs Is * animal * ; these, therefore— that Is to say, the species *man* and the genus * animal* — are termed secondary substances*7 It Is Important at this point to note Aristotle's dis­ tinction between definition," "essence," "property," "genua," and Occident": V/e must now say what are 'definition, ' 'property', 'genus,* and 'accident*' A 'definition*

^eath, Euclid, the Elements* p* 147*

^Aristotle, Oategorlae. pp. 2* 11 - 2* 18* 16 Is a phrase signifying a thing's assanoa. . . • A ’property* Is a pradloata whioh does not Indloata the assanoa of a thing* but yet belongs to that thing alone, and is pradioatad oonvartlbly of it* Thus it is a property of man to be capable of learning grammar; for if A be a man, than ha Is capable of learning grammar, and if ha be capable of learning grammar, ha is a man. . . . A 'genus* is what is predicated in the category of essence of a number of things exhibiting differ­ ences In kind* We should treat as predicates in the category of essence all such things as it would be appropriate to mention in reply to the question, *What is the object before you?* * . . An 'accident* is (1) something which, though it is none of the foregoing— i.e* neither a definition nor a property nor a genus— yet belongs to the thing: (g) something whioh may possibly belong or not belong to any one and the self-same thing, as (e.g.) the 'sitting posture* may belong or not belong to some self-same thing.8 We can see from the foregoing quotations from Aristotle that the "genus and differentiae” method of defining served two purposes: (1) it gave some assurance that the defini­ tion would be in terms absolutely prior, and (8) it pro­ vided a formal means for finding the essential nature of a term* A critique of these purposes will be provided in Chapter 17 of this study* Let us examine now the place that the above method of defining has in modem geometry textbooks. The following eighteen, widely-used textbooks state specifically that definitions in geometry should be in the

®Aristotle, Tonics, pp. 101^ 37 - 102^ 8, 17 form of "genus et differentlee," or olass and difference : 1. Auerbach and Walsh, Plane Geometry, p. S4* £♦ Austin, A Laboratory Plane Geometry, p. 107, 3, Barber and Hendrix, Plane Geometry, p. 5. 4» Blaokhurst, Humanized Geometry, p. 82, 5, Halsted, Elementary Geometry, p. 2. 6, Henrlol, Elementary Geometry, p, 2. 7, Keniston and Tully, Plane Geometry, p. 56, 6, Leary and Shuster, Plane Geometry, p. 487. 9, Leonhardy, Joseph, lloLaary, New Trend Geometry. P* 12, 10, Mallory and Oakley, Plane Geometry, p, 1, 11, Rosskopf, Aten. Reeve, Mathematios A Second Course, p, 13, 12, Sohnell and Crawford, Plane Geometry, p, 86, 13, Skolnik and Hartley, Dynamic Plane Geometry. p. 20. 14, Smith, Reeve, Morss, Text and Tests In Plane Geometry, p, 38, 15, Stone and Mallory, M o d e m School Geometry, p, 11, 16, Trump, Geometry - A First Course, p, 45, 17, V/elchons sind Krlokenberger, Plane Geometry. (1952) p, 30. 18, r/elkowltz, Sltomer, Snader, Geometry. Meaning and Mastery, p, 36, In addition to the foregoing textbooks which advocate explicitly the genua et differentiae, or olass and differ­ ence, method of defining, there are many more that say 18 nothing about the method, but actually use it in most of their definitions. Out of the sixty-three textbooks examined, none proposed any method of defining whioh differed from the Aristotelian method described in this chapter. It is therefore safe to conclude from this evi­ dence that the Aristotelian method of defining by class and difference is firmly entrenched in contemporary geometry textbooks to the almost total exclusion of alternate methods.

2. Definitions should use terns which are prior and more intelligible and not in obscure language. Prior and more intelligible terms. In the following passages Aristotle presents his point of view on this praotice whioh occurs with a variety of modifications in the textbooks examined: Whether, then, a man defines a thing cor­ rectly or incorrectly you should proceed to examine on these and similar lines.9 Bat whether he has mentioned and defined its essence or no, should be examined as follows: — first of all, see if he has failed to make the definition through terms that are prior and more intelligible, for the reason why the defini­ tion is rendered is to make known the term stated, and we make things known by taking not any random terms, but such as are prior and more intelligible.

g Aristotle is referring to previous passages where he discusses ambiguity, metaphorical expression, and redundancy. 19 as la dona in danonstrations (for so it is with all taaohing and laaming) ; aooordingly it is olaar that a man who doas not daflna through tarns of this kind has not dafinad at all«10 In tha following passaga it is quits olaar which ona of two intarpratations of "prior and mora intalligihla" Aristotla prafars: Tha Stat amant that a dafinition has not baan mada through mora intalligihla tarns may ba undar- stood in two sansas, aithar supposing that its tarms ara absolutaly lass intalligihla, or suppos­ ing that its tarns ara lass intalligihla to us: for aithar sansa is possible* Thus absolutaly tha prior is mora intelligible than the posterior, a point, for instanoa, than a line, a line than a plana, and a plana than a solid; just as also a unit is mora intelligible than a number; for it is tha primus and starting-point of all number* Like­ wise, also, a letter is more intelligible than a syllable* Whereas to us it samatines happens that the converse is tha case: for tha solid falls under perception most of all— more than a plane— and a plana mora than a line, and a line mora than a point ; for most people leam things like tha former earlier than the latter; for any ordinary intalliganoa can grasp them, whereas tha others require exact and exceptional understanding* Absolutely, then, it is better to try to make what is posterior known through what is prior, inasmuch as such a way of procedure is mora scientific Of course, tha modem pragmatic view would disagree with tha possibility of making any distinction between absolute degrees of intelligibility and degrees of intel­ ligibility relative to human understanding* In fact, ona

10 Aristotle, Topics. 141® . 22-52* ^^Ibid*. p. 141^ 3-16* so of the arguments today for teaohing solid geometry along with plane geometry is based on the belief that people are more acutely aware of solids earlier in life than they are of planes and that the only basis for ranking intelligi­ bility is psyohological, rather than absolute* Aristotle proceeds by making a psychological concession and then connects the practice of using terms which are "prior and more intelligible" with the "genus et differen­ tiae" method of defining: Of course, in dealing with persons who cannot recognize things through terms of that kind, it may perhaps be necessary to frame the expression through terms that are intelligible to them# Among definitions of this kind eure those of a point, a line, and a plane, all of which explain the prior by the posterior; for they say that a point is the limit of a line, a line of a plane, a plane of a solid# One must, howerer, not fail to observe that those who define in this way cannot show the essential nature of the term they define, unless it so happens that the same thing is more intelligible both to us and also absolutely, sinoe a correct definition must define a thing through its genus and its differentiae, and these belong to the order of things which are absolutely more intelligible than, and prior to, the species. Concerning the "psychological concession" in the first part of the above quotation. Heath considers this to be the justification for Euclid's supplementing Definition 3, Book I, of the Elements:

IS Aristotle, Tonica. 141^ 16-27# SI • • • and we have here the best possible explanation why Euelid supplemented his definition of a point by the statement in I, Def• 3 that the extremities of a line are points and his definition of a surface by I. 5#f. 6 to the effect that the of a sur­ face are j^^ges. The supplemeniary explanations do in fact enable us to arrive at a better understanding of the foxmal definitions of a point and a line respec­ tively • • .15 The Aristotelian rule that definitions should be in terms prior and more intelligible appears in nineteen m o d e m geometry textbooks, but not always according to the intent of Aristotle. The variety of interpretations can be seen In the follwing citations: 1. Birïchoff and Beatley: Sinoe a good definition uses only ideas that are simpler than the idea that is being defined, some of the simplest ideas must remain undefined.1* ^ h e notion of undefined terms and the principle of the undesirability of circularity will be discussed after the present oitatlonsj^ B. Cook: The principal words in any careful statement of reasoning are of two kinds: (i) those left without specific meaning, e.g. "point** or "line" in geometry, (ii) those which are explicitly defined, e.g., a "circle" as a locus of points, "(iiy* are defined in terms of "(i)."15 3. Henrioi: IVhenever we introduce a new figure we have to define it. This can only be done by

^^eath, Euclid. The Elements, pp. 147-148.

^^Birkhoff and Beatley, Basic Geometry, p. 13

15 Cook, Geometry for Today, p. 115. 28 reference to simpler figures out of which it is formed 4. Herberg and Orleans: The above descriptions of straight line, point, and plane are not really complete définitifs of the terms, sinoe the des­ criptions involve ideas more complicated than the texms themselves$17

5. Keniston and Tully: In a logical chain of rea­ soning, however, such as geometry, the defini­ tions are phrased so as to be entirely clear if preceding terms have been mastered. For example, consider the terms line, broken line, polygon. triangle. Each of these was defined in terms of the one immediately preceding it, with the excep­ tion of the first word, line, which was undefined because of the difficulty of explaining it in simpler tezms than itself.18 6. Hawke3, Luby, Touton: A certain order is often essential in stating definitions. For example, the word degree cannot easily be defined before the terms angle and right angle are understood.1^ 7. Leonhardy, Joseph, McLeary: The term must be defined in terms more familiar than the one being defined.80

8. Kyberg: "^hile we all know what a plane is, most of us might find it difficult to give an exact definition of it. In the study of geometry it is necessary to give exact definitions of every figure. We could define plane as follows: A plane is a surface such that a straight line joining any two of its points will lie

^^Henrioi, Elementary Geometry, p. 33.

l^Herberg and Orleans, A New Geometry, p. 3. ^®Keniston and Tully, Plane Geometry, p. 56. l^Hawkes, Luby, Touton, New Plane Geometry, p. 22. 20 ^Leonhardy, Joseph, and McLeary, New Trend Geometry, p. 12. S3 wholly on that surface. • • • This definition of a plane assumes that we know or have defined what line Is. If we first agree that "A stral^t line Is the shortest path between two points'* then the above definition of a plane Is satisfactory. In defining line, the word oath Is used rather than the word dlstyioe because distance means the length of the line. v/e might now ask: How do we know what path Is the shortest? It Is Impossible, hear ever, to define every word. Some simple words must be left undefined. In each topic of this book, therefore, we must agree on certain fundamental facts and then base the definitions on these facts and on previous definitions.^^ 9. Rosskopf, Aten, Reeve: A good definition must also be simpler than the term defined.^8 10. Schnell and Orawford: A definition should use only previously defined terms. 11. Slgley and Stratton: The foundations of geometry depend upon the definitions of basic terms which make use of these elementary terms (point, line, plane). . . . It Is very necessary that they (definitions) be expressed concisely and In language that will be easily understood.^^ IS. Skolnlk and Hartley: A definition states the meaning of a word or term by using other words or terms. Uhless we understand these other words, the definition Is of no use to us* These other

®%yberg. Plane Qeometry (enlarged), p. 3. ^^osskopf, Aten, Reeve, Mathematics. A Second Course, p. 14. * S3 Schnell and Crawford, Plane Qeometry. p. 86. ^^Slgley and Stratton, Plane Geometry. p. S and p. 17. 24 words must be looked up until we find their meaning given in words that we do understand. All definitions depend on the fact that we do know the meanings of some words without having to look them up. . . . Having agreed on the undefined terms, we then define others using the terms that we had agreed to accept without deflnition.26 13. Smith (Eugene R.}: Some things are of such simple nature that it is difficult, if not impossible to define them in terms still simpler, and in such cases an explanation in regard to them may well take the place of a definition. 14. Stone and Mallory; The terms used must be simpler than the term defined.^" 15. Strader and ^hoads: In a good definition the term defined must be described in better known terms than the thing itself.B8 16. Trump: Good definitions are always as brief as possible and are stated in words as precisely understood as possible. In defining any word we must use other words. You have had the experience of looking up a word in the dictionary and then having to look up the meanings of still other words used in the definition. In defining a new word we try to define it in terms of words previously defined.S® 17. l/elchons and Kirckenberger: A definition should be expressed by words which are simpler

85 Skolnik and Hartley, Dynamic Plane Geometry. p. 20,

^®Eugene R. Smith, Plane Geometry, p. 22. ©7 Stone and Mallory, Modem Plane Geometry, p. 11. Strader and Rhoads, Plane Geometry, p. 20. 29 Trump, Geometry - A First Course. p. 46, p. 47. £5 than the term desoribed*^^ 18. V/elkowltz, aitomer, Snader: The teohnioal terms that are used should be only those that have been previously defined.^^ To summarize, we should try, In defining a word, to place it in a more inclusive class. . . . This class, in turn, is Included in a larger class, and so on. This successive Inclusion of one class within another exemplifies the property of sequence in defini­ tions. For exanq>le the following terms form a sequence; geometric figure, polygon, triangle, right triangle, isosceles right triangle. 19# vYorkman and Cracknell: All accurate knowledge is based upon certain fundamental concepts, that is to say upon certain ideas which cannot be explained by reference to any more simple ideas.33 An analysis of the above statements indicates that four textbooks held that definitions should be in terms of "simpler ideas," four in terms of "simpler terms," four in terms of "more familiar terms," one in terms of "simpler figures," four in terms of "previously defined terms," and two in terms of "undefined terms," Particular attention should be paid to the variation of terminology such as, ideas, terms, and figures ; and simpler, more familiar, and

^^Welchons and Erickenberger, Plane Qeometry. p. 30.

^^V/elkowitz, Si tome r, Snader, Geometry. Meaning and Mastery, p. 36.

^^Ibid.. p. 38. 33 Workman and Cracknell, Geometry Theoretical and Practical, p. 65. S6 prevlottsly defined. The varied terminology la auggeatlve of the oenturles-old oonfusion between real (of things) and nominal (of names) definitions. These statements were also examined as to whether they explicitly or Implicitly expressed the Aristotelian dictum that definitions should be In terms of the absolutely prior and more Intelligible or whether they should be In terms of the relatively prior and more Intelligible, that la, more Intelligible to the Individuals for whom the definition Is Intended. When the quotation In Its wording conveyed the Impression of an Independent hierarchy of complexity which the definition must take Into account, the former dictum (Aristotelian) was Identified with the quotation for the purposes of the follcsrlng classifications of texts. When the quotation conveyed the Intention that human understand­ ing and familiarity with terms controlled the form of the definition, the latter point of view (relatively prior and Intelligible) was accredited to the quotation. (Quotations 5, 4, 5, 6, 8, 13, 18 and 19 appeared to express preference for the definition In terms of the absolutely prior and Intelligible. Quotations 7, 11, IS, 15, and 16 seemed to prefer definitions In terms relative to Individual famil­ iarity and understanding. (Quotations 1, £, 9, 10, 14, and 17 were not easily Identifiable under either position on the basis of the Internal evidence of their wording alone. 27 Obaoure Ianggage. In regard to the rule against the use of obsoure language Aristotle is quite speoifio and analyzes the prob* lem with some detail: Inoorreotness falls into two branches: (1) first, the use of obsoure language (for the lan­ guage of a definition ought to be the very clear­ est possible, seeing that the whole purpose of rendering it is to make something known); (2) seoondly, if the expression used be longer than is necessary: for all additional matter in a defini­ tion is superfluous. . . . One commonplace rule, then, in regard to obscurity is. See if the meaning intended by the definition involves an ambiguity with any other, e.g. 'Becom ing is a passage into being,' or 'Health is the balance of hot and cold elements.' Here 'passage' and 'balance' are ambiguous terms: it is accordingly not clear which of the several possible senses of the term he intends to convey. Likewise also, if the term defined be used in different senses and he has spoken without dis­ tinguishing between them . . . Another rule is, See if he has used a meta­ phorical expression, as, for instance, if he has defined knowledge as 'unsupplantable,' or the earth as a 'nurse,* or temperance as a 'hamony.' For a metaphorical expression is always obscure. . . . Sometimes a phrase is used neither ambiguously, nor yet metaphorically, nor yet literally, as when the law is said to be the 'measure' or 'image* of the things that are by nature just. Such phrases are worse than metaphor . . Of all the Aristotelian practices with regard to definition, the rule against obscure language as stated and analyzed in the above quotation is probably the most

^^Aristotle, Tonica. 139^ 10 - 17. 28 defensible in the light of modern logical and pedagogical theory. Yet of the ten modern textbooks which list specific rules for definition, got one has a specific rule against the use of obscure language* The ten textbooks, including a reference to the page containing the specific rules, are as follows: 1* Keniston and Tully, Plane Geometry, p* 56. 8. Leonhardy, Joseph, and McLeary, New Trend Geometry, p . 12 • 3. Rosskopf, Aten, and Reeve, Mathematics A Second Course, p. 13. 4. Schnell and Orawford, Plane Geometry. A Clear Thinking Approach, p. 86-6V. 5. Skolnik and Hartley, Dynamic Plane Geometry, p. 20. 6. Stone and Mallory, Modem Plane Geometry, p. 11. 7. Strader and Hhoads, Plane Geometry, p. 20.

a. Trump, Geometry - A First Course, p. 45. 9. V/elchons and Erickenberger, Plane Geometry, p. 30. 10. >‘/elkowitz, Sitomer, and Snader, Geometry. Meaning and Mastery, p. 36. Of course many of these may have assumed that the rule against obscurity was implied by the rule demanding the use of terms which are prior and more intelligible, simpler, or previously defined. (See quotations pages 21-25.) But it can readily be seen that the latter rule may be followed while at the same time the foxmer is violated. 29 Cohan and Nagel make an important point in regard to the relation between unfamiliar and obsoure language. The injunction that the definiens should not be obscure expresses the psychological motives for definitions. . . . However, the occurrence in the definiens of terms unfamijiejr to most readers does not make the definition obscure. In physics, the defini­ tion of "the action of a system of particles," is given as "the sum for all the particles of the mean momentum for equal distances multiplied by the distance traversed by each particle." This definition is by no means obscure to the compe­ tent student of analytical dyneanics. whatever it may appear to be to the untrained.3* Definition through opposites, circular definition, and définitton by coordinate species. According to Aristotle, violation of the rule for the use of prior terms results from one of these three defec­ tive definitions: Of the failure to use terms that are prior there are three foms: (1) The first is when an opposite has been defined through its opposite, e.g. good through evil: for opposites are always simultaneous by nature. . . . (2) Another is— if he has used the term defined itself. This passes unobserved when the actual name of the object is not used . . . (3) Again, see if he has defined one coor­ dinate member of a division by another, e.g. 'an odd number' as 'that which is greater by one than

^^Cohen and Nagel, ^ Introduction to Logic and Scientific Method. p. 241. 30 an even number»* For the oo-ordinate members of a division that are derived from the same genua are simultaneous by nature, and 'odd* and *even' are suoh terms: for both are differentiae of number. A few passages later, Aristotle qualifies the first of these defective forms: **Xt may be that in some oases the definer is obliged to employ a negation as well, e.g. in defining privations# For * blind* means a thing which cannot see when its nature is to see»**^ The first two definitions of Suclid are like type (1): **A point is that which has no part. **^ **A line is breadth- less length.**^^ Regarding their negative aspect. Heath comments: Euclid's definition itself is of course practically the same as that which Aristotle's frequent allusions show to have been then current. . • . . . the definition has been over and over again criticized because it is purely negative. . . . he (Aristotle) seems to accept as proper the negative element in the definition of a point, sinoe he says (De anima III. 6, 430 b 80) that "the point and every division, and that which is indivisible in this sense, is exhibited as privation." . . . This definition (of line) may safely be attributed to the Platonic School, if not to Plato himself. Aristotle (Topics vi. 6, 143 b 11) speaks of it as open to objection because it "divides the genus by negation," length being necessarilv either breadth- less or possessed of breadth . •

Aristotle, Tonica. 148» 88 - 148b 10. ^^Aristotle, Tonica. 143^ 3 3 - 3 6 . 38 Heath, Euclid. the Elements. Vol. 1, p.155. ^^Ibid.. p. 158. ^ Ibid.. p. 156, 158. 31 Since none of the m o d e m tezthooks examined warn against the defective definitions described in (1) and (3) above, a separate category will be used only for "circular definitions." "Circular definitions." Only one mo d e m textbook men- tioned circular definitions specifically, using the same example that Aristotle employed: Wilkowitz, Sitomer, Snader: We must be care­ ful in a series of definitions not to violate the principle of sequence. For example consider this set of definitions: The sun is a steur which shines in the daytime. Daytime is the time when the sun shines. Here we have committed the error of forming what are called circular definitions.^1 Just as in Aristotle, circularity is used in two senses by modem logicians, namely, circularity in definition and circularity in proof: In particular, the constant to be defined, or any expression previously defined with its help, must not occur in the definiens; otherwise the defini­ tion is incorrect, it contains an error known as a vicious circle in the definition (just as one speaks of a vicious oimle in the nroCT. if the argument meant to esxablisn a certain theorem is based upon the theorem itself, or upon some other theorem pre­ viously proved with its help). Although the term "circularity" is applied to both

^^elkowitz, Sitomer, Snader, Geometry Meaning and Mastery, p. 37. ^®Alfred Tarski, Introduction to Logic, p. 35. 38 definitions and proofs, the term "petltlo prlnclpii,** or "begging the question, " seems to X«fer aalaly to proofs. Cohen and Nagel oonment upon this and the Inherent olrou- larlty of soienoe: It la also a fallaoy to olalm to have proved a proposition at Issue if It has been smuggled, in some more or less disguised form. Into our premises. (This Is called begging the question, petltlo frlnolnll.) To assume a proposition as a premise 8 not tne same as to prove It. 1. A special form of this fallacy la called arguing In a circle. It consists In Introducing Into our premises a proposition that depends on the one at Issue. Thus It would be arguing In a circle to try to prove the Infallibility of the Koran by the proposition that It was composed by God's prophet (Mahomet), If the truth about Mahomet's being God's prophet depends upon the authority of the Koran. There Is a sense In which all science Is circular, for all proof rests upon assumptions which are derived from others but are justified by the set of consequences which are deduced from them. Thus we correct our observations and free them of errors by appeal to principles, and yet these prin­ ciples are justified only because they are In agree­ ment with the readings which result from experi­ ment. . . . But there is a difference between a circle consisting of a small number of propositions, from which we can escape by denying them all or setting up their contradictories, and the circle of theoretical science and human observation, which Is 30 wide that we cannot set up any alternative to It. 43

Although only one modem textbook— that by Welkowltz, Sitomer, and Snader, Geometry. Meaning and Mastery— makes explicit the rule against circularity, many Infer the rule

^^Morris R. Cohen and Emest Nagel, An Introduction to Logic and Scientific Method. p. 379. 33 Implloltly from aom» form of the Aristotelian diotim that we must define the posterior in terms of the prior# The many forms of this dictum among the nineteen textbooks which express it have been enumerated on page 28. Sequences of definitions. This dictum, or rule that we must define by means of terms which are prior and more intelligible, manifests itself in modern geometry textbooks by statements about sequences or hierarchies of definitions. There are actually three separate types of sequences in Euclid and modem geometry textbooks«( 1) sequences of theorems which orig­ inate in the foundations consisting of primitive terms, definitions, axioms and postulates; (2) sequences of con­ structions (called "problems" in Euclid) which originate in the fundamental postulates of constructions; and (3) sequences of definitions which originate in the primitive or undefined terms. In different modem textbooks each of these separate types of sequences occur with variations of order. The possibilities of legitimate variation in the order and beginnings of such sequences have been denied by Aristotle, Euclid,and their successors for centuries. Even today, after the revisions of Legendre, the development of non-Euclidean geometries, the growth of postulational set theory, the proposals of Nunn and the British Mathematical 34

Assoolation,^^ the work of Blrkhoff and Beatley, and the variations in sequenoes currently exhibited in textbooks, the concept and possibilities of legitimate variation in sequences is not widely realized either in current textbooks or methods books. Many textbooks today convey to students the impression that sequences are invariant and absolute. The following textbooks make specific reference to sequence in definitions: 1. Blackhurst: Having defined a polygon, it becomes the classification by means of which we may define triangles, quadrilaterals, etc. For example, A triangle is a polygon having three sides only. By means of a triangle as a class, we may define scalene triangles, equilateral triangles, acute triangles, etc.^® 2. Henrici: We have in this book only to treat of classes relating to geometry, principally of figures. Whenever we introduce a new figure we have to define it. This can only be done by reference to simpler figures out of which it is formed. These simpler figures, again, are defined by aid of still simpler ones, till at last we arrive at the simplest figures possible, which from their very sim­ plicity cannot be defined geometrically.

^ote: It is very questionable whether rectangle, rhombus, and square could be ranked in order of absolute simplicity as suggested by Henrici^

*^ h e Teaching of Geometry in Schools. A Report pre­ pared for the Uathematioal Association. London : O. Bell and Sons, Ltd., 1923. Pp. 35-44, 56-59.

Herbert Blackhurst, Humanized Geometry, p. 82. ^^enrici. Elementary Geometry, p. 33. 35 3. Keniston and Tully, Plane Geometry, p. 56. 4. Hawkes, Luby, Touton, New Plane Geometry, p. £&• 5* Nyberg, Plane Geometry, p. 3. 6« Symon: The present work is a modified reprint of the earlier book and follows rigidly the sequence and the reoonmendations of the I .A. A.M. and E.I. 3.47

^ote: Although "sequenoe" probably refers to theorems here, the educational implications of the word "rigidly" in this context do not appear wholesome^ 7. Skolnik and Hartley: Because a definition des­ cribes a class within a class, it is advisable to give definitions in a definite order or sequence.49

8. Trump: A square is a rectangle with two adjacent sides equal. A rectangle is a parallelogram with one right angle. A parallelogram is a quadrilateral with each pair of opposite sides parallel. A quadrilateral is a polygon with four sides. A polygon is a closed broken line lying in a plane. A broken line is a series of straight line segments joining successive end joints of the segments and not lying in a straight line. A straight line segment is the portion of a stral^xt line between two points of the line. To continue this series of definitions we would have to define straight line and point. Yfe

47 I.A.A.M. stands for Incorporated Association of Assistant Masters. E.I.S. stands for Educational Institute of Scotland.

^®Symon, The New Plane Geometry Complete, preface, p. v. 49 Skolnik and Hartley, Dynamic Plane Geometry, p. S£. 36

must, In the nature of things, he content to stop somewhere#50 g# v/elkowitz, Sitomer, Snader, Geometry. Mmanin^ and Mastery, p. 36, Undefined terms. Just as the Aristotelian rule that definitions should be in prior and more intelligible terms led to the rule against circularity and to the notion of sequences of definitions, so it also leads to the concept of undefined terms. There is a rather widespread Impression that the necessity for undefined terms in geometry textbooks is a recent development in mathematics pedagogy in response to mathematical developments on the nature of postulational systems# This false impression is strengthened by textbook trends from 1850 to the present# The false impression of the recency of the concept of undefined terms in plane geometry is reflected in the fol­ lowing statement from a doctoral study by Shibli: Thirty years ago the texts gave elaborate definitions of all the terms used. , • # The tendency in recent times is to leave elementary concepts undefined#51 Another quotation in the same vein is the following by

50Trump, Geometry - A First Course, p# 48#

®^J, Shibli, Recent Developments in the Teaching of :, (ig32) p# 85# 37 Kinney and Purdy: To simplify the work, there has been an inoreasing tendency to develop certain concepts intuitively, and thus to avoid explicit statement of definition whenever possible. For example, Suolld gave a rather involved definition of straight line that m i ^ t not be understood by some people who know full well what a straight line is.°^ Actually, the concept of the necessity for undefined terms is not entirely a modem development but originated with Aristotle as the following comments by Heath indicate The difficulties connected with the definitions of the most elementary things in geometry, the point and the line, could hardly be more lucidly ?ut than they are in the long passages just quoted. Topics, 143 b 11 - 144 a 4) Definitions should, in the first place, be in terms of things absolutely prior, or prior in the order of thought, to the things defined. There being nothing in geometry prior in the absolute sense to a point, a point must be defined, if at all, either by means of a posterior term or by negation. The first method is illustrated by the definition of a point as an extremity of a line, the second by Euclid's defini­ tion of a point as 'that which has no part.' . . . The difficulty in Euclid's definition is the very one pointed out by Aristotle in 142 b 85 (Topics); the words 'that which' in 'that which has no part' almost invite the question, what is it that you mean by 'that*? As Aristotle says, when you define a body as 'that which has three dimen­ sions, ' you do not say what it is that has three dimensions; you do not state a genus as you ought to do. But what would the genus bein this case? Aristotle does not say. Periiaps 'figure' might be suggested; but there is no proper definition of 'figure,' and we are no better off.53

®®Kinney and Purdy, Teaohing Mathematics in the Secondary School, p. 189. 53 Heath, Mathematics in Aristotle, pp. 86-89. 38 Of course Aristotle defended the ooneept of indefin­ able terms on the basis of his metaphysical system of absolutes ; Nothing, then, which is not a species of a genus will have an essence— only species will have it, for in these the subject is not thought to partici­ pate in the attribute and to have it as an affec­ tion, nor to have it by accident ; but for every­ thing else as well, if it has a name, there will be a formula of its me suing— viz. that this attribute belongs to this subject; or instead of a simple formula we shall be able to give a more accurate one; but there will be no definition nor essence. In contrast with this, the m o d e m mathematical atti- tude--aocording to Robinson— is that indefinability is relative; If an idea cannot be analyzed into any function of a given set of primitive ideas, it is indefinable relative to that set. This is a perfectly real sort of indefinability, which mathematicians investi­ gate with as much care as they investigate whether a certain proposition is 'deoidable* on a certain set of postulates. But two points need to be remem­ bered about this sort of indefinability. First, it is relative to a set of ideas; and an idea which is indefinable relative to one set of primitives may be definable relative to another.®® Robinson also points out that indefinability is a notion that applies to the construction of logical systems and not to the lexical definitions one finds in diction­ aries:

®^Aristotle, Metanhvsica. 1030* 12-17. 55 Richard Robinson, Definition, p. 196, 39 In oonstruotlng logical systems, people asually wish that as many as possible of the simple symbols In the system shall have their meaning uniquely determined by the propositions in the system, and they express this by saying they want to minimize the number of 'undefined terms* In the system. In this enterprise we cannot avoid either having some * undefined terms* or making the system a circular set of definitions. If this Is what Is unambiguously meant by the statement that 'there must be Indefin­ able words,* It Is true; but It has nothing to do with lexical definition. Lexical definition Is our name for the enterprise of teaching some man the meaning actually borne by some word In some society; and there are no lexically Indefinable words for the simple reason that It Is possible to teach a man every word in the language. Before the treatment accorded the concept of unde­ fined terms In geometry textbooks Is discussed, a final comment by Robinson on the matter Is particularly illuminating; In a work of logic or mathematics the terms called undefined are often defined In our sense; that Is, the author means something by them and he uses expressions from which a suitable reader can dis­ cover what he means by them. But he calls them undefined terms because the expressions which explain their meaning do not consist wholly of terms belonging to his system, but are drawn at least In part from the general stock of common language. If a logical or mathematical system Is to be Inter­ preted, that Is, applied to some reality other than Itself, every term In It must be defined. This will be done by giving each of the 'undefined terms' what Is sometimes called an 'interpretative* definition In this connexion, that Is, simply a nominal definition In the present sense, a state­ ment connecting the term to something else as that which the term Is to mean. Or. Sigmund Koch called

^*^R1 chard Robinson, Definition, p. 44. 40 it 'ooordinating definition* in the Pgyoholo^ejJ. Review for 1941, heoause it correlates empirioal oonstruotions to the formal terms of the postulate- set, and thus transforms an abstract system into an empirioal one.57 The concept of undefined terms was investigated in modem textbooks on the basis of what reasons, if any, are given for them. The following citations are therefore grouped under the various categories of reasons offered. Sinoe books that mention undefined terms without reasons are listed in the last category, those books which do not appear in the following citations do not mention undefined terms in any way. Undefined terms are terms left without meaning. 1. Oook: The principal words in any careful state­ ment of reasoning are of two kinds: (i) those left without specific mejming, e.g. "point" or "line" in geometry . . Uhdefined terms are terms (Ideas or things) too simple or fundamental to^ e defined in simpler terms. 1. Birkhoff and Beatley: Since a good definition uses only ideas that are simpler than the idea that is being defined, some of the simplest ideas must remain undefined.5* fi. Breslich, Plane Geometry, p. 47. 3. Henrioi, Elementary Geometry, p. 33.

57 Richard Robinson, Definition, pp. 44-46,

®®Gook, Geometry for Today, p. 115. 59 Birkhoff and Beatley, Basic Geometry, p. 13< 41 4. Keniston and Tolly, Plane Geometry, p, 56. 5. Nyberg, Plane Geometry, p. 3. 6. Sohultze, Seyenoak, Schuyler: The notion of a straight line Is suoh a simple and fundamental one that It Is praotloallv Impossible to give a good definition of It.GO 7. Seymour, Plane Geometry, p. 3. 8. Eugene R. Smith, Plane Geometry, p. 22. 9. Stone and Mallory: We have left undefined the terms point, line, surface, and space, for the Ideas they represent are so fundamental that they cannot be defined In simpler terms. 10. Herb erg and Orleans: These are. In fact, terms, since they are too simple to be defined logically.G2 11. Leonhardy, Joseph, McLeary, New Trend Geometry, p. 14. 12. Wells and Hart (1915, 1926), Progressive Plane Geometry, p. 24, 27. 13. Workman and Cracknell, Geometry Theoretical and Praotloal. p. 65. Undefined terms are too difficult to define. 1. Hawkes, Luby, Touton: The next problem Is to choose the geometric terms needed and to give their meaning. Where definition Is difficult, as with the terms point and line, the Ideas

fiO Schultze, Sevenoak, Schuyler, Plane Geometry, p. 2. 61 Stone and Mallory, Modem Plane Geometry, p. 38.

^^erberg and Orleans, A New Geometry, p. 3, 48 whloh they denote oan be brought out by lllaet rat Ion • 63 8* Haertter: Some tezma understood* We start with some very simple notions in geometry* Many are extremely difficult to define, but easy to understand. 3* Carson and Smith: The idea of a straight line and of a segment of a straight line are familiar to everyone— so much so that these ideas are not easily put into words.65 Undefined terms cannot be defined precisely in simple terms. 1. Schnell and Crawford: In geometry there are some words which we understand and use but which we do not define because a precise definition cannot be given in simple terms.6® Undefined terms are necessary because defining must begin somewhere* 1* Barber and Hendrix: Just as the dictionary cannot define all words, but must have some words with which to begin , . * so geometry has to have some statements on which to base its reasoning* These foundation statements are assumptions and definitions.6? ^ote: This passage leaves some doubt as to whether or not the authors are talking about undefined terms. This point will be discussed and the passage analyzed in context at the end of this series of citations^

®®Hawkes, Luby, Touton, New Plane geometry, p. 28. ®^aertter. Plane Geometry, p* 7* A fC Carson and Smith, Plane Geometry, p* 92* A A Schnell and Crawford, Plane Geometry, p* 85* 67 Barber and Hendrix, Plane Geometry, p* 6* 43 £• tfellcowltz, Sitomer, Snader: The teohnioal terma that are used should be only those that have been previously defined. For example, you may not use the word "polygon" in defining a quad­ rilateral unless the word "polygon" has been previously defined. However, sinoe a beginning has to be made somewhere, certain terms have to be aooepted undefined.^ Some are undefined beoauae everyone knows their meaning. 1. Skolnik and Hartley: We have already seen that in geometry it is necessary to assume certain propositions without proof in the same way we must accept certain terms without defining them; that is, we shall agree that we know their meanings. Those terms that we take for granted are called undefined terms. Having agreed on the undefined termi^ we then define the others using the terms that we had agreed upon to accept without definition. An undefined term is one whose meaning we take for granted without describing it in words. 2. Trump: We must, in the nature of things, be content to stop somewhere. In geometry we assume that certain words such as line, plane. point. space mean the seme thing to each of us and we do not attempt to define them.^O 3. Smith and Marino, Plane Geometry, p. 1. 4. Williams and Williams, Pleine Geometry, p. 259.

,/elkowitz, Sitomer, Snader, Geometry. Meaning and Mastery, p. 36. go Skolnik and Hartley, Dynamic Plane Geometry, p. 20. 70 Trump, Geometry - A First Course. p. 48-49. 44 SoniB words are left undefined beoyiae they have the same meaning in geometry sgid ordinary language. 1* Skolnik and Hartley: We aooept without defini­ tion words like is, draw, extend, flaire, between, relationship, and so on. These words have the same meaning in geometry as in ordinary language.”^ 2. Major: Many more words could be put into this first vooabulary, and many textbooks do this. But some of them are part of your everyday vooabulary. You may not always be able to define these terms, but you can use them intelligently because you can recognize and describe the things they represent.”8 Some terms are left undefined because they cannot be defined. 1. Barnard and Child, A New geometry for Schools. P# 7. 2. Welchons and Krickenberger: We all know what points and lines are but we cannot define them.^^ The undefined term is an idealized concept and therefore undefinable. 1. Farnsworth: We must do nothihg short of imagining that which it is impossible to repre­ sent on the paper, the idealized or geometric line. Such a line will be wholly free from the imperfections, variations, and other encumbrances of the material line. . . . Now to give a precise definition of such an idealized concept as a geometric line Is

71 Skolnik and Hartley, Dynamic Plane geometry, p. 80. n p “Major, Plane geometry, p. 2. 73 Welchons and Krickenberger, Plane geometry, p. 7. 45 manifestly impossible. Instead of attempting to define a geometrie line, whether straight or oarred, we shall merely state oertain proper­ ties whioh our ideal lines are assumed to possess.^^ 2. Barber and Hendrix: (Ideal figures and imaginary superposition are discussed, but not in rela­ tion to undefined terms. 3. Wentworth: It must be distinctly understood at the outset that the points, lines, surfaces, and solids of Geometry are purely ideal, though they are represented to the eye In a material way.76 4. Strader and Rhoads: If we sketch or draw a straight line, a broken line, or a point, these sketches are only representations of the true or ideal geometric figures. • • • So all the sketches or drawings we make are only representations of geometric or ideal magnitudes. They are like the cartoons of men ill at we see in the daily papers in that they emphasize and portray the principal characteristics of the subjects.”” No reason given. Flat statement that no attempt will be made to define certain terms. 1. Rosskopf, Aten, and Reeve: No attempt will be made to define the following terms. However, the following explanations, which are generally accepted, may help to make sure that everyone working with geometry is thinkir^ of the same things when the terms are used.7B

^^Famsworth, Plane Geometry, p. 6, 75 Barber and Hendrix, Plane Geometry and Its Reasoning. p. 11. ^^Wentworth, Plane Geometry, p. 2. 77 Strader and Rhoads, Plane Geometry, p. 11. 78Rosskopf, Atan, and Reeve, Mathematice A Second Course, p. 12. 46 2. Slglsy and Stratton: No attempt will be made to give formal definitions for the elemental texms point. line, and plane. The foundations of plane geometry depend upon the definitions of basic terms whioh make use of these elemen­ tary terms.79 Several comments should be made concerning the above quotations. The books by V/entworth, Strader and Hhoads, Barber and Hendrix and by Farnsworth, when they discuss the notion of an "idealized'* or "geometric" line, are apparently expressing the ancient Platonic concept of "universal ideas." Fortunately, elaborate attempt to distinguish between material or physical figures on the one hand and geometrical or ideal figures on the other hand are not very common in contemporary books, ^ith geometry considered as a science of space, the best approach, for example, to straight lines seems simply to examine and consider the lengths of physical lines without regard to any width they m i ^ t have. The following table comparing publication date with the number of textbooks which do or do not mention undefined terms clearly indicates a recent trend toward mentioning undefined terms. Since twenty-six out of the twenty-eight texts below not only mention undefined terms but also pre­ sent varying reasons for their need, one might say that

70 Sigley and Stratton, Plane Greometrv. p. 2, 47 the table ahoirs a trend toward Justifying undefined terms,

TABLE I NUMBER OF PLANE GEOMETRY TE3CTB00ES» BY PUBLICATION DATE, V/HICH DO OR DO NOT MENTION UNDEFINED TERMS

Publication Date Do Not Do

Before 1900 ••••.. 1 1900 - 1909 ...... 3 1910 - 1919 ...... 3 1920 - 1929 ...... 3 1930 - 1939 ...... 5 1940 - 1949 ...... 10 1950 - ...... 3

Total ...... 28

"Informal descriptions of undefined terms." Of the twenty-eight textbooks whioh mention undefined terms, six­ teen texts offer informal descriptions or explanations of the undefined terms* Of these sixteen, seven appear to des> cribe with the explicit or implicit use of postulates or geometric axioms, while the other nine present informal, empirical descriptions. Following are quotations from the seven textbooks which described undefined terms with postulates or geometric axioms: 1. Barnard and Child: No satisfactory definition has ever been given of a straight line, but all admit the following characteristics: 48 (1) One and only one straight line oan be drawn between two given points. (ii) Two straight lines cannot intersect in more than one point. (iii) *nvo straight lines cannot enclose a space. (iv) Every part of a straight line is itself a straight line. (v) The straight line Joining two points is the shortest distance between them.80 2. Carson and Smith: The idea of a straight line and of a segment of a straight line are familiar to everyone— so much so that these ideas are not easily put into words. It will be suffi­ cient for our purposes if we state a few of the properties of a straight line which are so well known as to be accepted by everyone. . . . One straight line and only one can be drawn through two given points. . . .

* ^ 0 straight lines cannot intersect in more than one point. . . . A straight line is the shortest path between two points.31 3. Farnsworth: Instead of attempting to define a geometric line, whether straight or curved, we shall merely state certain properties which our ideal lines are assumed to possess. ASSUMPTIONS REGARDING STRAIGHT LINES a. Throu^ two given points one and only one straight line oan be drawn. b. A straight line oan be extended indefinitely. c. A straight line of limited length has one and only one middle point. d. The shortest line that can be drawn between two given points is the straight line join­ ing them.

®^Barnard and Child, A New Geometry for Schools. p. 7. 81 Carson and Smith, Plane Geometry, p. 92. 49 e* Two straight lines in a plane either inter­ sect, if sufficiently extended, or are parallel, f, *l\ro straight lines can intersect in not more than one point. g. Through a given external point one and only one straight line can be drawn parallel to a given line.Sfi 4, Henri ci : V/e may say a ^flat surface is called a plane," and call this a definition of a plane. But this defines the class "plane" by aid of the class "flat," What "flat" is remains undecided. But then there is the Axiom 7 (Through three points which do not lie in a line, one, and only one, pleine may always be drawn,), which tells us what we do understand by flat, It enables us, if we apply it to a surface and a copy of it, to decide whether a surface is a plane, and thus it teüces the place of a definition. The same is true for the other Axioms, All six axioms together define s p a c e .83 5, Herberg and Orleans: A straight line is the shortest distance between two points, A straight line has length, but no width and no thickness, A point is the place where two straight lines cross, A point has no length, no width, no thick­ ness ; it has only position, A plane is a flat surface of two dimensions, length and width. The above descriptions of straight l^e. point.and plane are not really complete deflni- tions of the terms, since the descriptions involve ideas more complicated than the terms themselves,84

®^Farnsworth, Plane Geometry, p, 8, ®®Henriol, Elementary Geometry, p, 11,

®^erberg and Orleans, A New Geometry, p, 3, 50 6» Leonhardy, Joseph, MoLeary: In plaoe of an ideal definition for such terms (point and line), it is oustomary to give the properties whioh distinguish them and give them meaning in our thinking. . . . "Rfo straight lines oan interseot at only one point ; or two interseot- ing straight lines determine a point.85 7. Trump: Sometimes those who wish to be com­ pletely "logical" make sure that there is agreement about how undefined terms are to be used by assuming oertain things about them, as for example, agreeing that: One and only one straight line oan be drawn through two points, or Two straight lines lying in a plane must either interseot in a point or be parallel. y/hile these are not definitions they are statements whioh limit the properties possessed by the things we choose to oall points and lines.86 Birkhoff and Beatley, while classified with those giving informal descriptions of undefined terms, are actu­ ally intermediate: 1. Birkhoff and Beatley: part of our undefined notion of straight line and of plane is that each of these is a collection of points; also, that a straight line through any two points of a plane lies wholly within the plane, v/e shall assume also that a straight line divides a plane into two oarts, though it is possible to prove this.8" The remaining eight describe "point" as having position but not dimensions, as represented by a dot, and as

36 Leonhardy, Joseph, MoLeary, New Trend Geometry, p. 14. SÔTnimp, Geometry - A First Course, p. 49.

87 " Birkhoff and Beatley, Basic Geometry, p. 39. 51 separating one part of a line from another* They desorihe "line** as represented by the edge of a ruler, by a stretched string, by a straight mark, and by the boundary between two portions of a surface* Following is a list of these text­ books : 1* Rosskopf, Aten and Reeve, Mathematics. A Second Course, p* 12. 2* Seymour, Plane Gaometry. p* 2* 3* Eugene R* Smith, Plane Geometry Developed by the Syllabus Method. p* 22. 4* Smith and Marino, Plane Geometry, p* 2* 5* Stone and Mallory, M o d e m Plane Geometry, p. 38. 6* Welchons and Krickenberger, Plane Geometry, p. 7* 7* Welkowitz, Sitomer, and Snader, Geometry. Meaning and Mastery, p. 14*

8 * Wells and Hart, Progressive Plane Geometry, p. 6*

Summary o£ findings regarding t ^ rule that definitions shoulduse terms which are prior andT^re intelligible and not in obscure language* (1) Of the two interpretations of **prior and more intelligible** Aristotle preferred the absolute rather than the relative interpretation* Nineteen m o d e m textbooks con­ tained a rule that definitions should be in terms prior and more intelligible, while, of these, eight preferred the absolute interpretation, five preferred the relative, and six could not be identified in either category* se (e) Out of the ten mo d e m textbooks whioh present speolfio rules for defining, not one contained the Aristotelian rule against obsoure language. Likewise none of the modem textbooks contained a rule against definition through opposite terms or through coordinate division. The rule in favor of prior and more intelligible terms leads to rules against opposite, circular, and coordinate definitions and leads to the notion of sequences of definitions and the idea of undefined terms. (3) Only one modem textbook mentioned circular definitions specifically. Nine modem textbooks make specific reference to the idea of sequences of definitions. Three separate types of sequences occur in plane geometry— sequences of definitions, sequences of theorems, and sequences of constructions, and these may legitimately vary among different systems. In regard to undefined terms it was shown that, although textbook trends show increasing emphasis, the idea of undefined terms goes back at least to Aristotle. Twenty- seven out of sixty-one modem geometry textbooks mention the necessity of undefined terms, but these offer nine varying reasons why they are necessary. Four books refer to '^idealized'* or "geometric" lines and figures as opposed to the "material" line or figure drawn on paper. Out of the twenty-seven textbooks which mention undefined terms, 53 fifteen offer deeoriptions or explanations of them— seven by means of postulates or geometric axioms and eight by means of Informal, empirical descriptions. 5. Definitions should not contain redundant or superfluous Information, Aristotle states a rule against redundancy In defini­ tions and then discusses five aspects of It. 1. The redundant part might be true of everything In general. If, on the other hand, he has phrased the definition redundantly, first of all look and see whether he has used any attribute that belongs universally, either to real objects In general, or to all that fall under the same genus as the object defined: for the mention of this Is sure to be redundant. For the genus ought to divide the object from things In general, and the differentia from any of the things contained In the same genus. Now any term that belongs to everything separates off the given object from absolutely nothing, while any that belongs to CLll the things that fall under the same genus does not separate It off from the things contained In the same genus.®® S. The redundant part may be struck out and the remainder still defines. Or see If, though the additional matter may be peculiar to the given term, yet even when It Is struck out the rest of the expression too Is pecul­ iar and makes clear the essence of the term.8* 3. The redundant part does not apply to the whole species and therefore the definition Is too narrow.

Aristotle, Toploa. 140^ 24 - 140* 31.

®^Arlstotle, Toplca. 140* 33 - 140* 35. 54 Moreover, see If anything oontained in the defini­ tion fails to apply to everything that falls under the same speoles: for this sort of definition is worse than those which Include an attribute belong­ ing to all things universally• For In that case. If the remainder of the expression be peculiar, the whole too will be peculiar for absolutely always, if to something peculiar anything whatever Is true be added, the whole too becomes peculiar# Whereas if any part of the expression do not apply to everything that falls under the same species. It Is impossible that the expression as a whole should be peculiar: for It will not be predicated convertIbly with the object • • #^0 ^^ote: The last sentence above In effect states that a definition Is not reversible If It Is too narrow# "A polygon having three equal sides Is called a triangle," Is an example of the above# The word "equal" causes the definition to be too narrow and consequently It Is not reversible: "A triangle Is a polygon having three equal sides." Of course the latter Is not acceptable only If we have already agreed that the species of triangle Is broader than the equilateral triangle^ 4# The redundant part might be mere repetition# Again, see If he has said the same thing more than once, saying (e#g#) 'desire* is a 'conation for the pleasant#' For 'desire* Is always 'for the pleasant,' so that what Is the same as desire will also be 'for the pleasant#' Accordingly our defini­ tion of desire becomes 'conatlon-for-the-pleasant for the pleasant': for the word 'desire* Is the exact equivalent of the words *conation-for-the- pleasant ,* so that both alike will be 'for the

90 Aristotle, Toplca, 140^ 16 - 140^ 24. 55 pleasant* . . • Absurdity results, not when the same word Is uttered twice, but when the same thing is more than once predicated of a subject . • .*1 5. The redundant part might be a special case of a universal already mentioned. Again, see if a universal have been mentioned and then a particular case of it be added as well, e.g. * Equity is a remission of what is expedient and just*; for what is just is a branch of what is expedient and is therefore included in the latter term: its mention is therefore redundant, an addition of the particular after the universal has been already stated.

The following thirteen contemporary textbooks gave some form of the rule that definitions should not be redundant: 1. Barber and Hendrix, Plane Geometry, p. 122. 2. Hayn, A Geometry Reader, p. 185. 5. Keniston and Tully, Plane Geometry, p. 56. 4. Leonhardy, Joseph, MoLeary, New Trend Geometry, p . 12. 5. Nyberg, Fundamentals of Plane Geometry, p. 17. 6. SchnoU and Crawford, Plane Geometry, p. 87. 7. Schorling, Clark, Smith, Modem-School Geometry, p. 74. a. Skolnik and Hartley, Dynamic Plane Geometry, p. 21. 9. Strader and Rhoads, Plane Geometry, p. 21. 10. Trump, Geometry - A First Course, p. 46. 11. ,/elohons and Krickenberger, Plane Geometry, p. 30.

®^Aristotle, Topics. 140^ 27 - 141® 6. 92 . Aristotle, Topics. 141® 15 - 141® 18. 56 12« VYelkonritz, Sitomer, Snader, Geometry. Meaning and Mastery. p. 36. 13. Workman end Craoknell, Geometry, p. 66. An examination of the statements about redundanoy in these thirteen textbooks indioates that only aspect **Number 2" of the five discussed by Aristotle is considered by modern textbooks, namely, "The redundant part may be struck out and the remainder still defines." In commenting on this aspect, Aristotle is explicit in pointing out that the additional or redundant matter may be peculiar to the given term. Modem texts interpret this to mean that the addi­ tional matter is redundant because it is deducible from the remainder of the definition. For example, the definition "A reotangle is a parallelogram containing four right angles" is redundant according to modem textbooks because the information in the definition that there is more than one right angle is deducible from the facts that the figure is a parallelogram and contains one right angle. This interpre­ tation of modern textbooks is corroborated by the fact that they nearly all define rectangle as a parallelogram contain­ ing one right angle. The textbook by Trump was the only one to avoid redun­ dancy completely with regard to similar triangles: "Similar triangles are triangles in which two angles of one are equal respectively to two angles of the other." Barber and Hendrix discussed the danger of redundancy in connection with 57 defining similar triangles and actually chose a definition which was free of redundancy* However, when they said, **We oan then base our definition on either definition * . *," they did not point out that the other definition, namely, "Two triangles are similar if the three angles of one equal the three angles of the other," would be redundant* Schorling, Clark, and Smith are the only other authors who recognize that the conditions for similarity at the defini­ tion stage could be less for triangles than for polygons in general. Congruency was handled like similarity in those text­ books which gave redundant definitions for both terms* The excess conditions were first presented by definitions and these were then narrowed in subsequent theorems. In the case of six of the above books the excessive conditions in the definition of congruent were implied by the word "coin­ cide," which requires that two triangles be equal in angles and sides. The other six books explicitly defined congruent triangles in terms of the equality of six parts* There is a question whether definitions of similarity in terms of "the same shape" and congruency in terms of "coincide" would con­ vey meanings beyond what would be obtained by defining them in terms of the equality of parts and the ratio of sides* The difficulty lies in the fact that the former definitions in terms of "shape" and "coincide" do not provide us with a 58 oompletely usable stairtlng point for deducing other theorems* None of the texts uses a definition of similarity in terms of "shape," while those whioh use a definition of congruency in terms of "coincide" are forced to use the much-criticized proof by superposition* Unlike their treatment of the definitions of similarity and congruence* the authors of the thirteen textbooks seem to be aware of their rule of redundancy in the case of the definition of rectangle and the square* the rhombus* and the isosceles triangle* Only one textbook* by f/orkman and Craoknell* defined rectangle redundantly: "A rectangle is a quadrilateral in whioh each of the four angles is a right angle*" However the error was obscurely corrected 91 pages later in a footnote* This practice does not seem very desirable* 4* Definitions must be reversible* Before the Aristotelian precedents of the above rule are given* it is necessary first to illustrate* explain* and discuss what is meant by the rule* In the first place Aristotle used the term "convertible" to mean the same thing that modem textbooks mean by the term "reversible* " When a definition is converted or reversed* the term to be defined and the defining expression are interchanged. For example * the definition "A polygon having three straight-line sides is called a triangle" can be reversed as follows: "A 59 triangle la a polygon having three atralgbt-llne aides*” The rule above means that in all solentlflo definitions, both the definition and Its reversed form must be true with regard to the meaning we have already agreed to attach to both the term and Its defining phrase. At this point It will be useful to Introduce two technical logical terms designating parts of the definition* ”Definiendum** Is employed In logic to refer to the term being defined, while "deflnlens” Is employed to refer to the defining phrase * Thus, In the definition of **trl angle” above, the word ”triangle” is the definiendum, while the phrase ”a polygon having three straight-line sides” Is the definiens* In Aristotle the definiendum Is the species being defined and the definiens is the phrase containing the genus and the differentiae* According to Orgejn, Aristotle employs the terms "speoles,” "genus,” and "differentiae” but not "definiendum" and "definiens*”®® One other word usage in Aristotle must be explained before citations concerning the rule of reversibility are made* Aristotle uses the phrase "predicated of" in the following manner: in the sentences, "A triangle is a polygon” and "Man Is an animal,” "polygon” Is predicated of "triangle” and "animal” la predicated of "man.”

®®Troy Wilson Organ, ^ Index to Aristotle, pp. 40-41. 60 Following are some statements by Aristotle whioh express a rule of reversibility for definitions and also throw light on some other matters. For every predicate of a subject must of necessity be either convertible with its subject or not: and if it is convertible, it would be its definition or property, for if it signifies the essence, it is the definition; if not, it is a property: for this was what a property is, viz. what is predicated convertibly, but does not signify the essence. If, on the other hand, it is not predicated convertibly of the thing, it either is or is not one of the terms contained in the definition of the subject: and if it be one of those terms, then it will be the genus or the differentiae; whereas, if it be not one of those terms, clearly it would be an accident, for accident was said to be what belongs as an attribute to a subject without being either its definition or its genus or property.®^ The following quotation from Aristotle indioates he believed that, since a definition must be peculiar to its subject, it must therefore be convertible with it. . • . for the definition put forward must be predicated of everything of which the term is predicated, and must moreover be convertible, if the definition rendered is to be peculiar to the subject. The following statement from Aristotle relates the question of convertibility with class inclusion: //hereas if any part of the expression do not apply to everything that falls under the same species, it is impossible that the expression as a whole should be peculiar: for it will not be predicated convertibly with the object ; e.g. 'a walking

®^Aristotle, Tonica. 103^ 8-19.

®®Aristotle, Tonica. 154» 39 - 154^ 2. 61

biped animal six feet high*: for an expression of that kind is not predicated oonvertibly with the term» because the attribute *slx feet hl^* does not belong to everything that falls under the same species.00 As mentioned in a note on page 54 of this study, the above quotation is equivalent to stating that a definition is not reversible if it is too narrow, i.e. if the denota­ tion of the definiens is less than the definiendum. Howard, in a study representing a pragmatio interpretation of the Aristotelian logic, supplements this idea: A good definition of **man** ought to be equiva­ lent in denotative capacity to the term defined; neither wider nor narrower. The test of a definition, in truth, is whether it is or is not "convertible with" the term defined. Thus: A man is a rational animal, converts to, 5 rationaT animal Xs a man. In a formal definition, we intend that the definition shall have the same extension, the same denotative range, as the term defined."” Discussions on convertibility are scattered throughout the logical works of Aristotle and most of them deal with convertibility of propositions. However, the above state­ ment, as far as Howard is concerned, is probably an author­ itative interpretation of Aristotle's position on the con­ vertibility of definitions, since he says in his preface:

®®Aristotle, Tonica. 140^ 21 - 26. 97 _ Delton Thomas Howard, Analytical Sylloaistica. p, 152. 6S **ETerythliig of Importance that I have to a ay about the Aristotelian logic is presented in the main body of the text . . , This explanation of convertibility of defi­ nitions by class inclusion is illustrated on the following page by the logical device of -Ruler's Circles. Vfhether the explanation of convertibility of definitions by class inclu­ sion is the only explanation of convertibility will be dis­ cussed in a later chapter when the treatment of definitions in current books on logic is taken up. (See next page now.) Defining-phrase number one, "a polygon," for the term "triangle" has a denotation whioh is broader than and includes the denotation for the term "triangle." The condi­ tion, "a polygon," is necessary or indispensable to a "triangle," In other words, if you do not have a polygon then you do not have a triangle. The contrapositive or logically equivalent expression to this is: if you have a triangle then you have a polygon, whioh is the converse of the following: if you have a polygon, then you have a triangle. In like manner, defining-phrase number four, "a three­ sided figure," is a necessary or indispensable condition. Defining-phrase number three, "a polygon of three equal sides," has a denotation whioh is narrower than the term

90 Ibid.. p. vii. 63

Explanation of Gonvartiblllty in Definition by Representing the Denotation of Term and Defining Phrase with Class Circles (Denotation means the class of examples to whioh a term refers,)

Defining Phrase

A polygon

Term Tria ngleA polygon of TriangleA 3 sides —

A polygon of 3 equal sides

4, A three-sided figure ---

Figure I 64 "triangle** and is included within it* This phrase presents a sufficient condition for having a triangle* The converse does not hold: If you have a triangle then you have a polygon of three equal sides* Defining-phrase number two, "a triangle of three sides," has the same denotation as the term "triangle." It repre­ sents a set of neoessGLry and sufficient oonditions for having a triangle. The phrase "a polygon of three sides" oan be used as a definition of "triangle" since the denota­ tive range of both is the same. This follows from the fact that a defining phrase must enable us to select those things whioh are or are not named by the term being defined. The defining phrase and the term being defined must denote the same things. It oan be seen from the circles that the defining phrase always has a denotation whioh is the logical product of all of the necessary, but logically independent, condi­ tions for having a term. The "logical product" of two class circles is defined as the area of overlap of the two cir­ cles, "Logically independent" oonditions mean conditions whioh cannot be deduced from each other. The denotation of triangle, since it is a logical product of the denotation of "polygon" and "three-sided figure," therefore is the class of figures whioh possess both the property of being a poly­ gon and of being three-sided. 65 The logical product of all of the logically-independent, necessary conditions makes up a sufficient condition. When the definition Is overloaded with an independent but unnec­ essary condition, as In the case of **a polygon with three equal sides," the denotation is too narrow, but the condi­ tions are still sufficient. What, then, la meant by the verb "is called" In the definition: A polygon of three sides Is called a triangle. In the context of this discussion It means that the denotations of "triangle" and "polygon of three sides" are the same. From the foregoing discussion It can be seen that when the defining phrase is a necessary condition for the term being defined (Numbers 1, 2, and 4 above), the defining phrase may always be substituted for the term. When the defining phrase Is a sufficient condition for the term being defined (Numbers 2 and 3 above), the term may always be sub­ stituted for the defining phrase. When the defining phrase Is both a necessary and sufficient condition for the tezm (Number 2 above), the defining phrase and the term are mutually substitutable, l.e, the definition holds both ways In terms of the denotations agreed upon. The above treatment of definition Is supported by Stabler: 66 A mathenLatloal or logioal définition oan be Interpreted aa a necessary and sufficient condi­ tion, but usually the form of the definition does not Indicate this» For example, when we say "triangle ABC Is defined to be equilateral sides AB, BC, CA are equal,” we surely understand "If and only If" In plaoe of "lf.«®9 Before the treatment of convertibility or reversibility of definitions in modem geometry textbooks Is presented, a final point must be made regarding the explanation of con­ vertibility on the previous page. The property of converti­ bility In definitions constitutes a test for the equivalence of definiens and definiendum» Tarski makes the follcwing comment regarding the necessity for this equivalence in definitions: "It may be added that the fona of an equiva­ lence Is not the only form In which definitions may be laid down.The possibilities of alternative forms is a sub­ ject discussed In Chapter IV» Of the slxty-one textbooks in geometry examined, only sixteen mention reversibility, while only five of these attempt to explain the reason for the reversibility prop­ erty In definitions.

99 E, R* Stabler, Introduction to Mathematical Thought. p. 64.

Alfred Tarski, Introduction to Logic, p. 36. 67 Reversibility of definitions is stated with reasons. 1. Hart and Feldman: A necessary and sufficient test of the completeness of a definition is that its converse shall also be true# Hence a definition may be quoted as the reason for a converse or for an opposite as well as for a direct statement in an argument.101 2. Rosskopf, Aten, and Reeve: A good definition has other requirements. It is reversible; that is, it holds true if the subject and predi­ cate are interchanged.102

The definition of two parallel lines can be restated as follows: If two lines in the same plane are cut by a transversal so that two corresponding angles are equal, the lines are parallel, . . . Recall that the converse of a definition is also true; that is, a definition is reversi­ ble. Hence, we can make the statement: if two parallel lines are cut by a transversal, the pairs of corresponding angles are equal.103 Notice that the statement that describes a locus is reversible. Recall that in a defini­ tion a statement and its converse are both true; that is, a definition is reversible* When a proposition or its converse are both true, we say that the proposition satisfies a necessary and sufficient condition; hence when you prove a locus you will have two statements to prove the necessary condition and the sufficient con­ dition.104 3. Skolnik and Hartley: The second property of a definition is that it must include all the things that it defines but exclude all others. A

IQlRart and Feldman, Plane and Solid geometry, p. 45.

Rosskopf, Aten, and Reeve, Mathematics A Second Course, p. 14. ”*

^°^Ibid.. p. 104, ^Q^Ibid.. p. 228, 08 definition muet be limiting. ;Vhen something is defined, it must be described well enough to distinguish it from the other things in the group that are not being defined. For example. It must be stated how an island is different from all other tracts of land. Suppose it was said that an island is a tract of land surrounded with water. This statement would be correct as a fact, but it could not be used as a definition because it would include other tracts of land such as the continent of North America. There is a simple test that enables us to tell whether or not we have excluded all the things in the larger group from the inner group. In this test we first write our definition in the "if-then" form, and then turn it around. Thus we mi^t test the poor definition in the last paragraph by writing it first as follows: If a tract of land is an island, then it is sur­ rounded by water. Then turning this around, we write: If a tract of land is surrounded by water, then it is an island. The first of these statements is true. But the second proposition, which is called the converse of the first, is not true. In a definition, both must be true. A definition must be reversible. This means that the converse of a definition must always be true. The converse of a proposition is a new proposition formed by interchanging the hypothe­ sis and the conclusion. A reversible proposition is a proposition whose converse is eq.ually valid. A definition is reversible.^^® A necessary and sufficient condition that two triangles be congruent is that they agree in all their parts. This, in fact, is our defini­ tion of congruent triangles. Every definition

105 Skolnik and Hartley, Dynamic Plane Greometry. pp. 2 0 -2 1 . 69 expresses a necessary and sufficient condltlon,^^ 4, Swenson: ^^Ince every definition gives both necessary and sufficient conditions, Its con­ verse Is true* Hence: If two polygons are similar, then (a) their corresponding angles agree and (b) the ratios formed by their correspond­ ing sides are equal*107 A definition gives both necessary and suffi­ cient conditions, whereas a statement of fact gives only sufficient conditions, unless the con­ verse of the statement Is also proved* Hence, If a definition Is stated In the If-then form It Is always reversible* This Is very seldom the case with an ordinary statement* For example, consider the following definition: A mammal Is an animal which suckles Its young* Since this Is a definition, It Is equivalent to the two statements : 1* If an animal suckles Its young, then the animal Is a mammal* 8* If an animal Is a mammal, then the animal suckles Its yuung*106 5* Jelkowltz, Sitomer, Snader: Requirements of a good definition* * • * 4* In algebra, after you learned the meaning of "^exponent, " you could replace the expression x x by x^, eind the symbol x2 could be replaced by x x* %ese substitutions are permissible because of the reversibility property of definitions* * • * Thus, we may observe that In definitions, a word (or a short form) and a set of properties

^Q^Ibld*. p* 119* John A Swenson, Integrated Mathematics. p* 867* 106 Ibid.. p* 461 (In supplement). 70 are oonsldered aa equivalent to eaoh other, and, henoe, either may replace the other.Iv9 ^ote; The first paragraph justifies substitutability by the assumed reversibility property of definitions. The second paragraph Justifies substitutability by the assumed equivalence property of definitions. The authors make no attempt to relate reversibility and equivalence^/ Reversibility of definitions is stated and explained, but no reasons offered. 1. Keniston and Tully: As you have learned in your English classes, a good definition is reversible; that is, it is equally true if turned around by interchanging the subject and predicate. In fact, there is a flaw in a definition if, when you reverse it, it no longer makes sense. % e n reversed, the defi­ nition of a triangle given above reads, "A polygon with three sides is a triangle." Since either form defines the triangle, you may state the definition in either way. Note that, though good definitions are reversible, many statements are not reversible.HO An import Em t use of definitions: As we said on page 82 definitions may be used to prove facts such as the equality of lines and angles, 'fhen writing a definition as authority for such conclusions, we usually change the wording to make its order correspond to the order of the steps in the proof. . . . Notice that we have interchanged the main and subordinate clauses of the definition of an

109 Velkowitz, aitomer, Snader, Dynamic Plane Oeometry. p. 37 . 110 Keniston and Tully, Plane Geometry, p. 56, 71 isosceles triangle. Je cell such a statement a converse of the definition. Use the converse of a definition when proving facts which are the conditions of the definition. 2. Leonhardy, Joseph, MeLeary: The requirements of a good definition. 4. The definition must be reversible. 3. Barber and Hendrix: Every good definition includes its own converse. Thus the definition "If one straight line makes equal adjacent angles with another, the lines are perpendicular" may also be stated as follows: "If one straight line is perpendicular to another, they make equal adjacent angles." And the definition of a ri^t triangle as a triangle having one right angle contains both the statement "If a triangle has a right angle it is a right triangle" and the statement "If a triangle is a right triangle it has a right a n g l e . 4. Major: Jhen one straight line meets another straight line so that the adjacent angles formed are equal, the lines are said to be perpendicular to each other. The angles formed by two perpendicular lines are called right angles. Each of the above definitions is reversi­ ble. Tj^erefore we have the statements : .Then two straight lines are perpendicular, equal angles are formed ; and, Jhen two lines meet at right angles, they are perpendicular. 5. Schnell and Crawford: Every good definition is reversible, that is. it can be turned around and still be correct. Consider the definition:

^^^Ibid.. p. 84. ^^^I-eonhardy, Joseph, Mo Leary, New Trend Geometry, p. 12.

^^^Barber and Hendrix, Plane Geometry, p. 48. 114 Major, Plane Geometry, pp. 7-8. 72 A biped is a oreature with two feet. Reversing it: Creatures with two feet are bipeds. The first definition is used to describe a biped while the second is used to describe a two- footed creature. In one sense there are two entirely separate and distinct definitions because each is used for a different purpose. But it should be noted that each is really the reverse of the other. Therefore, for each definition given in this chapter and in Chapter 17, you are entitled to use its reverse whenever the need arises.

6. Schorling, Clark, Smith: A definition is reversible. If we know that the statement, an isosceles triangle is one with two equal sides, is a definition, we can turn it about and say: A triangle with two equal sides is isosceles. Since congruent triangles are defined as tri­ angles whose corresponding parts are equal, it is correct to state the reverse: Corresponding parts of congruent triangles are equal. Not all statements are reversible as defi­ nitions are. For example, all right angles are equal, but not all equal angles are right angles.11® 7, Stone and Mallory; A definition must be reversi­ ble. Thus, "A triangle is a polygon" is not a definition because a polygon is not always a triangle.117 3. Strader and Hhoads: The definition must be reversible. As a pupil once expressed it, "A definition must be in easy words which say just enough and

115 Schnell and Crawford, Plane Geometry, p. 86. 116 Schorling, Clark, Smith, Modem-Sohool Geometry. p. 74. 117 Stone and Mallory, Modem Plane Geometry, p. 12. 73 not too muoh, and you must be able to turn the whole thing around* 9. Trump: If we say, "A square is a rectangle with one pair of adjacent sides equal,^ it follows that all the angles of a square are right angles and all of its sides are equal, v/e find that: 1* All rectangles which satisfy the defini­ tion are figures which we choose to call squares* 2* All figures we choose to call squares are figures which satisfy the defini­ tion *1%9

^ote: It is significant that the above quotation appears in connection with one of two essential characteristics of a good definition: "B* The second part of the definition tells enough, but no more than enough. to distinguish the object from others in its class*" This point will be dis­ cussed later^ 10* .Velchons and Krickenberger: Any correct definition is reversible. Thus the definition: "A square is a rectangle having equal sides" can be stated: "A rectangle having equal sides is a square,"120

11, Palmer and Taylor: A definition of an object is a description given in such a way that the object is distinguished from all other objects. The converse of a correct definition is always true. For example, an. acute angle is an angle which is less than a right angle. And

ll®3trader and Rhoads, Plane Geometry, p, 12. ^^®Trump, Geometry - A First Course, p* 46, 120 ’.Volchons end Krickenberger, Plane Geometry, p. 30. 74 oonrersely, an angle whloh Is less than a right angle la an acute angle.121 A logical difficulty arises from using reversibility as a criterion in framing a definition and then later using this definition as a justification of reversibility. This Is like saying, "This surface reflects because it is a mirror rather than because It was made to reflect." In like manner, definitions are reversible not because they are definitions but because they were made to be reversible. As will be discussed later, definitions are made to be reversi­ ble because their role in all logical systems Is based on the mutual substitutability of deflniens and deflnlendum. Mutual substitutability Is achieved while framing a defini­ tion by making sure that it is reversible.

5. There is only one definition of a thing. Just as Aristotle*s theory of knowledge, which assumed that each thing had a universal essence, led him to the adoption of the "genus et differentiae" method of defining, to the total exclusion of other methods, so this theory led him to conclude that there is only one definition of a thing. This point of view, In spite of the broadening influences of modern mathematics and logic regarding the nature of postulational systems, Is still generally held

121 Palmer and Taylor, Plane G-eometry. p. 84, 75 by the authors of oont amporary geometry teztboolcs, whether explioltly stated or Implied by their treatment of defini­ tions. The relation between Aristotle*s theory of knowledge and this point of view Is evident In the following quota­ tion: • . • aooordlngly, it Is clear that a man who does not define through teims of this kind has not defined at all, ^ h e terms referred to are "prior” and "more Intel 11 gib le, Jy Otherwise, there will be more than one definition of the same thing: for clearly he who defines through terms that are prior and more Intelligible has also framed a definition, and a better one, so that both would then be defi­ nitions of the same object. This sort of view, however, does not generally find acceptance: for of each real object the essence Is single: If, then, there are to be a number of definitions of the same thing, the essence of the object will be the same as It Is represented to be In each of the definitions, and these representations are not the same. Inasmuch as the definitions are different,128 This Idea Is even more forcibly presented in the fol­ lowing statement by Aristotle:

For If a definition Is an expression signifying the essence of the thing and the predicates contained therein ought also to be the only ones which are predicated of the thing In the category of essence; and genera and differentiae are so predicated In that category: It Is obvious that If one were to get an admission that so and so are the only attri­ butes predicated In that category, the expression containing so and so would of necessity be a def­ inition; for It Is Impossible that anything else should be a definition, seeing that there Is not

188 ^ . Aristotle, Tonlca. 141^ 31 - 141° 1, 76 anything aIsa pradloatad of tha thing in tha oata- gory of assanca*123

Examination of aaoh of tha avallabia oontamporary gaomatry books with raspaot to this point indicatas that all but Sevan taka tha Aristotelian position that there is only one definition of a thing. These seven vary slightly in their approach. Most of them state that a definition is an agreement with respect to the meaning irtiich will be attached to a thing or word in the textbook. The impression is therefore given the pupil that this agreement may vary at the beginning, but once made, must be kept until another agreement alters the first. Not all of these, however, state that a later alteration is permissible. It is the privilege of the person developing a deductive system to improve a definition if it is not functioning satisfactor­ ily. This point of view is in keeping with modem mathe­ matics and logic, namely, that a choice is open to us in laying down our foundations in a postulational system emd that our sequence of theorems may vary but remain logically sound by changing our definitions and assumptions. There­ fore the remaining textbooks which convey the impression to pupils that only one definition of a thing is possible are instrumental in continuing an outmoded tradition.

T53------a Aristotle, Tonics. 153® 13 - 153® 23. 77 Fifteen out of the remaining textbooks whioh seem to perpetuate the Aristotelian tradition on this point do so with speoifio statements about adhering to and memorizing the speoifio words of the definitions presented in the text. 3reslioh,and Sohnell and Crawford (1938) even go so far as to make the Aristotelian distinction between definition and property. Those textbooks which say nothing about defini­ tions or the process of defining but present arb itrary lists of definitions to be accepted without question by pupils and memorized are also certainly conveying the impression to pupils that there is only one definition of a word or thing. Follov/ing are the seventeen books which seem to follow the Aristotelian tradition regarding one definition because of implications in their statements: 1, Auerbach and .Valsh: The wording of our defini­ tions may differ, but the content must be the same,124

2. Austin: Let us not be satisfied with anything short of a complete and correct statement of a definition, for in so doing we shall eliminate much of the difficulty in seeing the origin of a theorem and the necessity of its proof.125

124 Auerbach and V/alsh, Plane Geometry, p. 24. 125 Austin, A Laboratory Plena Geometry, p. 107. 78 3. Breslioh: For example, it la also true that the base angles of an isosoelas triangle are equal. However, this is a property of the isosceles triangle and it cannot be used in place of the definition. Likewise it is true that the oppo­ site sides of a parallelogram are equal, but this fact is merely one of the properties of all parallelograms and it is not to be used as a definition.126

^ote: When Breslioh states that equal base angles cannot be used in place of the definition, he is implying that properties cannot be used in place of a particular defining characteristic^ 4. Biikhoff and Beatley: V/here there is a choice of two expressions, each equally simple, we should always choose the more aoouxrate one. Often we must choose between a simple expression and a more involved expression that is also more accurate. In such oases we may use the simple expression of everyday speech so long as there is no danger of confusion. For example, the word "circle" means strictly a curved line (in a plane) every point of which is equ i distant from a fixed point called the center of the circle.127 ^fote: The fact is that "circle" has no essential meaning that must be stated by every definition of it. "Circle" has also been successfully defined as a particular kind of conic section_j7 5. Edwards: Definitions describe, in as simple a

^^^Breslich, Plane Geometry, p. 291.

1 P7Birkhoff and Beatley, Basic Geometry, p. 14. 79 manner as possible, the objects with which we have to deal.128 6. Hawke8, Luby, Touton: Applied to the study of geometry this means that the definitions of the words used must be clearly understood if we expect to understand the subject. The idea conveyed by a definition must be sharp and clear, so that we shall recognize the thing defined and be able to give in our own words a concise and correct definition.189 7. Lougheed and Workman : Other facts that we have at our disposal with which to begin operations, are those involved in the definitions of the terms applied to geometric figures. Thus if we are investigating the properties of the paral­ lelogram, the definition of that figure justi­ fies us in accepting and using the fact that its opposite sides are parallel. Definitions of geometrical terms are extremely important. .7ith some of these we are already familiar. These will be recalled and others introduced in the text as they are required, and they should be carefully memorized.130 0. Ivlajor: .Then you have used such words often enough to become acquainted with the special and technical twists they sometimes have, you should then satisfy yourself of your ability to provide correct définitions.131 9. mine: Students may sometimes be allowed to express definitions, axioms, and theorems, etc., in their own language, but as a general rule their expressions are inaccurate and faulty. The teacher should in such instances call

^^Edwards, Elements of Geometry, p. xv. (This case is doubtful.) 129 Hawkes, Luby, and Touton, New Plane Geometry, p. 21.

^^Lougheed and Workman, Geometry for High Schools, p. 48, 131liajor, Plane Geometry, p. 2. 80

attention to the errors and require oonoise and aoourate statements* It will then be discovered that they approximate very closely those given In the book*l32 10. Schorling, Clark, Smith: In mathematics a defi­ nition Is limited to the most precise statement possible of the meaning of a term* 133 If each word had an exact meaning and everyone understood this meaning, the communication of knowledge would be a simple matter Indeed* In geometry you have been able to draw valid con- elusions because you used terms with exactness*^^* 11. Skolnik and Hartley: A definition states the meaning of a word or term by using other words or temis*^3® From experimental work, as well as from exercises that we proved, we realize that If a figure Is a parallelogram. It Is necessary that these four conditions be true: (1) The opposite sides are parallel. (2) The opposite sides are equal* (3j The opposite angles are equal, (4) The diagonals bisect each other* From the definition of a pareüLlelogram, we know that the first condition Is true* We shall now prove that the other three conditions are also true In every parallelogram*136 j^ote: The definition of parallelogram In Skolnik and Hartley is given on page 61 In their text without any sug­ gestion of possible alternatives. Nowhere Is It hinted that

^^^.Illne, Plane Geometry, p. 5. ^^^Schorling, Clark, Smith, Modern-School Geometry, p. 74. ^^^Ibld.. p. 432. 135 Skolnik and Hartley, Dynamic Plane Geometry, p* 20. ^^®Ibld., p. 118. 81 any of the above four oondltions could serve as a defini­ tion of the kind of quadrilateral in question. Here, again, the Aristotelian distinction between definition, property, and accident is evident 12, Stone and Mallory: V/hile you probably know in a general way what such geometric figures as cir­ cles, triangles, etc, are, it will be a good rule always to make certain that you know the definition of the mathematical terms you use.137 13, Strader and Rhoads: It is not necessary to state the definition of a teim in the words of this book, but it may be done in the pupil's own language, • . , Since pupils usually have more or less difficulty in forming good definitions, it is suggested that the more importemt definitions be committed to m e m o r y , 138 14. Sykes and Comstock: The fundamental character­ istic of parallelograms is stated in the defi­ nition, namely: The opposite sides are paral­ lel, 13Ô The definition of a parallelogram is the fundamental test for parallelograms. Other tests are dependent primarily upon the funda­ mental one,140 15. Workman and Cracknell: The expleuaation of the exact meaning of any term is called a defini­ tion,141

137 Stone and Mallory, Modem Plane Geometry, p, 11,

1393trader and Rhoads, Plane Geometry, p. 20. l^^Sykes and Comstock, Plane Geometry, p. 80, 140. Ibid.. p, 82, 1^1•.Workman /n-pVman aand Cracknell, Geometry Theoretical and Practical, p. 66. 82 The definitions of the various teims will be given from time to time as the need arises. In all oases they should be l eamt by heart. The following books convey the impression to the reader that there is freedom in the choice of the content of a definition when constructing one; 1. Barber and Hendrix: Since each of the foregoing conditions (equal angles and proportional sides) is sufficient to make the triangles similar, we do not need to put both of them into the defi­ nition: to do so would be to make a redundant definition. Vîe can then base our definition on either condition, but the second one will be more convenient for our later work.1^3 2. Cook: The meaning of words in reasoning is a matter of agreement or contract between persons, viz., we agree to say that a line "a” is per­ pendicular to a line "b" if the adjacent angles made by "a" with "b" are equal. The agreement is binding upon the contracting parties unless a new agreement is m a d e . 144 3. Dupuis, Synthetic Geometry: A statement which explains the sense in which a word or phrase is employed is a definition. A definition may select some one meaning out of several attached to a common word, or it may introduce some technical term to be used in a particular s e n s e . 145 4. Sohnell and Crawford; It is very important, how­ ever, that we define all technical words about which there would be misunderstanding or those

142 Ibid.. p. 67. 143 Barber and Hendrix, Plane Geometry, p. 122, ^*^Cook, Geometry for Today. p. 115.

145Dupuis, Synthetic Geometry, p. 1. 83 whloh ooald too defined in more than one way, For example, if we define a rectangle aa a quadrilateral whose opposite sides are equal, we must recognize that the definition imposes restrictions on us whenever we use it. .Then a rectangle is given in a protolam, we would know that its opposite sides are equal and only that. If, however, we define a rectangle as a parallelogram with one right angle, we would toe entitled to say only that in such a figure one of the right angles is a right angle, V/e could not add that the opposite sides are also equal unless we proved that fact. Once we agree to accept a definition we must always thereafter adhere strictly to it.146 ^ o t e : The above quotation is from the 1953 edition. The 1938 edition changes the last sentence to read: V/e have considerable freedom in making our defini­ tions but once we have agreed to accept a definition, we must always thereafter adhere strictly to it,147 ^ote continued: This is one of those textbooks which point out that a definition is an agreement. However, it does not point out, as does Cook, that this agreement may be changed or revised at a later point. Unfortunately the 1938 edition contains a serious inconsistency in a state­ ment in a list of criteria for definitions three pages later:

(6) A definition must describe the object rather them discuss its properties.148

146 Schnell and Crawford, Plane Geometry, p. 85. , (1938) p. 98.

^ ^ Itoid.. (1938) p. 101. (This quotation does not appear in the 1953 edition.) 84 ^ote oontlnued: In separating the act of describing the object from that of discussing its properties, one can hardly avoid the conclusion that the latter quotation makes the Aristotelian distinction between essence and property. If such is the case, then the freedom referred to in the former quotation vanishes_%y 5. Eugene R« Smith: Suggestions to teachers: As far as possible, let the pupils frame the defi­ nitions for themselves, and lead up naturally to the simple deductions from them, so that the pupils can begin to discover these truths from the very beginning,149 A definition is such a description of the thing defined as will distinguish it from all other things; it might be said to be an agree­ ment as to what a term shall be used to indicate,150 j^te : Smith * s statement that a definition is an agreement as to what a term shall indicate seems to say that a defi­ nition associates a term with a defining property which is known as the connotation. The above quotation seems to confuse denotation with connotation^ 6, Trump: In order to know exactly what we mean when we make any statement we must agree on the definitions of the words we use. Such definitions sometimes require long and careful wording, yfhen a law fails to cover some such point adequately, then someone may accidently or purposely take advantage of it, and the law must be amended.

^^®Eugene R, Smith, Plane Geometry Developed by the Syllabus Method, p. 6. ISOibid,. p, S2. 85 .(ords have different meanings for differ­ ent people, and at different times, //hen we use the dictionary, you find that many words have various accepted definitions.151 7. ./elkowltz, Sitomer, Snader: It has already been pointed out that it is necessary to agree on the definitions of the terms which we are to use in geometry.158 The Aristotelian distinction between genus and property la made in a quotation from Topics on 16 of this study. Aristotle believed that ’’genus" and differentiae together were the essential and fundamental characteristic of a thing, while "property" was a less fundamental attribute, but, like genus and differentiae, was uniquely related to a thing. In geometry this meant that the genus and differ­ entiae or essence would be stated in the definition of a thing, while the properties would be deduced from the defi­ nition in the form of theorems. For example, according to Aristotle, the essence of parallelogram would be "quadri­ lateral with opposite sides parallel," Properties of parallelogram would be: (1) opposite sides equal, (2) opposite angles are equal, (3) the diagonals bisect each other. According to Aristotle, none of the properties could be used to define with. From the modem point of view, any

151 Trump, Geometry - A First Course, p. 43. 152 v/elkowitz, Sitomer, ^nadar, Geometry. Meaning and Mastery, p. 36. 86 of the throe properties oould be satisfactorily used in the definition instead of "opposite sides parallel,” Four of the fifteen textbooks which supported the tradition that there is only one definition of a thing promoted this idea by making the Aristotelian distinction between genus and property. These four were as follows: Breslioh, Lougheed and Vworkman, Skolnik and Hartley, and Sykes and Oomstook, The 1938 Sohnell and Crawford textbook revealed an incon­ sistency in stcting that there is freedom in defining, yet the definition must describe the object, rather than discuss its properties. However, two of the seven textbooks which supported the possibility of choosing among several alterna­ tive definitions of a thing were speoifio in stating that in many oases any one of a number of properties could serve as a definition. These two were: Barber and Hendrix, and Schnell and Crawford (1953), Note that between the 1938 and 1953 editions, Schnell and Crawford cleared up this point,

6, Definitions state what a thing is but not the fact that the thing exists. In commenting on Aristotle with regard to the question of definition and existence. Heath is quite definite : Let us now first be clear as to what a defini­ tion does not do# There is nothing in connexion with definitions which Aristotle takes more pains to emphasise than that a definition asserts nothing as to the existence or non-existence of the thing defined* It is an answer to the question 87 what a thing is, and does not say that it is,153 From the second book of Aristotle's Posterior AnalTtioB. whioh Robinson desoribes as "inoredibly obscure,** it is possible to select a few passages whioh support Heath: Then too we hold that it is by demonstration that the being of anything must be proYed— unless' indeed to be were its essence ; and, since being is not a genus, it is not the essence of anything, Henoe the being of anything as fact is matter for demon­ stration ; and this is the actual procedure of the sciences, for the geometer assumes the meaning of the word triangle, but that it is possessed of some attribute he proves. • . • Moreover it is clear, if we consider the methods of defining actually in use. that defini­ tion does not prove that the thing defined exists . . .154 According to Heath, the question of existence provides the basis for the distinction between real and nominal . definitions : What is here so much insisted on is the very fact which Mill (System of Logic, Bk. I, ch, viii) pointed out in his discussion of earlier views of Definitions, where he says that the so-called real definitions or definitions of things do not consti- tute a different kind of definition from nominal definitions, or definitions of names ; the former is simply the latter plus something^else, namely a covert assertion that the thing defined exists. This covert assertion is not a definition but a postulate. . . . This statement (by Mill) really adds nothing to Aristotle's doctrine: it has even the slight disadvantage, due to the use of

l^^Heath, Euclid, the Elements, p. 143.

l^^Ariatotle, Analytics Posteriors. II 92^ 12 - 92^ 20. 88 the word "postulate" to desorlbe the "oovert assertion" in all oases, of not definitely point­ ing out that there are oases where exlstenoe has to be proved as distinct from those where it must be assumedliSS

Heath olaims that liill^s aooount of the distinotion between real and nominal definitions was fully antioipated by oaocherl in his Logioa Damonstrativa; • . • Saooheri lays down the olear distinction between what he calls definitionea quid nominis or nominales, and definitiones quid rei or realea. namely that the former are only intended to explain the mean­ ing that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of con­ structing it.156 Heath continues by discussing the methods of changing nominal definitions into real definitions and then oites examples from Euclid : Definitions quid nominis are in themselves quite arbitrary,and neither require nor are capable of proof: they are^merely provisional and aw,only^ intended to be turned as quickly as possible into definitiones quid rei. either (1) by means of a postulate in which iT is asserted or conceded that what is defined exists or can be constructed, e.g. in the case of straight lines and oiroles. to which Euclid’s first three postulates refer, or {£) by means of a demonstration reducing the construction of the figure defined to the successive carrying- out of a certain number of those elementeury con­ structions, the possibility of which is postulated. Thus definitiones quid rei are in general obtained as the result of a series of demonstrations. Saooheri gives as an instance the construction of a

^®®Heath, Euclid, the Elements, p. 144,

^^^Ibid.. pp. 144-145, 69 square In Euolid !• 46 (Book one, 46th theorem). Suppose that it is objected that Euolid had no right to define a square, as he does at the begin­ ning of the Book, when it was not certain that such a figure exists in nature; the objection, he says, could only have force if, before proving and making the construction, Euclid had assumed the aforesaid figure as given. That Euclid is not guilty of this error is clear from the fact that he never presupposes the existence of the square as defined until after I, 46.1®” Robinson agrees with the above statement, except that he does not mention tha existence postulate:

It is a legitimate and useful activity to point out that a certain character occurs, and to give it a name or report its accepted name. On the other hand, it is mere loss to call this activity ”real definition*; for that can only conjure up the wraith of essence or at least confuse the activity with some of the other activities called "real definition. ** The proper description of it is that it is an existential proposition plus a nominal definition. The nominal definition assigns a short name to the character described at greater length in the existential proposition.!®® Euclid did not specifically establish the existence of points by postulation, but the existence of straight lines was established by postulates one and two: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line,l®9

1 *57 P. 145. ^^^RiChard Robinson, Definition, p. 157. 159 Heath, Euclid, the Elements, p. 154, 90 These two postulates establish the oonstruotlbillty, and therefore exlstenoe of straight lines, aooordlng to Euolid, and at the same time provided the authority for the use of the straight edge in the system to produce straight lines* Euolid also established the oonstruotlblllty and there­ fore the exlstenoe of oiroles and provided the authority for the use of the compasses in postulate three: 3* To describe a circle with any centre and distance,l&O Throughout Uuclld's Elements no figure whioh had been defined is assumed to exist and then used In a theorem until Its construction had been demonstrated by the use of the above construction postulates or by other constructions based on these postulates* This unnecessary degree of rigor was relaxed by the great French mathematician Adrien Marie Legendre In his Elements de Geometrle of 1794, Legendre made use of "hypo­ thetical constructions," sometimes called "existence postu­ lates," which consisted of postulating the exlstenoe of the midpoint of a stral^t line segment, perpendiculars, angles, the bisector of an angle, and parallel lines without Initially shewing the method of construction* These hypo-

^^°Ibld*. p. 154, 91 thetloal oonstruotlons or exlstenoe postulates were postu­ lated for the purpose of simplifying some oomplloated proofs of theorems before it was demonstrated how these oonstruo- tlons could be made. This procedure has been accepted by mathematicians in the cases of those theorems whose proof depends only upon the existence of the figure used and not upon the possibility or manner of its construction by com­ passes and straight edge. Sometimes a figure does not exist in a system because it contradicts the postulates defining the space of the sys­ tem, such as parallel lines or similar triangles in Riemannian geometry. On the other hand some figures do not exist in a system because some of the attributes in the definition of the figure are incompatible, such aa a quadrilateral with opposite aides parallel and unequal or a concave inscribed quadrilateral. These examples show that there is not com­ plete freedom in postulating existence by hypothetical con­ struction. The danger of error in this respect is greater with increasing complexity of the figure, as Heath in a quotation from Saooheri points out: Confusion between the moninal and the real defini­ tion as thus described^ i.e.the use of tne former in demonstration before it has been turned into the latter by the necessary proof that the thing defined exists, is according to âaocheri one of the most fruitful sources of illusory demonstration, and the danger is greater in pTOportion to the 92 "oomplexlty" of the definition, i.e. the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible. i.e. the simultaneous presence of ehich in a given figure can be proved, by means of other postulates etc. forming part of the basis of tke science, to be ii^ossible. The same thought is expressed by Leibniz also. . . . Leibniz* favourite illustration was the **regular polyhedron with ten faces,** the impossiv bility of which is not obvious at first sight. The following examples, suggested in conversation by Lazar, shows that existence or non-existence of a des­ criptively named figure in plane geometry is not imiediate- ly evident: (1) an Inscribed equiangular, unequal-sided pentagon and (2) an inscribed equiangular, unequal-sided hexagon. Definitions of both of these figures based on the foregoing descriptive names can be stated without difficulty. However the definitions as thus stated would not have anything to do with the question of whether either or both of these figures existed. In fact these examples are cited because it would not be immediately apparent from their definitions whether or not these figures existed< One had better be extremely cautious in assuming the exis­ tence of either one of the figures. It can be proven that the above hexagon exists, while the pentagon does not . Fur­ thermore, the hexagon is constructible with compasses and

^^^eath, Euclid, the Elements, p. 145. 93 straight edge while the pentagon of oourse is not, beoause the conditions imposed cannot occur together in the same figure. The conclusion whioh can be drawn from the preceding discussion of the question of existence and definition are as follows: 1* Neither Aristotle nor Euolid assumed the existence of anything by defining it* £* In Euolid the exlstenoe of a defined figure was established by a postulate or a problem shewing the possibility of construction* 3. In modern mathematics the existence of a defined figure is established by a (l) postu­ late, (2) a temporary postulate of existence known as a "hypothetical construction," (3) a construction theorem, or (4) an existence theorem* 4. The use of hypothetical construction requires some mathematical judgment and should be done cautiously. A total of thirteen modem geometry textbooks specifi­ cally mentioned either the problem of existence or hypo­ thetical constructions, sometimes called "logical postu­ lates." ^hose which refer to existence are listed first* Existence. 1* Birkhoff and Beatley: Long before you began this oourse in demonstrative geometry you learned to think of "parallel lines" as lines

^ t can also be shown that an inscribed equiangular irregular even-sided polygon exists, but an odd-sided polygon does not. 94 whioh lie in the seme plane and do not meet, however far extended. Probably it never occurred to you to Inquire, first of all, whether such lines exist or not. In this development of geometry, however, we prefer to make sure that such lines exist before we define them or discuss any of their relation­ ships. Consequently our first theorem in this chapter. Theorem 13, states that through a given point not on a given line there is one and only one line whioh does not meet the given line. Then, having proved this theorem, we describe such lines, by definition, as parallel. In general we prefer not to define anything until we have first shown that it exists. How­ ever, to follow this practice without exception throughout our study of geometry would make us consider many difficult and annoying details. It is desirable, nevertheless, to mention this ideal and to illustrate it with this theorem.103 2. Halsted; V/hat are celled problems of construc­ tion have a double import. Theoretically they are really theorems declaring that the existence of certain points, sects, straights, angles, circles, etc., follows logically by rigorous deduction from the existences postulated in our assumptions. Thus the possibility of solving such problems by elementary geometry is a matter absolutely essential in the logical sequence of our theorems. So, for example, we have shown (in 101) that a sect has always trisection points, and this may be expressed by saying we have solved the prob­ lem to trisect a sect. Now it happens that a solution of the problem to trisect any angle is impossible with only our assumptions. Thus any reference to results following from the tri­ section of the angle would be equivalent to the introduction of additional assumptions.1^4

^^^Birkhoff and Beatley, Basic Geometry, pp. 105-106.

^^^alsted. Rational Geometry, p. 62. 95

3. Seymour: Slnoe the truth of oonolusions arrived at through a oourse of reasoning should not depend on the aoouraoy of the draw­ ing used, It does not destroy the validity of a proof to assume the oonstruotlon of certain lines and angles, provided that It Is shown that such lines ana angles actually exist,

Hypothetical Constructions, 1, Durell and Arnold: Besides the postulates which are used In the actual construction of figures, there are certain other postulates which are used only In the process of reasoning. Thus, for purposes of reasoning, a given angle may be regarded as divided into any convenient number of equal parts, V/hether It Is possible actually thus to divide this angle on paper by use of the ruler and compasses. Is another question,^®® 2, Hall and Stevens: (In the Preface) Theorems and Problems are arranged In separate but parallel courses. Intended to be studied pari passu. This arrangement Is made possible by the use, now generally sanctioned, of lypo- thetloal Constructions, These, before oelng employed In the text, are carefully specified, and referred to the Axioms on which they depend.^®" Hypothetical Constructions Prom the Axioms attached to Definitions 7 and 18, It follows that we may suppose (I) A straight line to be drawn perpen­ dicular to a given straight line from any point on It, (II) A finite straight line to be bisected at a point.

^®^3eymour. Plane geometry, p, 35.

^^^Durell and Arnold, New Plane geometry, p. 24. 167 Hall and Stevens, A School geometry, p. v. 96 (111) An angle to be bisaoted by a line.iôQ 3. Hayn: (Beneath the theorem: Base angle of an isosoeles triangle are equal*) Here we have another Instanoe, In whioh we learn to oonstruot a line whioh we did not have at the beginning of the disoussion* It is customary to use, at this point, an axiom of a single bisector of an angle * «Whether the construction of this line is known, or not, it exists and forms a pair of equal triangles.1^9

4. Ilajor: A note on hair-splitting: It is assumed that an angle can be divided into two equal parts and that this can be done in only one way. Add this to your list of postulates on page 19* Later you will learn that the proof that the line which bisects an angle can be drawn— not that it exists— depends upon a theorem which in turn depends upon the theorem you are now prov­ ing* Therefore we may not say: Draw CD, the bisector of angle C* But we may let CD repre­ sent this bisector, for we know it exists*^'^^ 5* Playfair; Such demonstrations, it must, however, be acknowledged, trespass against a rule which Euclid has uniformly adhered to throughout the Elements, except where he was forced by necessity to depart from it; This rule is, that nothing is ever supposed to be done, the manner of doing which has not already been taught, so that the construction is derived either directly from the three postulates laid down in the beginning, or from problems already reduced to those postu­ lates* Now, this rule is not essential to geo­ metrical demonstration, where, for the purpose

^ ® ® I M d * , p* 6*

^^^Hayn, Geometry Reader, p. 98, 170 Major, Plane Geometry. p* £0* 97 of discovering the properties of figures, we are certainly at liberty to suppose any figure to be constructed, or any line to be drawn, the existence of which does not Involve an impossi­ bility. The only use, therefore, of Euclid's rule is to guard against the introduction of impossible hypotheses, or the taking for granted that a thing may exist which in fact implies contradiction; from such suppositions, false con­ clusions mlgbt, no doubt, be deduced, and the rule is therefore useful in as much as it answers the purpose of excluding them.l?! Ô. .Eugene R. Smith: In the theorems of Book I, auxiliary lines have been added to the given figure, but only as representations of existing lines about whioh it was necessary to reason in order to establish the proof. The question of the accuracy of the drawing of these lines had no bearing on the truth of the theorem, for the reasoning was entirely about the figures that the lines were taken to represent.i72 7. Smith and Marino : Besides the postulates which are used In the actual construction of figures, there are certain other logical postulates. which are used only In the processes of reason- Ing. Thus, a given angle may be regarded as divided into any convenient number of equal parts. ‘•'hether It Is possible actually to divide this angle on paper by use of the ruler and compasses makes no difference. 3. /ells and Hart (1926): The fundamental construc­ tions have been placed early In the text. By this order of propositions; the only hypothet­ ical construction Is the one used on page 32. . . (The one referred to Is the angle bisector in order to prove "Base angles of an isosceles triangle are equal,")174

^"^^Playfalr, Elements of Geometry, p. 286, 1 7p ^Eugene R. Smith, Plane Geometry, p. 94. 173 Smith and Marino, Plane Geometry, p. 20. 174 ./ells and Hart, Modem Plane Geometry (1926 Edi­ tion), p. V. 96 Jells and Hart (1943): Some terms of geometry are neoessarily undefined; those whioh are defined appear after an informal introduction of the ooncepts by means of drawing exercises or discussions, or after formal proof of their existence.

9. '.Viliiams and Williams: It has been assumed thus far that any construction needed in a demonstra­ tion could be made, and a figure has been drawn, somewhat informally, to represent the condi­ tions. But sufficient progress has now been made to enable the student to employ exact methods of construction based on the principles brought out in the preceding propositions. 10. Workman and Craoknell: (Preface) The usual hypothetical constructions are of oourse admitted and justified. The student is carefully warned against illogical hypothetical constructions-- such as **Let Â5CD be a square equal in area and perimeter to the triangle EFG"— a very neces­ sary precaution in regard to the solving of riders. In the preceding proposition a "hypothetical con­ struction" was used; that is to say we supposed that a certain line (viz. EH) was drawn to satisfy a certain condition, and without having proved in a previous proposition, that such a line could be drawn. This particular construc­ tion was justified in a footnote. A hypothetical construction should not be used where it can be reasonably avoided, and must never be used unless it can be fully justi­ fied. It is illogical to suppose a line or point drawn so as to satisfy certain conditions unless it is certain that such a line or point exist s.

175 Wells and Hart, Progressive Plane Geometry (1943), p. vii. Williams and Williams, Plane Geometry, p. 99. 177 Workman and Craoknell, Geometry, Preface, p. viii. 178 IbjLd., p. 92. 99

7• Purposes of definitions given by textbooks» Slnoe the purposes for definitions indicated by geome­ try textbooks will be orltlolzed in Chapter IV, the aotual survey of these purposes will be provided here. Those books not included In the following summary are those In which no purpose for definition could be found stated. I, Definitions are an explanation of the meaning of words. 1. Dupuis, Synthetic Geometry 2. Sohorllng. Clark. Smith. Modern-School Geometry 3. Skolnik and Hartley, Dynamic Plane Geometry 4. v/elchons and Krlckenberger, Plane Geometry 5. l/orkman and Craoknell, Geometry Theoretical and Practical II. Definitions are an agreement as to the meaning of words. 1» Cook, Geometry for Today 2. Sohnell and Crawford. Plane Geometry 3. Eugene R. Smith, Plane Geometry 4. Trump, Geometry - A tfirst Course 5. Vïelkowltz, Sltomer, Snader, Geometry. Meaning and Mastery III. Definitions enable us to recomize objects and to distinguish them from each other. 1» Pallor, Plane Geometry 2. Hawkes, Luby, Touton, New Plane Geometry 3. Palmer and Taylor, Plane C^ometry 4. Schorling, Clark, Smith, ^ d e m - S o h o o l Geometry 5. Slgley and Stratton, Plane Geometry 6. Eugene R. Smith, Plane Geometry 7. Sykes and Comstock, ÿlane Geometry 17. Definitions enable us to place objects in classes. 1. Henricl, Elementary Geometry 100 Y, Definitions describe the objects of geometry# 1# Sdwarda, Elements of Geometry VI. Definitions explain the sense In which a word or phrase Is employed. 1. Dupuis, Synthetic Geometry 2. Schorling, Clark, Smith, M o d e m -School Geometry 3. Skolnik and Hartley, Dynamic Plane Geometry 4. Welchons and Krlckenberger, Plane Geometry 5. i;orkman and Craoknell, Geometry VII, Definitions settle arguments and remove differences of opinion. 1. Barber and Hendrix, Plane Geometry 2. Herb erg and Orleans. A kew Geometry 3. Kenlston and Tully, Plane Geometry 4. Hawkes, Luby, Touton, New Plane Geometry 5. Leonhardy, Joseph, Lie Leary. Mew Trend Geometry 6. Schnell and Crawford, Plane Geometry 7. Schorling, Clark. Smith, M o d e m -School Geometry 8. Trump, Geometry - A Plrstl^ourse 9. ï/elkowltz, ^Itorner, Snader, Geometry. Meaning and Mastery '/III, Studying geometry Is like studying a language— you need to know some words first. 1. Major, Plane Geometry IX. The premises In the early part of geometry rest on definitions. 1. Auerbach and V/alsh, Plane Geometry 2. Bernard, Plane Geometry 3. Lougheed and ,/orkman. Geometry for High Schools 4. Nyberg, Plane Geometry 5* Rosskopf, Aten, and fteeve, Mathematics. A Second Course 6. Slgley and Stratton, Plane Geometry 7, Seymour, Plane Geometry X. Definitions are abbreviations. 1. Slaught and Lennes, Plane Geometry 101 XI. Dafinitlonfl are important for reaeonlng. 1. Brealloh, Plane Geometry.

Chapter Sygfflaja* This ohapter presented in detail the extent to whioh contemporary geometry textbooks continued the following six defining practices advocated in the writings of Aristotle, not all of which, of course, are objectionable. 1. Definitions should be in the form of "genus et differentia." 2. Definitions should not use obscure language and should use terms which are prior and more intelligible. 3. Definitions should not use redundant or super­ fluous information. 4. Definitions must be reversible. 5. There is only one definition of a thing. 6. Definitions state what a thing is but not the fact that the thing exists. A seventh topic was included in this chapter dealing with purposes of definitions given by textbooks. For each of the above practices, supporting passages from Aristotle's writings were first presented and dis­ cussed. The manner and extent to whioh m o d e m textbooks followed the practice were next included in the form of appropriate quotations or references from each textbook which advocated the practice or referred to it in any way. The quotations or references were then summarized and 102 dlsoussed* This treatment resulted in a relatively complete survey of defining praotioes in contemporary geometry text­ books* The principal conclusion whioh one might draw from this survey is that Aristotelian defining practices are deeply imbedded in contemporary geometry textbooks to the almost total exclusion of modem pedagogical and logical recommendations• The next chapter will analyze the treatment accorded the subject of definition in books on the teaching of mathematics and in other pedagogical sources* CHAPTER III

TREATMENT OF DEFINITION MATHEMATICS METHODS BOOKS

The previous chapter surveyed a large sampling of oontemporary geometry textbooks with respect to the treat­ ment of six defining practices and documented the occur­ rence of these practices in the works of Aristotle* This chapter will present a similar survey of twenty-four books on the teaching of mathematics and close with a discussion of several other pedagogical sources* These books are listed in Part II of the Bibliography. Ideas about defi­ nition which are not included in the following survey are those which the books on mathematics teaching do not dis­ cuss. Similarly, any book in Part II of the bibliography which is not mentioned with regard to a given aspect of definition can be assumed to be one which does not discuss this aspect. 1. Per genus et differentiam method of defining. Five books out of the twenty-four surveyed proposed the genus et differentia, or class and difference, method of defining: 1. Davis: A good definition should have the fol­ lowing properties:

103 104

1« The thing defined is named. £. It is placed in the smallest known class to whioh it belongs. 3. Those characteristics whioh are sufficient to distinguish it from the other members of its class are given.1 Davis* phrase "smallest known class" means the same as the commonly used phrase "proximate genus." For example, **polyeon," rather than "plane figure," would be the proxi­ mate genus of "triangle." 2. Hassler and Smith : The pupils should also see that an ideal definition tells the immediate class to which a thing belongs and then tells enough and only enough about its particular characteristics to distinguish it from all others of its class.& Besides presenting the Aristotelian method of defining, the above statement contains a rule against redundancy in the phrase "only enough." 3. I.linnick: A good definition involves three steps. First, the thing defined is named; second, it is placed in the smallest known class; and third, the characteristics whioh distinguish it from other members of that class are given.^

^David Davis, The Teaching of Mathematics, p. 130.

%assler and Smith, The Teaching of Secondary Math­ ematics. p. 368. g J. H. Minnlok, Teaching Mathematics in the Secondary School, p. 226. 105 4» 3ohultze: The nuinber of ways of defining the term **definition" is as large as the number of treatises on logio and the number of diction­ aries. For the purposes of elementary mathe- matlos, however, the old soholastie definition, although little used by modem writers on logio, is exceedingly useful, viz. A definition is the desimation of the pmximate genua and the specific difference?^ The above statement by Schultze deserves considerable study and interpretation. Superficially, it would seem to imply that Schultzs recognized alternative ways of defining, but ohose to confine himself to the Aristotelian method in elementary mathematics. However, a careful examination of his words indicates that he did not say this. He did not say that any one of the treatises on logic proposed a number of different types of definition. In fact, his words "as large as" imply that each treatise proposed a single meaning of definition and that he chose the classical meaning of definition as the single type to be used in elementary mathematics• 5. /estaway: It will be observed that, for framing definitions, we have used the old device per genus et differentiam.5 It should be noted in the case of '.^estaway and in the other four citations above that the use of the Latin phrase

4 Arthur Schultze, The Teaching of Mathematics, p. 66. 5 F. V/. Westaway, Craftsmanship in the Teaching of Elementary Mathematics, p. 855. 106 for class and difference Is found only occasionally as a means of designating the Aristotelian method of defining. 2. The Question of whether there is only one, or more than one, definition a~term. This question may be resolved into three positions as follows ; 1. There is only one definition of a term because the definition must state the essence, not the property, and there is only one essence. 2. There is more than one type of Aristotelian, or class-and-dlfference, definition of a term, because the distinction between essence and property is not valid. The choice of defining property is arbitrary, just so long as the rest of the properties of the term are deducible. 3. There is not only more than one type of Aristo­ telian definition of a term, but there are other types of definition, besides the Aristotelian, which are usable in geometry, such as the genetic and operational definitions. Only five books dealt directly, or by implication, with the above question. 1. Christofferson: . . . definitions, like postu­ lates, are arbitrary. That is, one author may define trapezoid one way and another author may define it differently; yet each, if consistent, would have an equally rigorous geometry,® This statement seemed to present position two above. There is no evidence that Christofferson accepted or recognized non-Aristotelian definitions.

g H, C. Christofferson, Ceometry Professionalized for Teachers, p. 31. 107 2. Minnlok: It le new neoossary to formulate a definition (of parallelogram) as a basis for proof* A parallelogram and a general quadri­ lateral are drawn on the board. The pupil is asked what makes the parallelogram different from the general. He readily responds that it is the fact that the opposite sides are parallel.^ Immediately before this passage Minnlok has explained that the pupil already is familiar with the various proper­ ties of the parallelogram. The question might be raised, then, as to why Minniok uses the phrase "readily responds" and does not suggest that the pupil might also readily respond that "the opposite sides are equal," One answer to this question is that Minniok felt that the essenoe of parallelogram, whioh he mi^t have assumed was the "opposite sides are parallel," would be immediately evident to the pupil, causing him to "readily respond." Although this may be only a oonjeoture, it is partially corroborated by the faot that Minniok did not suggest alternative ways of defining parallelogram. 3. Smith and Reeve: Human liberty may lead us to say, "I shall define a straight line as crooked," and I shall assume that the results of adding equals to equals are always unequal." If we do this, we can proceed and see what results will follow. Since we very soon get tangled in a mass of absurdities, we are usually glad to abandon our ideas of independence and come back

7 J. H. Minniok, op. oit.. p. 230, 108 to what the world generally calls a state of oonmon sense* This common sense leads us to say that definitions and axioms are general agreements as to the meaning of words that we use and as to what we assume to be fundamental truths In any science.8 As many mathematicians and logicians have pointed out, the nominal, stipulatlve definitions of mathematics have no truth value. This Is not to be confused with the truth value of definitions which report a usage or convention. Apparently Jmith and Reeve have assumed the usual connota­ tions of "straight line" and "crooked" and then have attempted to equate the two in a definition. This is not what takes place in the case of stlpulative definitions In geometry. Take, for example, the collection of figures described by the connotative phrase "triangles with two equal sides." V/hen this bulky collection of words proves unwieldy in repeated use, the abbreviated term "isosceles triangle" Is chosen to stand for the longer phrase. It Is meaningless to say that the resulting definition, "A tri­ angle with two equal sides is called an isosceles triangle," is true or false. It has been arbitrarily agreed that the extension of figures to which the words "triangles with two equal sides" refer shall be the same extension to whioh the term "isosceles triangle" shall henceforth refer. Now in

®Smith and Reeve, The Teaching of Junior High Mathe­ matics . p. 252. 109 the case of the definition olted above by Smith and Reeve, If it is assumed that "straight line" shall refer to the same extension to which "crooked line" refers, then their following discussion about a "mass of absurdities" does not appear to follow, Smith and Reeve happen to have chosen the most arbitrary aspect of defining, that of naming, in which to deny freedom, 4, The Teaching of Geometry In Schools : './here several definitions of one concept are all current it is usually better to ask for a proof that one specified property A implies another specified property Z than to ask for a proof that the concept, specified only by name, does possess the property Z, This avoids the diffi­ culty that some students may use Z to define the concept, and the further difficulty that the passage to Zfrom A may be much harder, or much easier, than the passage to Z from some third property B also In use for a definition,® The above statement not only recognizes the possibility of various acceptable definitions of a concept, but also suggests procedures for examiners to follow in order to avoid the special problems arising because of the actual occurrence of various definitions, 5, iVestaway: '.7e might define a square as a Quadrilateral with four equal sides and four rigkt angles. But all that this definition tells us is that: 1, The four sides are equal, 2. The four angles are right angles. It is quite a good definition, but it does not

9 The Teaching of Geometry in Schools, British Mathe­ matical Association, 1923, p, 68, 110 tell as that the square Is a parallelogram, and therefore it does not tell us that the opposite sides are parallel. Henoe this property is one we should have to find out (perhaps by oongruenoe) ^ we used the new definition. Let us decide not to use it.10 Among the above five books on the teaching of mathe­ matics which contain passages relating to the question of one or more than one definition of a term, V/estaway pro­ vides the clearest example of the nature of the choice which is available in many oases of definition in geometry. Three of the above five sources implied freedom in the defining process, but none suggested other types of defini­ tion besides the Aristotelian, 3. Use of simpler or previously defined terms in a definition. Five books made reference to this rule of definition, but only one used the phrase "previously defined," while the rest employed the word "simpler," Aristotle’s version of this rule Involves the words "absolutely prior and more intelligible," The books whioh use the word "simpler" do not indicate how the relative simplicity of two terms is to be judged, 1. Breslioh; A definition of a concept must be stated in terms simpler than the concept,

/estaway, o p , oit,. p, S67,

^^2mst Breslioh, Problems in Teaching Secondary Llathematios. p. 37, Ill

B» Davis: The wording is clear and concise, and contains only terms whose meanings are known without the aid of the concept being defined. 3. Kinney and Purdy; Furthermore, it was concluded that, to be of value, definitions needed to be stated in simpler, more commonly understood words. 4. Minniok: The wording of a definition should be adapted to the development and past exper­ ience of the pupil. It should always be in language simpler than the term defined. 5. Reeve : The essential features of a definition are that the term defined shall be described in words that are simpler than, or at least better known than, the term itself . . 6. Second British Report ; The straight line is in fact one of the primary ooncepts which cannot be defined in terms of others more fundamental still. Hence no * definition' of it is possi- ble.lS The last statement implies that definitions are "in terms of others more fundamental still." However, the phrase"primary concepts which cannot be defined" implies that the status of being undefined is an absolute one, rather than an optional one. If this interpretation is

12 David Davis, The Teaching of Mathematics. p. 130.

^^Kinney and Purdy, Teaching Mathematics In the Secondary School, p. 106.

^^innick, 0£. oit.. p. 227. 15 W. D. Reeve, Mathematics for the Secondary School. p. 200. 1 6 A Second Report on the Teaching of Geometry in Schools. British Mathematical Association, 1938,p. 178. 112 correct, then it carries with it the additional implication that the order of terms being defined is absolute which is based upon the assumption that some terms are absolutely more fundamental than others. The matter of order, or hierarchy, in terms being defined is mentioned by one book: Furthermore, he can be shown that there must be certain undefined words and at the same time see the "hierarchy" of definitions by some such series as: Rectangle, parallelogram, quadri- lateral, polygon, broken line, and straight line. Christofferson, Schultze, Shibli, and De Morgan might be considered four additional references to the above list of five on the use of simpler terms in definition, De- Morgan uses a somewhat different terminology: Definition is the explaining a term by means of others, which are more easily understood, and thereby fixing its meaning . . , Christofferson, Schultze and Shibli each refer to simpler words while defining in connection with undefined terms, which will now be discussed. Undefined terms. The need for undefined terms is usually recognized as the consequence of two rules of defining: (1) definitions should use simpler, or prior.

17 Hassler and Smith, The Teaching of Secondary Mathe­ matics . p. 3Ô9,

18 Augustus DelJorgan, On the Study and Difficulties of Mathematics. p. 11, 113 or previously defined tems and (2) definitions should not be circular, i.e. the tenu being defined should not appear in the defining phrase or be used to define terms in the defining phrase. In observing these two rules, undefined terms constitute a recogiition that we must begin defining somewhere. In Aristotle, undefined terms were considered to be absolutely indefinable because they are basically more primitive and simpler (prior) than other terms. The modern view is that the choice of undefined terms is optional and that there are no Inherently indefinable terms which are absolutely simpler than all the rest. The following books discuss undefined terms in either the ancient or the m o d e m sense : 1. Breslich: Indeed, a considerable number of oon­ cepts is indefinable. They should be used with­ out definitions.19 2. Butler and ,/ren: In the last analysis there neoessarily will be certain undefined terms which are accepted as established elements of common knov/ledge. For example there is no clarification of concepts in setting up formal definitions of point, line, plane, and space in approaching the study of geometry.20 3. Christofferson: The foundation of geometry con­ sists . . . of terms which are accepted without rigorous definition, such as point, line, dis­ tance, area, straight, direction, erect, draw.

19 Breslioh, 0£. cit.. p. 85.

20 Butler and Vren, The Teaching of Secondary Mathe­ matics. p. 424, 114

These terms may be described or explained but really cannot be satisfectorily defined by the use of concepts more simple than themselves,21

4, Davis: In geometry the undefined elements are usually listed as point, line, plane and space. Concepts associated with these names are gained through frequent usage in which certain proper­ ties are ascribed to each of them. . • , It is clear that these geometric elements have no material properties and exist merely as objects created by the Imagination, Our concept of each of these elements continues to enlarge as its properties unfold throughout the develop­ ment of the subject, , , , ./hether these elements are accepted as the undefined basic elements or are replaced by another set of primitive elements, the resulting postulational pattern is the same. That is, in mathematics as in other fields of thought, a starting point must be established by means of certain primitive concepts which cannot be defined in terms of simpler ideas, àuch con­ cepts must remain undefined, but their proper­ ties serve as coordinating factors in the exposi­ tion of other i d e a s , 22 5, Delvlorgan: './e have seen that there are terms which cannot be defined, such as number and quantity. An attempt at a definition would only throw a difficulty in the student's way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, whioh are to be found in treatises on that subject. , . , Now let any one ask himself whether he could have guessed what was meant, if, before he began geometry, any one had talked to him of "that which has no parts and which has no magnitude," and "the line which lies evenly between its extreme points,” unless he had at the same time men­ tioned the words "point” and "straight line," which would have removed the difficulty? In

21 Christofferson, op.. oit.. p. 30, 22 Davis, op, oit.. pp, 127-128, 115 this oasa the explanation is a great deal harder than the term to be explained, whioh must always happen whenever we are guilty of the absurdity of attempting to make the simplest ideas yet more simple.«3 The m o d e m approach in the statement above by Davis, which considers the choice of undefined terms to be optional, should be contrasted with DeMorgan’s older approach cited above, which considers the choice of unde­ fined terms to be governed entirely by the question of whioh set of terms is absolutely the simplest,

6, Hassler and Smith: Logically, it makes little difference which words are chosen as undefined; but from the standpoint of simplicity, it is wise to choose those which are the most primi­ tive,24

Hassler and Smith seem to have elements of both approaches in that they recognize, on the one hand, an option in selecting the undefined t e m s and imply, on the other hand, that relative primitiveness can be determined, 7, Kinney and Purdy: If we start to use a dictionary to trace definitions of terms, we discover that the attempt is made to describe words in terms of simpler, more commonly understood terms. There comes a time, however, when certain words must remain undefined; or if they were defined, are defined with the words from which we started. Similarly, in geometry we must take certain fundamental ooncepts as undefined,25

23 DeMorgan, op* oit., p. 12, 24 Hassler and Smith, op, cit,. p, 9, 25 Kinney and Purdy, op.* oit.. p, 129, 116 3« Minnlok: Mathematioal conoepts may be thought of as being arranged roughly in a series from the most elementary to the most complex• Those lying at either extreme are difficult to define. The difficulty in defining the elementary con­ cept lies in the fact that there is no known class in whioh to place i t , On the other hand the complex concept is difficult to define, because when the thing defined is placed in the smallest known class, it is not possible to find characteristics which distinguish it from other members of that class.*® Schultze was the only other writer who referred to the difficulty of defining at both ends of a series of defini­ tions . 9. Fawcett: The terms that were to remain unde­ fined were selected and accepted by the pupils . . .27 Fawcett, in keeping with modem practice, develops the meaning of the undefined terms with pupils informally and without definition, 10. Reeve: It is evident that satisfactory definitions are not always possible ; some tezms must go unde­ fined, for since the number of words in our language is limited, there must be at least one term that is as simple as any other, and this cannot be described in terms simpler than itself. Such terms, for example, as point. line, plane, end angle are good examples of terms that should go undefined.2o

26 ‘Minniok, op. cit.. p. 226, 27 Fawcett, The Nature of Proof. p. 42. 28 Reeve, op. oit.. p. 340. 117 11. Sohultze: Two classes of terms are most diffi­ cult to define, viz. general class names, as mathematics, geometry, algebra, functions, etc., and very fundamental terms, as line, direction, plane, etc. The former can be formulated end appreciated only by advanced students who really know what these terms mean; and their importance for secondary schools is consequently very small. The latter we shall consider at somewhat greater length. . . . Just as all theorems rest finally upon a number of supposedly self-evident propositions or axioms, so all definitions ultimately must be based upon a few exceedingly simple ones. Hence there exist a number of terms, such as space, boundary, position, direction, straight line, plane, etc., which are either not capable of definition or which are hard to define from "the difficulty of finding ideas more simple and intelligible than the ones to be defined, 12. Shibli: A definition of a term is a description that distinguishes it from all others. It is essential that the description be in terms that are simpler than the term defined. Therefore, in order to avoid a vicious circle in the defini­ tions, a set of one or more terms must be left entirely undefined since there can be no simpler terms to describe them.^0 This concludes the material in the books on the teach­ ing of mathematics concerning undefined terms. A closing remark will now be made about an important logical idea expressed by Davis in the foregoing series of quotations when he said, "Our concept of each of these elements

29 Schultze, on. cit.. pp. 68-69. 30 Shibli, Recent Develonments in the Teaching of Geometry, p. 84. 110 continues to enlarge as its properties unfold throughout the development of the subject.** This idea was expressed in Aristotelian terminology by Katsoff: The postulate system in which the concept occurs is clearly the class to whioh it belongs. Henoe the postulate system is the genus of the concept. Further, since the concept is not the system but a part of the system, the relation of the concept to other conoepts would give the difference. It follows that the definition by postulate system gives the genus and the differ­ ence— not, it is true, in the form of a single proposition, but the form in whioh the genus and difference Is stated Is of little consequence.31 The idea stated by Davis and Katsoff is sometimes expressed in the following words: "The system defines the undefined teims," 4. The relation of meaning to definition. It is very common in the textbooks of geometry to read statements to the effect that the purpose of a definition is to provide the meaning of a term, or the meaning of a term is not known until it is defined, or definitions enable us to place objects in classes, or definitions enable us to recognize objects, and, in at least one case, undefined terms are terms left without meaning. Now it may be sur­ prising to l o a m that, in contrast with the geometry text­ books, out of the eight books on the teaching of mathe-

31 L. 0, Katsoff, "Undefined Conoepts in Postulate Sets," Philosophical Review. Xl t II (îÆay, 1938), p. 299. 119 matios whloh make some reference to this topic, all but the book by DeMorgan state that meaning precedes definition, rather than Is the result of It. 1. Brésilch: The contention has been made In the foregoing pages that understanding of a mathe* matlcal concept should be the outgrowth of experience, and that the definition, If given at all, should be taught when the pupil knows the meaning. The test of understanding Is not the correct reproduction of a memorized state­ ment but the ability of the pupil to use It correctly In a variety of situations.32

2. DeMorgan: Definition Is the explaining a term by means of others, whloh are more easily understood, and thereby fixing Its meaning, so that It may be distinctly seen what It does Imply, as well as what it does not.33 3. Grodfrey and Slddons: In the early stages a boy should not be bothered with formal defini­ tions; he wants to get a working knowledge of the language of geometry; he must understand the meaning of words and expressions rather than be able to define them In set terms.34

4. Mlnnick: The more recent authors have wisely Introduced definitions throughout their books, where they are needed. The mathematical pro­ gram for a pupil should be so arranged that he will have a well-developed concept of a mathe­ matical element before a definition Is needed.35

32 Breslich, op, cit.. p. 88.

3 3 DeMorgan, op. olt.. p. 11.

Godfrey and Slddons, The Teaching of Elementary Mathematics, p. 255.

35Minniok, op. pit., p. 230. ISO 5» Fearoett: Perhaps it should he emphasized that definitions were made by the pupils after a reoogiition of the oharaoteristios of the con­ cept to be defined.36 6. Reeve : Unless a beginner in geometry knows what an angle is before he reads the definition in a textbook, he will not know it from the defini­ tion itself. . . . Do not formally define a term until the student knows what it means.37 7. Sohultze: In a large number of oases, the giving of a definition is simply the verbatim repet1- tion of a number of words. To be a logical exer­ cise, it would be necessary to make the students themselves formulate these definitions. Even assuming that the average student were able to do this, it is evident that he could do so only after he had acquired a clear notion of the thing to be defined, 8. Second Report on the Teaching of geometry in Schools : Hence it is relevant to inquire Eow the concepts of distance and angle, and of direction if there is one, do in fact arise or how they can be made to arise in minds in which they are not yet present. Clearly, like many of our concepts, they arise from our bodily experiences, especially from our movements.3® The last quotation is about how the meaning of an undefined term may be acquired, and is in keeping with a principle described by Peirce in the pioneering essay "How to Ivlake Our Ideas Clear":

36 Fawcett, 0£. cit.. p. 43. 37 Reeve, 0£. cit.. pp. 340, 343, 38 Schultze, 0£. cit.. p. 73. 39 A Second Report. p. 173. 121 V/hen I Just said that thought Is an aotion and that It consists in a relation, although a person performs an aotion but not a relation, which can only be the result of an action, yet there was no inconsistency in what I said, but only a grammatical ragueness* • . . I only desire to point out how impossible it is that we should have an idea in our minds which relates to anything but conceived sensible effects of things. Our idea of anything is. our idea of its sensible effects; and if we fancy that we have any other we deceive ourselves; and mistake a mere sensation accompanying the thought for part of the thought itself. It is absurd to say that thought has any meaning unre­ lated to its only function, The point of view expressed above has considerable implication for putting into practice the teaching principle that the meaning of a term should be developed before a formal definition is attempted. The development of the meaning of a terra with pupils should therefore involve the bodily experiences of turning, laying down successively a unit while measuring, drawing, stretching a thread, examin­ ing physical figures, and using instruments. Of course, some meaning can be developed by other words and synonyms, but ultimately the meanings of words will originate from sensible effects. 5, The problem of existence and definition. Only two of the books on method discuss the problem of existence:

40 Charles S. Peirce, Pragmatism and Prapmatioism. Collected Papers. Vol. V, Cambridge: Harvard t/. Press, 1934. xii 455 pp. (The quote from pp. 255, 253.) 122 1, DeMorgan: No definition ought to be introduoed until it is certain that the thing defined is really possible. Thus though parallel lines are defined to be "lines which are in the same plane, and which being ever so far produced never meet," the mere agreement to call such lines, should they exist, by the name of parallels, is not sufficient ground to assume that they do exist. The definition is therefore inadmissable until it is really shown that there are such lines which being in the same never meet 2« Reeve : It should also be understood that a definition makes no assertion as to the exist­ ence of the thing defined. If we say that a tangent to a circle is an unlimited straight line that touches the circle in one point, and only one, we do not assert that it is possible to have such a line; that is a matter of proof. Not in all oases, havever, can this proof be given, as to the existence of the simplest concepts. We cannot, for example, prove that a point or a straight line exists after we have made a somewhat futile attempt to define these concepts, We therefore tacitly or explicitly assume the existence of these fundamentals of geometry. On the other hand, we can prove that a tangent exists, and this may properly be considered a legitimate proposition or exercise of elementary geometry. In relation to geometric proof it is necessary to bear in mind, therefore, that we are permitted to define any term we please; for example, "a seven-edged polyhedron” or a "ten- faced regular polyhedron," neither of which exists. Strictly speaking, however, we have no right to use a definition in a proof until we have shovm or assumed, tacitly (as in the case of such elementary concepts as a point or a line) or formally, that the thing defined has an existence,^2

41DeMorgan, 0£, cit,. p, 207, 42 Reeve, op_, c it.. p, 343. 1£3 Although this Is a rather lengthy statement on exist­ ence, as far as the books on the teaching of mathematics are concerned, there are several important points which are not discussed, i'irst, Reeve does not indicate how existence must be established. He does not say whether construction by straight edge and compasses is the only way or one of several ways of establishing existence in geometry. This is a very important question in deciding whether problems deal­ ing with angle trisectors would be acceptable in Euclidean geometry, i^uolid, who never dealt with a geometric figure unless he v/as able to establish oonstruotlbility by com­ passes and straight edge, would most likely not accept angle trisectors into his system. Yet, certainly, the non­ existence of angle trisectors in the sense of non-construotL- bility is not the same kind of non-existence as the "seven- edge polyhedron" mentioned by Reeve, which cannot be con­ structed by any means. The non-existence of a "seven-edged polyhedron" is co mparable to the non-existence of a "parallelogram with unequal opposite sides." Secondly, when Reeve uses the phrase "or assumed . . . formally, that the thing defined exists," he does not point out that he is referring to what is known as "the hypothetical construction, ’ which was not employed by Euclid, but is generally accepted today. 184

ô. The question of reversibility and definition. This topic is given the same inadequate treatment in the books on method as in the geometry textbooks. Only two of the modem books on method refer to reversibility of definitions and neither offers an explanation of this property: 1. Hassler and Smith: It should be made clear that a definition from its very nature is reversible. If we know that a triangle has two equal sides, we may call it isosceles. If we are told that a triangle is isosceles, we know immediately that it has two equal sides. If the nature of a defi­ nition is understood, this point causes no difficulty; otherwise it may be confused with the important knowledge that the converses of true statements are not always true,43 2, Fawcett: The idea of "converse statements" also developed from this discussion, and the point was made that when one accepted a definition he also accepts that definition when turned around. This was finally stated as "A definition when turned around is acceptable authority" and was accepted as an assumption.44 The next chapter will analyze the logical justifica­ tions for reversibility of definitions and will describe two kinds of justification which it seems could be under­ stood by high school students. 7, Redundancy or overloading in definitions. There are two kinds of redundancy, often called

^^Hassler and Smith, ££. cit.. p. 368, 44 Fawcett, ££. Pit.. p. 56. 125 "overloadinc ’: (1) the excess conditions follow from the other conditions, thus the whole conbination can be shown to exist or be possible, (2) the excess conditions are incon­ sistent with the other conditions, thus the whole combina­ tion can be shewn not to exist or be impossible* The prin­ cipal type that arises in elementary geometry textbooks is the first of these. For example, a rectangle is a paral­ lelogram with four right angles* It is sufficient to say that a rectangle is a parallelogram with one right angle. Another example is: 'A parallelogram is a q.uadrilateral with the opposite sides equal and parallel." The main objection to this type of redundency in ele­ mentary geometry is based on efficiency* To prove that a figure is a rectangle or parallelogram, using the redundant definition, requires the establishment of more conditions than in the case of the non-redundant definition. For this reason a rule of redundancy is based on a very practical consideration, rather than on an arbitrary, logical restric­ tion which would not be understood by high school students* 3ix of the books on method refer to a rule of redun­ dancy, but only one, «'estaway, urges a relaxation of it: 1. Breslich: Redundancies in definitions should be avoided.45

45 Breslich, or. cit.. p. 86. 126 2. Hassler and Smith: The pupil should also see that an ideal definition tells the Immediate Glass to whloh a thing belongs and then tells enough and only enoufdti about its partloular oharaoteristios to distinguish it from all others of its class. 3. Llinniok: Definitions should be as oonoise as is consistent with the pupil's development. Unnecessary words and redundancies should be eliminated. The definition, "An isosceles triangle is a triangle having two sides and the angles opposite equal respectively," is unsatisfactory because the angles will be equal if the sides are equal. The statement that the angles are equal adds nothing to the definition. The increased wordiness tends to obscure the essential fact. On the other hand, understanding should not be sacrificed for the sake of brevity.

4. Schultze: I:either more nor fewer differences must be given than are necessary to determine precisely the meaning of the term. The state­ ment, "An inscribed polygon is a polygon whose vertices lie in a circumference, and whose sides are chords," is redundant, since two differences axe given, each of which is the consequence of the other, . . . Redundancy, hov/ever, is particularly objec­ tionable if the differences which are given are not obviously compatible, ouch a definition either involves a theorem or it defines an impossible thing. "A parallelogram is a quad­ rilateral whose opposite sides are equal and parallel" involves a theorem; while the state­ ment, "A spherioel square is a spherical quad­ rilateral whose sides are equal and whose angles are right angles," gives differences that are incompatible, and hence refers to a figure that cannot exist.^

^%assler and Smith, op_. cit.. p. 369. ^"^Llinnick, op. cit.. p. 228, 48 Schultze, £p. cit.. p. 67. 127 Sohultze*s example of the parallelogram In the second paragraph does not seem to belong there, but rather in the first paragraph* In the second paragraph Schultze is refer­ ring to definitions where the conditions are not compatible. His definition of parallelogram in the second paragraph contains redundant conditions which are compatible. The only difference is that in the case of the "inscribed poly­ gon" the compatibility of the "sides being chords" and "the vertices lying on the circumference" is established by a definition of chord, while in the case of the "parallelo­ gram" the compatibility of the "equality of sides" and the "parallelism of sides" is established by theorem, 5, Smith: Textbooks are also liable to err on the side of redundancy in definition, as in the statement, "A rectangle is a parallelogram all of whose angles are right angles," It would be thought absurd to say, "A rectangle is a four­ sided p ,irai 1 elogram whose opposite sides are equal and parallel, and all of whose angles are right angles," because of the manifest redun­ dancy, But if the definition is given at the proper place, it suffices to say, "If one angle of a parallelogram is a right angle, the parallelogram is called a rectangle," The same criticism applies to one of the common defini­ tions of a square, "A rectangle whose sides are all equal"; it suffices if two adjacent sides are e q u a l , 49

49 David 2, Smith, The Teaching of Elementary I la the- matios. pp, 260-261. 120 6, ,/estaway: In particular, do not worry about "redundant” definitions. In the early stages they are inevitable; they are then almost to be encouraged. It is much better to let a young boy say that "a rectangle is a right- angled parallelogram" than "a rectangle is a parallelogram with a right angle." A beginner naturally regards the latter with suspicion,50 It should be noted that none of the preceding books on method pointed out the justification of a rule of redun­ dancy on the basis of efficiency and economy of effort in certain proofs. a. Other pedagogical sources on definition in geometry. Outside of books on the teaching of mathematics, there are two other pedagogical sources of literature on defini­ tion in geometry, namely, periodical literature and research studies. This literature is not large, consisting of three articles and one research study, which will now be reviewed. Periodical articles. The first of these articles was by T. ’,7. Smith in 51 Johool Science and I.Iathematics and was delivered as a talk in 1910 for the old Ohio Association of Science and Llatheniatics Teachers at Ohio State University, Smith is very much opposed to variety in definitions: "The striking dissimilarity in the wording of fundamental 50 Westaway, 0£. cit.. p. 255. 51 T. Smith, "Definition in Geometry, " School Science and Mathematics. XI (December, 1911), pp. 794-801. 129 definitions is wholly out of aooord with a suhjeot over two 52 thousand years old." In advocating a direction definition of parallel lines,

Smith remarks: It is amusing, to say the least, to see an author using *left to left' and 'right to right* in a mad endeavor to avoid the term 'direction,' and in a footnote on the same page have him speak of 'similarly directed lines. Smith's attitude tov/ard the nature of definition is quite conservative: Here for the first time in the pupil’s experience it becomes more than a defining process; not only does it separate the object in mind from all other concepts in the universe, but It also takes the added nature of substantial authority in proving theorems, and as such takes its place beside the axiom or assumption as an absolute fact, back of which we agree not to l o o k . 54 Smith describes having a class look up the definition of circle in various books, which discloses three different definitions. He then has the class state what a definition is. Smith's statei.jent at this point is surprisingly m o d e m after the quotation above: Just a little steering will cause them to conclude that the definition is an agreement, made by per­ sons interested in the subject, concerning a certain

^^Ibid.. p. 796. S'T Ibid.. p. 797.

^^Ibid.. p. 797, 130 set of oondltions whloh shall he called by a oertain name, and the statement of whloh must be carefully wordgd as to shut out all other ideas In the universe,®® The following statement by Smith Is suggestive of the outlook of John Dewey: Let them understand also that they themselves have both the ability and the right to make agreements, provided they stand willing to take all consequences of the agreement. '^his of course presupposes the teacher to be guiding them Intelligently while they form this judgment.®» Smith concludes with a recommendation that a committee be set up to draft a set of definitions for all geometry textbooks. The other two articles will be taken as a group because one is a reply or criticism of the other. The first of these articles was by 3ugene Illohols®^ in The llathematlcs Teacher and the second by Robert Fouch®® in the same journal about a year later, Nichols, In a series of six criteria of a good defini­ tion, commits himself in favor of the Aristotelian genus et

^^Ibld.. p. 801.

Ibid", p, 301,

®’^ü;ugene Nichols, "How I Teach Understanding of Defi­ nition," The Mathematics Teacher. iXVII (April, 1954, pp. 274-75, ®®Robert Fouch, "Another View of the Process of Defini­ tion." The ilathematics Teacher, a LVTII (March, 1956), pp. 178, IS'ST" 131 differentia method of defining;, a rule of redundancy, a truth value for definitions by saying that the converse of a definition is true, and the use of previously defined terms in a definition. The rest of the two-page article concerned the implementation of these six criteria in the classroom, A year later, the article by Fouch appeared which criticized the points about definition in the I'Jiohols article :

In the April issue of this Journal Ivlr, Nichols presented a description of his method of teaching understanding of definition. It seems important to point out that in the history of western thought there have been many different views of this matter and that it has often been the subject of consider­ able philosophical controversy, I should like to describe here a view which I believe is more in keeping with contemporary conceptions of mathe­ matics as well as with such recent philosophical developments as semantic and logical positivism, The points which Fouch made are as follows; defini­ tions are about words and symbols, rather than objects or things; definitions are agreements to use a certain linguis­ tic form in place of another form which is usually larger; reversibility is a consequence of the term*s being an abbreviation for the longer phrase; definitions do not have a truth value ; definitions do not always have to be of nouns; redundancy is undesirable, but does not have serious

59_ Ibid.. p, 178, 132 logical consequences (where the excess conditions are deduoible); definitions should use previously defined terms. Fouch concludes by saying thet, to be consistent, he cannot say that Nichols* concept of definition is wrong, but rather *^.7e can only quarrel about the relative fruitfulness of our two definitions. A research study. Only one research study, by Miller,®^ on definition in geometry was found and examined. This thesis by Hiller does not investigate textbooks, books on method, or mathematical and logical books. The Miller thesis is rather directly based on the theories of Korzybski in semantics and proved of little value in this study. Chapter Summary In this chapter twenty-five books on the teaching of mathematics were reviewed with respect to the following seven topics: 1. Per genus et differentiam method of defining 2. The question of whether there is only one or more than one definition of a term 3. Use of simpler or previously defined terras in a definition

^°Ibid.. p. 186. Ô1 0. Sherwood I^ller, "A Theory of Definition and Mean- ing Applied to the Study and Teaching of Plane Geometry,** Unpublished Master’s thesis. University of Buffalo, 1940, 99 pp. 133 a. Undaflned texma 4. The relation of meaning to definition 5* The problem of ezistenoe and definition 6* The question of reversibility and definition 7* Redundancy or overloading in definition An eighth topic included in this chapter was the following: 8. Other pedagogical sources on definition in geometry. Five of the books on the teaching of mathematics, or about twenty per cent, stated that definitions should take the Aristotelian form of genus et differentia. Three books expressed the idea that a choice is possible when defining a term, while two implied that only one definition was possible. Six books stated that definitions should be expressed in simpler, more fundamental, or better known terms, while none stated that definitions should use pre­ viously defined terms, which is the mathematical point of view. Ten books expressed the idea that undefined terms were basically indefinable; one stated that the system defines the undefined terms ; and one stated that the stu­ dents selected them. Seven books stated that meaning should precede definition, while only one stated that defi­ nition gives the meaning of a term. Only two discussed existence and both books stated emphatically that definition does not establish existence. Only two books mentioned the reversibility property of definitions, but neither attempted 134 to explain it. Fire books expressed a rule against redun- danoy in definitions, while one reooramended a relaxation of the rule in the ease of rectangles and squares. The next chapter will undertake to discuss each of these points from the point of view of m o d e m mathematical and logical literature. CHAPTSR IV

TRSAT1ISI3T BY LOGIC AIJD MATHEMATICS BOOKS OF CERTAIN DIFFICULTIES CONCERNING DEFINITIONS FOUND HI CONTEMPORARY GE01.ÎETRY TE3CT300K3

Chapter II and Chapter III of this study have served the purpose of pointing out certain difficulties or unsound practices with regard to definition in contemporary plane geometry textbooks (Chapter II) and in mathematics methods manuals and related works (Chapter III). This chapter will take up each of these difficulties in turn and show the treatment accorded them in modem logic and mathematics books. Following this, eight major kinds of definition are presented and described* The relation of certain of these to mathematical systems and in particular, to plane geometry, is indicated* The chapter closes with a discus­ sion focused specifically on definition in mathematics* For the purpose of facilitating discussion, several of the following difficulties will be treated in related groups* The difficulties listed below constitute the ones which will be taken up in this chapter: 1* Difficulties arising out of the Aristotelian distinction between essence and property* a* The only way to define a term is by genus and differentia*

135 136 b. The purpose of definitions in geometry is to state the essenoe of a term. 0. There is only one possible definition of a tezm* 2, Diffioulties arising out of the Aristotelian notion that classes end rank of intelligibility are absolute. a. There is a fixed order of terms in defini­ tional sequences. b. There is only one possible set of undefined terms which are simpler them all others. 0. Undefined terms are too simple to be defined. 3* ,/e do not know the ueemir.r of a term until we define it. 4. Definitions can be true or false. 5. Definitions establish existence. 6. Diffioulties relating to the reversibility of definitions. 7. Other pedagogical points about definition. 1. Difficulties arising out of the Aristotelian distinction between essenoe and property. The following statement by Stabbing effectively des­ cribes and questions this distinction; The traditional theory of definition is based upon the theory of predicables. It cam be summed up in the rule; definition should be per genus et dif­ ferentiam (i.e. by assigning the genus and the dis- tinguishing characteristic). This rule expresses Aristotle’s view that definition states the essence of what is defined. Accordingly the traditional Logicians regarded definition as being of things of a certain sort, i.e. species, or concepts, not of names. Everything, it is assumed, has a determinate essenoe and there is one and only one definition appropriate to it, viz. that which expresses the essence. From this point of view definition may well seem to be the culmination of scientific inquiry. Definitions will be in no sense arbitrary; they will be determined by the nature of things. The 137

traditional oonoeption of geometry supported this view since geometrical figures were regarded as given in intuition by construction of the figure. The definition of triangle as plane figure bounded by three straight lines seemed to express tEê essence as no other definition would. Hence the acceptance of Aristotle's view that the distinction between essence end property (proprium) is absolute. So long, too, as there was belief in the fixity of organic species, it would seem that each species had an essence which must be stated in the definition. Modem theories of organic evolution have combined with modem theories of mathematics to destroy the basis of the Aristotelian conception of essence, and hence to throw doubt upon the traditional theory of definition,! The above quotation is one of the clearest attacks on the Aristotelian theory of definition to be found in modem logic books, otebbing emphasizes that the class-difference method of defining is not the only way: "It has been customary to give as the first rule that definition must be per genus et differentiam. This, however, involves an unduly narrow conception of the nature of definition. Besides per genus et differentiam. Stebbing lists three other methods of defining in mathematics: (1) ana­ lytic definition of an expression, (2) definite description, (3) genetic definition.^ v/hether or not these names or types of definition agree with other authorities is less

^L. S. Stebbing, A Modem Introduction to Logic, pp. 432-433.

^Ibid.. p. 425. ^Ibid.. p. 424. 138

important here than the fact that moat m o d e m authoritiea in logic and inathematic a agree that there are more ways to define than the Aristotelian per genus et differentiam method* A number of other v/ays are described in the latter part of this chapter* The idea, namely, that the Aristotelian method of defining is not the only way, can be documented by many other logical sources, Copi says; "Definition by genus and difference is regarded by many writers as the most important type of definition and by some as the only "genuine** kind* There is scarcely any Justification for the latter view «4 • • • Robinson comments in the same vein; The mathematicians have a tradition about rules of definition which appears to be unrelated to the Aristotelian tradition and much younger* It can be found, for example, in Valter Dubislav*s Die Definition, pp* 34-8, and in Alfred Tarski’s Introduction to Logic, pp* 35, 133, 150.® Dubislav’s work has not been translated into English, but Dewey and Bentley review its contents in an article for the Journal of Philosophy* According to Dewey and Bentley, Dubislav’s analysis yields five types of definitions: (1) Special rules of substitution within a calculus* (2) Rules for the application of the formulas of a

4 Irving Copi, Introduction to Logic, p. 110* 5 Richard Robinson, Definition, p. 148, 139 oaloulua to appropriate situations of factual inquiry. (3) Concept-constructions. (4) Historical and juristic clarifications of words in use. (5) Fact-clarifioations, in the sense of the deter­ mination of the essentials of things, these to be arrived at under strictly logical-mathemat­ ical procedure out of basic presuppositions and perceptual determinations, within a frame of theory; and from which in a similar way and under similar conditions all other assertions can be deduced, but with the understanding (so far as Dubislav is concerned) that things-in- themselves are excluded as chimerical.^ In a very important footnote Dewey and Bentley remark that Dubislav found four main historical "theories" of definition: 1. The Aristotelian essenoe and its successors to date 2. The determination of concepts 3. The fixation of meanings, historic and Juristic 4. The establishment of new signs? Tarski, referred to by Robinson above, presents the mathematical concept of definition: The phrase "if, and only if" is very fre­ quently used in laying down definitions, that is, conventions stipuleting what meaning Ts to be attributed to an expression which thus far has not occurred in a certain discipline, and which may not be immediately comprehensible. Imagii^ for instance, that in arithmetic the symbol " 3 " has not as yet been employed but that one wants to introduce it now into the considerations (looking upon it, as usual, as an abbreviation of the

°John Dewey and Arthur Bentley, "Definition," The Journal of Philosophy. XLI7 (May, 1947), p. 287. 7 Ibid.. pp. 206-287, 140 expression "is less them or equal to")* For this purpose it is necessary to define this symbol first, that is, to explain exactly its meaning in terms of expressions which are already known and whose meanings are beyond doubt* To achieve this, we lay down the following definition,— assuming that " > " belongs to the symbols already known: we say that x 6 y if, and only if, it is not the case that x >• y. . • • In short, by virtue of the definition given above, we are in a position to transform any s^ple or compound sentence containing the symbol " a " into an equivalent one no longer containing it; in other words, so to speak, to translate it into a language in which the symbol " S " does not occur. And it is this very fact which constitutes the role which definitions play within mathematical disci­ plines.® Tarski refers to the following rules of definition, although he does not attempt an exact formulation: 1* On the basis of a complete set of defining rules, every definition may assume the form of an equivalence, 2. The definiendum, should be a short, grammati­ cally simple sentential function containing the constant to be defined. 3. The definiens may be a sentential function of arbitrary structure, containing only constants whose meaning either is immediately obvious or has been explained previously. 4* The term being defined must not occur in the definiens, else the error of a vicious circle in the definition will be committed. 5. The conventional character of a definition is emphasized by the prefix "we say that," This

0 Alfred Tarski, Introduction to Logic, pp. 33, 34, 35. 141 distinguishes it from other statements whloh have the form of an equivalenoe,^ Later on, Tarski substitutes for "rules of definition" "Llethodologioal postulates of the formalization of defini­ tions," Concerning these Tarski has the following footnote: A very high level in the process of formaliza­ tion was achieved in the works of the late Polish logician S, Lesniewski (1666-1939); one of his achievements is an exact and exhaustive formulation of the rules of definition,(Unfortunately these sources were not available at this writing,) Any discussion of the mathematical tradition in defini­ tion would not be complete without reference to Peauao’s famous article "Le Definizioni in Metematica," Peano dis­ cusses the following rules of definition: 1, The form of a definition should be : definiens ■ definiendum, 2, In mathematics all definitions are nominal, 3, The rule of genus et differentia is not valid for all definitions, 4, The existence of that which is defined is not necessary, 5, Definitions o u ^ t to proceed from the known to the unknown, 6, There must be a set of primitive ideas. It is not possible to define all terms,H Further discussion of definition in mathematics may be found at the end of this chapter.

9 Ibid.. p, 35,

^‘^Ibid.. p. 133,

Peano, "Le Definizioni in liatematlca, " Bologna: Periodico Pi Uatematlohe. 6erie IV, Vol. I, No, 3, 1921, pp. 175-186, 142 Cohen and Nagel clarify the sense in which the dis­ tinction between genus and differentia is no longer valid: The distinction between genus and differentia was absolute for Aristotle, and was connected with his metaphysical views. But from a purely logical or formal point of view, the distinction is abso­ lute only within a specific context. For consider the definition, ”I.lan is a rational animal," Accord­ ing to Aristotle, the genus is "animal," the differ­ entia is "rational." But formally we may regard with equal right "rational" as the genus emd "man" as the differentia. . I . The logical function of the differentia is to limit or qualify the genus. And this function is performed by either term in the definition with respect to the other. A defini­ tion may, therefore, be regarded as the logical product of two terms. The Aristotelian distinction between essence and property gave rise to two fallacies in definition which per­ sist in many contemporary geometry textbooks : (1) that the only purpose of definition is to state the essence of a thing being defined, and (£) there is only one possible definition of a term. Regarding these matters, Robinson comments as follows: This activity of searching for essence is bad because there is no such thing as essence in the sense intended. . . . Yet there has been from Plato onwards a very strong and persistent tendency to believe in essence. In particular, it has been a tendency to believe in the distinction between essence and property. . . . That having the sum of its interior angles equal to half a turn is a property and not the essenoe of the triangle is merely a confused way of saying that

12Cohen and Nagel, ^ Introduction to Logic and Sci­ entific Method, pp. 235-236. 143 this oharaoteristlo is not assigned to be the mean­ ing of the word ’triangle,* but entails and is entailed by another charaoteristio which is assigned to be the meaning of the word 'triangle,* If we choose, there is nothing in the world to prevent us from defining the word ’triangle* to mean a figure whose interior angles sum to half a turn; and then this would be the essence of a triangle, while its having three sides would be a property,l3 C, I, Lewis corroborates the above statement: Traditionally any attribute required for application of a term is said to be of^ the essence of the thing named. It is, of course, meaningless to speak of the essence of a thing except relative to its being named by a particular term,14 Stebbing clearly shows how the unsound Aristotelian distinction between essence and property leads to the 15 fallacy that there is only one definition of a thing; Moreover, the distinction between propria (proper­ ties) and definition would appear to be relative, depending upon what definition is selected. This admission would be in accordance with modern views of mathematics. But Aristotle did not regard the distinction as relative, since essence is fixed and unalterable, ./e may conclude, then, that whilst the distinction between essential and acci­ dental characteristics is of fundamental importance, the distinction between definition as essence and propria as being demonstrable cannot be drawn in exactly the way in which Aristotle drew it,**- Cohen and Nagel c onfirm the above and emphasize more

^^Hichard Robinson, op, cit.. pp. 154-155, I, Lewis, ^ Analysis of Knowledge and Valuation. p, 41. ^^Aristotle called the subject defined a ’’thing,” by which he meant "a species,” expressed by a class-name, (^tabbing, p, 430.) ^^otebbing, 0£, cit.. pp. 431-432. 144

explioitly tiiat according to Aristotle there was only one

essence: The distinction between essence and property was regarded by Aristotle as absolute, for a sub­ ject has, according to him, only one essence. . . . Thus if we define a circle as the locus of points equidistant from a fixed point, we can formally deduce the property that its area is maximum with a given perimeter. On the other hand, if the circle is defined as the plane figure having a maximum area with a given circumference, it follows neces­ sarily that all its points are equidistant from a fixed point. The roles of definition and property are therefore interchangeable.17 In a recent article Dubs explains the persistence of the Aristotelian theory of definition and cites two points where modern theory disagrees: Definition is one of the subordinant parts of logic that was cultivated by the classic medieval Aristotelian logic perhaps more highly than by modern thinkers. The medieval account of defini­ tion has consequently been altered less by modern thinkers than any other part of classical logic. Definitions are made for use in connection with some purpose. They are subservient to other logical or practical interests. Hence there can hardly be only one definition that is "best". . . . In most oases there would seem to be several "best" definitions, each one being "best" for some particu­ lar purpose. The classic statement, that definition is a statement of a thing’s essence, must then be given up. IQ

17 Cohen and Kagel, qp_. cit.. p. 2^57. 1 q Homer Dubs, "Definition and its Problems," The Ihilosophioal Review. LII (November, 1943), p. 566. 145

IlaoKaye denies that there are proper meanings of words and states a rule of freedom in defining; It has been pointed out, however, that there are no suoh things as proper meanings of words, so agree­ ment oannot be reached by an appeal to authority as to what is proper. This seems to present a diffi­ culty, but by following a simple and easily adopted rule of intelligibility, it may readily be avoided. The rule is; Any person is free to stipulate any meaning he pleases for a word andHTls meaning sliall always be aooepted.l^ Lazar practiced this rule in his definition of con­ verse:

Since the definition of "converse" is no more than a declaration of intention to use that word instead of a longer statement, it really needs no justification. 2. Difficulties arising out of the Aristotelian notion that classes and rank of intelligibility are absolute. Stabbing remarks that in Aristotelian classification the distinction between species and genus is not relative but absolute. After an example of classifying the types of vehicles in London street traffic, she stetes: "This rough example will enable us to see the utility of arranging classes in a certain order. ;/hat order is selected will depend upon the purpose for which classificetion is under­ taken."^^

1 Q Jeunes I.:acEaye, The Logic of Language, p. 61.

a than Lazar, The Import anc e of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry, p. 16,

^^Stebbing, 0£* o i t .. p. 434. 146

aince definitions in ^?eonetry are often based upon hierarohies of classes and since the arrangement of classes may vary, it follows that such definitions may also vary. For example, Euclid has the following sequence of

definitions : 1. Undefined terms not specifically set forth part, no, breadthless, length, one, two, etc. 2. point, line, then plane surface 3. extremity 4. boundary 5. figure 6. rectilinear figure 7. trilateral figure 0. isosceles triangle and equilateral triangle (Defined independently of each other) Seymour and Smith’s, Plane Geometry, has the following sequence, which is somewhat different from Euclid’s: 1. solid, surface, line, straight line, point, encloses, portion (Set forth as undefined.) 2. line segment 3. broken line 4. closed line 5. polygon 6. isosceles triangle 7. equilateral triangle (Defined as a subset of isosceles triangle.) Concerning variations of sequence in definition, Stamper makes the following comment:

The sequence in three of Euclid’s definitions is not pedagogical, h© defines point, line, and surface in the order named, placing first the most remote from experience. This order has pre­ vailed to a large extent up to recent years. One marked exception is found in the geometry of Gerbert (later Pope Sylvester II), where the above 147 order is reversed, the definition of a solid being placed f i r s t . 28

The Aristotelian theory that there are absolute ranks of intelligibility and simplioity has led to the false idea that there is only one set of undefined terms. Cohen and Nagel disagree with the idea of only one set of undefined terms: '*.7e have already seen, in connection with the dis­ cussion of the nature of mathematics, that there are no intrinsically undemonstrable propositions or intrinsically undefinable terms. Robinson also voices this idea: There is no justification for believing in inde­ finable words ; but there is for believing in inde­ finable forms if and only if some forms are simple. However, no conclusive reason for believing in the occurrence of absolute simplicity has yet been found, dome have argued that ’if there are com­ plexes there must be simples.’ But the existence of complexes necessitates only the existence of relatively simple things, not also of absolutely simple things.84

otebbing, In discussing Russell’s concept of defini­ tion, not only explicates the modem notion of definition but also links it with the idea of undefined terms and freedom in their selection: Russell says, "A definition is a declaration that a certain newly-introduced symbol or

^^Stamper, o^. cit.. p. 135. 23 Cohen and Nagel, op. cit.. p. 237, 24 Robinson, op. oit.. p. 174. 148

oombinatlon of symbols Is to mean the same as a oertain other combination of symbols of which the meaning is already known. • • • Their importance, he says, is due, first, to the fact that the intro­ duction of a definition shows that the definiens is worth consideration; secondly, to the fact that *when what is defined Is (as often occurs) some­ thing already familiar, suoh as cardinal or ordinal numbers, the definition contains an analysis of a common idea, and may therefore express a notable advance .... * •7e can now consider I.Ir. Russell’s definition of "definition." Two points are important: (1) the definition must be expressed in terms of other expressions already defined; (2) which are the terms to be previously defined is dependent to some extent upon the choice of the person who offers the defin­ ition, The first point emphasizes the important fact that there must be terms which either are indefinable or are taken as undefined. The second point emphasizes the fact that there is some choice in the selection of the undefined terms. . . . V/hioh expressions the mathematician takes as unde­ fined will be largely determined by his climate of opinion and by the development of the subject by his predecessors. The choice of the initial concepts will determine the form of the s y s t e m , 25 Young explores somewhat further the arbitrary nature of the choice of undefined terms: Theoretically, there is no reason why we should not choose the more complex. There is absolutely no restriction upon our choice except that which has already been mentioned: The terms which we leave undefined must be such that every other term we use may be defined in terms of them, and the set of propositions which we leave unproved must be suoh that we can derive all the others from them by formal logic, without any further appeal to intuition.26

25 otebbing, op. oit.« pp. 400, 441, 442.

.V. Young, fundamental Concepts of Algebra and Geometry, p. 38. 149

The possibilities of legitimate variation j.n the hierarchies of classes of figures are most strikingly illustrated in the classifications of quadrilaterals, and their accompanying definitions, among a sampling of eight­ een geometry textbooks, ^ive different groups of classifi­ cations were found as follows: Group 1: Euclid quadrilaterals 1, Oblong - right-angled, non-equilateral 2, Square - right-angled, equilateral 3, Rhomb us - non-right angled, equilateral 4, Rhomboid - non-right angled, opposite sides and angles equal 5, Trapezia - all quadrilaterals besides the above Group 2: Slaught and Lennes, .Ventworth, Barber and Hendrix Quadrilaterals 1. Trapezium - no two sides parallel 2. Trapezoid - two and only two sides parallel 3. Parallelogram - both pairs of sides parallel a. Rhomboid - all angles oblique 1. Rhombus - all sides equal b. Rectangle - all angles are right angles 1. Square - all sides equal (Barber and Hendrix defined a square as a regular quadrilateral.) 150 Group 3: Birklioff and Geatley, Herb erg and Orleans G.gadr Hat erals 1, Trapezoid - two and only two sides parallel 2, Parallelogram - both pairs of sides parallel a* Reotangle - equiangular b. Rhomb us - equilateral 0, Gquare - equilateral and equiangular Group 4: Breslioh; Rosskopf, Aten, and Reeve ■quadrilaterals 1. Trapezoid - two or more sides parallel a. Parallelogram - both nalrs of sides paral­ lel 1, Re0tangle - right-angled a. I3quare - equilateral (redundant) 2, Rhomb us - equilateral a, Gquare - equiangular As shown above, some books define "square” in two ways: "A square is an equilateral rectangle," and "A square is an equiangular rhombus." Group 5: Bartoo and Osborn ; Durell and Arnold; Johultze; Mallory and Oakley; oohnell and Crawford; Gohorling, Clark, Smith; Seymour and Smith; Strader and Rhoads; './elohons and Kriokenberger; ./ells and Hart Quadrilaterals 1* Trapezoid - two and only two sides parallel 2. Parallelogram - both pairs of sides parallel a. Rhombus - equilateral 151 b. Reotangle - right-angled 1* 3qgare - equilateral (redundant) or two adjacent sides equal (not redundant) Many of the books which consider the rectangle a parallelogrum give a redundant definition by saying that all the angles are right angles. It is sufficient to say that only one angle is a right angle, ‘Similarly, the following definition is redundant: "A rectangle is an equiangular parallelogram." It is sufficient to say that two adjacent angles are equal. The criterion for selecting one of the foregoing classiflentions of quadrilaterals is not on the basis of which group presents the essence of the various quadri­ laterals, Mor can it be said that the ranking of the var­ ious quadrilaterals is rigid and absolute or that some are basically more complex or simple than others. It is more desirable that the choice of classification be based on per­ sonal preference and on convenience in working with them later on, 3, ^ ^ not know the meaning of a term until we define it, The point of view that v/e do not know the meaning of a term until we define it, or a similar one, has been shown to be widely held in contemporary geometry books by surveying in Chapter II the reasons they give why definitions are necessary. Over twenty books were shovra in this survey to 152 express in one form or another this fallacy* Related to the misconception that we learn the meaning of a term through its definition only is the point of view that definitions settle arguments and remove differences of opinion. This role is only one of the roles of definition in society and, in mathematics, this function hardly occurs at all. The argument-settling justification for definitions carries the assumption that v/e must always turn to a formal definition to learn the meaning of a term. The separation of meaning from definition is easily documented, otebbing discusses this point as follows: It must suffice to point out that to be able to define a word is already to know what it signifies. . , , Defining is not primarily a process of making our own thought clear; it is the signal that clarity has been achieved,27 Jtebbing puts this another way in an earlier work: But to be familiar with a concept and to know its analysis are quite different ; similarly to be familiar with a symbol and to know how to analyse it are quite different. The purpose of an analytic definition is not to explain the meaning of a sym­ bol which we already understand, but to give an analysis of it in terms of more primitive symbols

£7 Lizzie otebbing, Logic in Practice, p, 78, 28 Lizzie otebbing, A Modem Introduction to Logic. p, 442, 153

Pap explains the distlnotion between looanlng and defi­ nition in terras of children: To see how unassailable the semantic theory really is, we merely have to observe how children are taught the meaning of words. One never begins with abstract verbal definitions, but rather eluci­ dates the meanings of words by examples. In order to standardize the usage of words, one then pro­ ceeds to verbal definitions which serve as rules of correct application of words, Ayer is also among those who separate meaning and definition: xTevertheless, here again it is possible to indicate what types of relations must obtain between persons for the political statements in question to be true: so that even if no actual definitions are obtained, the meaning of the political statements is appro­ priately clarified,^0 4. Definitions can be true or false. Robinson makes some clear distinctions in this regard: Lexical definitions have a truthvalue but stipula- tive definitions have not, A lexical definition is an assertion that certain people use a certain word in a certain way, and is therefore either true or false, A stipulative definition, however, is not an assertion at all. Therefore, since asser­ tions are the only sentences that have a truthvalue, it has no truthvalue. It is more like a request to the reader that he will understand the word in a certain way, or a command ; and these, though signi­ ficant utterances, have no truthvalue.31

^^Arthur Pap, Elements of Analytic Philosophy, p, 96,

Alfred Ayer, Language. Truth and Logic. p, 24, 31 Richard Robinson, 0 £, cit,. pp. <32-63, 154 Copi agrees v/lth Robinson on these matters: The assi^iment of meanings to new symbols is a matter of ohoioe, and we may call the definitions which make the assignment stipulative definitions. . . . A stipulative definition is neither true nor false, but should be regarded as a proposal or resolution to use the definiendum to mean what is gg meant by the definiens, or as a request or command, A lexical definition does not give its definiendum a meaning which it hitherto lacked, but reports the meaning it already has. It is clear that a lexical definition may be either true or false.33 I.:aoKaye clarifies this problem with the following com­ ments : Consider for example, the follov/ing proposition: A circle is a rectangle. Anyone acquainted with the customary meanings of the terms in this prop­ osition would not hesitate to pronounce it untrue. But this is because it is interpreted to be a material, and not a definitive proposition. . . . It is assumed to be the quivalent of the proposi­ tion: "The customary meanings of the words circle and rectangle are the same," Thus interpreted, the proposition is obviously untrue. But suppose it is Interpreted as a stipulated definitive proposition, a proposition involving not tv/o, but only one mean­ ing, and expressible thus: A circle means a rec- t a n ^ e . . . . It is not a customary definition to be sure, but it cannot be untrue any more than any other identity, unless, independently of cus­ tom, by "untrue" is meant "uncustomary. **34

llax Black makes the seme points and distinctions as those above on this issue.

^^Copi, 22. oit.. pp. 94, 95, 33 IMd., p. 96. 34 James ilacKaye, oj^. oit.. pp. 101-102. 3o Llax Black, Critical Thinking, pp. 189-190. 155 Ther# la not muoh point in continuing this aerias of quotations, sine# moat logic booka are in agreement on the question of truth value and definition, furthermore, moat of them diaousa the queation in the aame manner aa above. Frequently textbooks which state or imply that there ia only one definition of a term also imply that the definition of a term ia true, while all others are false. Some textbooks bring in the question of truth by using the terms correct and incorrect with regard to definitions. It would be helpful if authors would specify that they mean "usage" in this connection. 5, Definitions establish existence. Aa was shown in Chapter II the queation of existence and definition was discussed by only three textbooks, while ten textbooks specifically referred to hypothetical con­ structions or existence postulates in relation to exist­ ence. Many textbooks seem to assume the existence of a term, once it is defined. It is not difficult to find refutations of this fallacy in modem mathematical and logical literature. The relation of existence, denotation, and connotation to definition ia very carefully discussed by Lewis: In the first place we should note the sense of meaning in which a definition may specify or determine a meaning, is the sense of the inten- sional modes of meaning. When one understands a 156 definition, it is the oonnotative signifioanoe of it whioh is so understood. The connotation or intension of an expression being fixed and known, the comprehension of it is thereby fixed and known, or can be determined without reference to particu­ lar and empirical facts, merely by reflection, . , • But no definition can medce known, or knowable by reflection, the denotation of an expression; since it cannot make us aware what things thus nameable exist and v/hat do not This position of Lewis with regard to existence leads him to question the traditional distinction between real and nominal definition: However no definitive statement asserts any exist­ ence, either palpable or mysterious, nor depends on any, Even explicative statements stop short v/lth the relations of classifications; of criteria; of meanings; and could not be made true or false by the existence or non-existence of anything whatever. The traditional distinction of real and nominal definitions is, thus, poorly taken.^7 Copi also points out the distinction between existence and definition, which he illustrates with an example: One point should be made clear, however, con­ cerning the question of "existence." ,/hether a definition is stinulative or lexical has nothing to do with the question of whether the definiendum names any "real" or "existent" thing. The follow­ ing definition: *The word "unicorn" means an animal like a horse but having a single straight horn projecting from its forehead*

37 Ibid,. p. 105. 157

is a "real" or lexical definition, and a true one, because the definiendum is a word with long estab­ lished usage and means exactly what is meant by the definition* Yet the definiendum does not name or denote any existent, since there are no unicorns. Carnap calls the (question of existence and definition a metaphysical one: ■7e have here absolutely nothing to do with the metaphysical question as to whether properties exist in themselves or whether they are created by defi- nition*39

'.laoKaye devotes over seven pages to the problem of the relation of meaning and existence. The following comment states that words may signify things which do not exist at all : Not only may words signify things, whioh from the limitation of nan's observing pwers, oannot be observed, but they signify things which do not exist at all. Definition does not imply exist­ ence,40

In Couturat’s chapter on logic in the Snoyolopedia of the Philosophical Sciences. his remarks about defini­ tion and existence agree with the treatment accorded this question in the last section of Chapter II in this study: A definition usually has a concept as its object, in whioh case we must guard against the belief that it implies the existence of the

33 Copi, cit. . p. 97, rig Rudolph Carnap, The Lop;loal Syntax of Language, p, 114. 40 James llacKaye, op, c i t ,. p, 51, 158 oorrespondlng olams, l#e. th# (logloal) azlatenoa of an objaot ocrraapondlng to tho oonoapt. In order to evoke this ezletenoe we muat either prove it or posit it ezplioitly. And since every theorem of existence is demonstrated in the last resort by another assumed existence, all existential judg­ ments amount, in the last instance, to postulates of existence.41

6. Difficulties relating to ^ reversibility of definitions. The difficulties in textbooks with respect to reversi­ bility of definitions center around the fact that out of sixty-nine textbooks examined, only five attempted explana­ tions, which were usually inadequate, of the reversible property of definitions, sixteen merely mention the property without any attempt at explanation or Justification, while all the rest of the textbooks do not even mention reversi­ bility. Tarski presents a mathematical treatment of this prob­ lem, which would seem to be easily adaptable to high-school plane geometry: If any two sentences are joined up by this phrase "if, and only if," the result is a com­ pound sentence called an equivalence. The two sentences connected in this way are "referred to as the left and right side of the equivalence. By asserting the equivalence of two sentences, it is intended to exclude the possibility that one is true and the other false ; an equivalence, therefore, is true if its left and right sides are either both true or both false, and otherwise the equivalence is false.

41 Louis Courturat, "The Principles of Logic," Ency­ clopedia of the Philosophical Sciences, p. 183. 159 The sense of an oquivnlence oen also be oharaoterized in still another way. If, in a conditional sentence, we interchange antecedent and consequent, ;ve obtain a new sentence whioh, in its relation to the original sentence, is called the converse sentence (of the converse of the given sentence). I I • Instead of joining two sentences by the phrase "if, and only if," it is therefore, in general, also possible to say that the relation of consequence holds between these two sentences in both directions, or that the two sentences are equivalent, or, finally, that each of the two sentences represents a necessary and sufficient condition for the other. The phrase "if, and only if" is vary frequently used in laying down definitions, thut is, conven­ tions stipulating what meaning ia to be attributed to an expression v/hioh thus far has not occurred in a certain discipline, and which may not be immediately comprehensible, Langer, like itussell, introduces a special logical symbol to stand for the relation of equivalence in defini­ tions : /herever we find one expression v/e may substitute the other if we like, for they mean the same thing. Now, "equivalence by definition" is a relation between propositions, and a relation between prop­ ositions is called a logical relation; so we have a new logical relation, v/hioh I shall denote by ft 2 df," "is equivalent by definition to,"43 Black points out in the following comment that the Aristotelian definition can be thought of as a necessary and sufficient condition for the use of the term being defined : 42 Alfred Tarski, op^, cit.. pp. 32, 33, 43 ousanne Langer, ^ Introduction to

"triangle” from all other species of the genus polygon. The method actually employed on page 63 is more modem in that it does not show the iuristotelian absolute ranking of genus

4 4 ,.:ax Black, "Definition, presupposition,_ and Asser­ tion,” Philosophical Review. LAI {October, 1952), p, 532, 161 and differentl a . From the foiregoing disoussion it can be seen that reversibility or oonvertibility can be justified from two points of view: 1* ilOilinal equivalence: the relation between term and defining phrase is an "if. and only if" relation, i.e. the defining phrase provides a set of necessary and sufficient conditions for the term, 2. Glass equivalence: the tern and defining phrase have the sane denotative range. In the case of the only textbooks whioh justify reversibility, Hart and Feldjnan; okolnlk and Hartley; Rosskopf, Aten, and Reeve; and Jv/enson are the only books out of the sixty-nine examined v/hioh use the first reason above for reversibility, ^kolnik and Hartley is also the only textbook which attempts an explanation by means of classes, but this falls considerably short of clarity and adequacy, ./elko .’itz, Sitomer, and 3nader attempt to relate reversibility to the mutual substitutability of left euid right members in an algebraic equation, but their treatment is not clear as to the basis of their justification, 7, Other theoretical points about definitions. Since this study argues that the Aristotejjan method of definition per genus et differentiam is not and should not be the only method or kind of definition in geometry, it would seem necessary to propose other acceptable kinds of 16£ definition* Classifioation of definitions is a very diffi- oult task for two reasons: (1) it is so easy to confuse names of purposes of definitions with names for methods of definitions and (2) the various classes and dichotomies overlap in many places. At this point an attempt will he made however imperfectly to outline the various kinds of definitions for which names have been offered. Much of the following organization was suggested by Robinson. The following distinction between purpose and method was made by Robinson: The purpose of a definition is what it is trying to do; and the method is the means which it adopts to achieve its purpose. Different definitions have different purposes; and they achieve them by different methods. V/e shall be much clearer if we know whioh of the names of a special sort of definition are names of a purpose of a definition, which are names of a method. and which are names of a specific purpose achieved by a specific method. For example, •nominal definition* is the name of a purpose, the purpose of explaining the meaning of a word; but *ostensive definition* is the name of a method, the method that makes use of pointing or physical introduction.^® a. Real and definition. According to Robinson, real definition is concerned with things in gen­ eral; but nominal definitions are concerned with symbols or words. Heath felt that nominal definitions in Euclid were converted to real definitions by means of constructions

45 Robinson, o p . cit.. p. 15. 163 which established the existence of the concept previously defined nominally, Robinson discusses many confusions revolving around the distinction between real and nominal definitions : The failure to distinguish all the time between the analysis of things and the nominal definition of words has been the cause of most of the common errors in the theory of definition,^® Vone of the errors mentioned dealt with considering real definitions to be a search for essence, a search for an identity of meaning amongst the various uses of an ambiguous word, a search for the cause of a thing, and others. î-ointing to the cause of these errors, Robinson writes Jut i/hy sliould so many activities have been confused under one name? A very large part of the cause of the birth and long life of this con­ fused concept, real definition, is surely the occurrence in language, or at least in Indo- huropean languages, of the question-form: * ./hat is X? * Real definition first appears in litera­ ture as the answer to questions put by Socrates having the form ’.That is x?* nnd the oonfusedness of the concept of real definition is an effect of the vagueness of the formula ’ .ihat is uince Rascal^® and Mill‘d® the tendency has been to

^^Ibid,. p. 177. ^'^Ibid.. p. 190. 40 t Blaise Pascal, ’'L'Bsprit de la deometrie,” in X ens^os. ed, havet, ^aris, 1381, vol. II, ^^John otuart Pill, A Bystem of Logic. London: Longmans, dreon and Co., 1919. xvi ^ 622 pp. 164 consider all definitions to be nominal. This has been espec­ ially true in laathematics and logic. b, ütinulatlve and lexical definitions, stipulative, sometimes called "legislative,” definition is the arbitrary setting up of a meening-relationship between a word and a longer phrase not containing it. Thus, one of its functions is abbreviatory, Robinson defines it as follows: By ’stipulative* word-thing definition, then, I mean the explicit and selfconscious setting up of the neaning-relation between some word and some object, the act of assigning an object to a name. . . . This is the kind of definition that ■Jhitehead and Russell had in mind when they wrote that a 'definition is a declaration that a certain newly introduced symbol, . .is to mean,’ etc.50 example of this type of definition in geometry occurs in the case of figures v/hioh are triangles with two equal sides. The examples are •oiowi. The defining properties are known in the phrase "triangles with two equal sides" which proves bulky v/ith repeated use. The short term "isosceles triangle" is then stipulated to mean the same as "trieungle with two equal sides," Stipulative definitions have no truth value because the question of v/hether they are true or false is meaningless. ,/hile most mathematical definitions are stipulative in nature, often the defining phrase is the result of an analysis of a con-

50 Robinson, on., cit.. p. 59. 165 oept which oan amount in some oases to actual concept- construction, (See descriptive definition, page 173) Lexical, or dictionary, definition is "that sort of word-thing definition in which we are explaining the actual way in which some actual word has been used by some actual persons,Since they report the conventional usage of a term, lexical definitions have a truth value. Definitions in geometry actually combine stipulative and lexical aspects. In so far as geometry definitions are the result of pupil analysis of a concept, are guided by the teocher, and involve construction of the defining phrase v/lth whioh they arbitrarily link a name, the definition is stipulative. In so far as geometry definitions are developed In the light of the conventional usage of a term in geometry, they are lexical. For example, if a geometry class, after analyzing the concept of parallel lines by examining examples of parallel lines, decides that any one of three defining phrases would be usable as a definition and then selects one of these in the light of current usage, it would seem that both stipulative and lexical considera­ tions have entered into this definition.

^^Ibid,. p, 35, 166

0 . Analytio and synthatlo definition# Analytic defi­ nition has traditionally been considered to be the Aris­ totelian definition by genus and differentiam# HnreTer, Robinson considers Aristotelian definition to be only one case of analytic definition#Even today an analytic definition is often considered to be one that states the intensional or oonnotative meaning of a term in the defining phrase# Intensional or oonnotative meaning is contrasted with eztensional or denotative meaning in that the latter refers to the class and the examples of the term being defined, while the former refers to those properties whioh are peculiar to the class of things named and which serve to distinguish that olass from all other classes# The diffi­ culties in classification can be seen in the fact that a stipulative definition in mathematics usually involves a defining phrase whioh is the result of an analysis of a concept# Synthetic definition, according to Robinson, is a method of defining in which the relation between the refer­ ent of the word being defined and other things is indi­ cated#^^ An example in physics Is the definition of a color

^^Ibid,. p# 96#

53 , p # 98# 1Ô7 In terms of a range of wavelengths of light. Conoeming synthetio definition Robinson writes: The synthetic method of word-thing definition indicates the thing meant by mentioning its relation to some other known things. It might therefore be called the * relational* method of definition.54

Special types of synthetic definition. Causal or genetic and operational definitions. Robinson refers to the causal or genetic definition aa a special type of synthetic definition which indicates the thing aigiified by mentioning how it is caused or whence it arises. He cites the example: **Thus 'circle* may be said to mean the figure covered by a line moving in a plane with one end fixed. **^5 prom this example, it appears that many of the figures in plane geometry could be defined causally or genetically. Another type of definition, which overlaps the genetic or causal definition in part, is the operational definition. Hart defines the operational definition as follows : Def. 11 An operational definition consists in: (Type P) predicting that specified operations will produce, isolate, or bring under observation the thing in question; (Type T; predicting the observations which will be experienced when

°*Ibld.. p. 99. 55 , p. 99. 168 spoolflad taftlng oparatlona ara appliad to tha thing dafinad; or (Tjpa 3} stating tha identity of tha term dafinad with soma other taxm or terms whioh have previously bean dafinad by Type F or Type T definition. . . • all operational defini­ tions ara predictions. . . .56 It can be seen that tha definition of "circle" offered by Robinson above is a "Type P" operational definition. An example of a '*Type T" operational definition in geometry is given by Williams and Williams in their high-school geometry textbook: A convex polygon is one no side of which, if produced» will enter the polygon. A concave polygon is one, some of whose sides, if produced, will enter the polygon.57 It is clear that in the examples given by Robinson and by Williams and Williams there is a break with the Aristotelian method of defining. In physics, under the impetus of the writings of P. //. Bridgman, operational definitions have also included the process of how we meas­ ure oertain basic physical quantities; The concept of length is therefore fixed when the operations by which length is measured are fixed: that is, the concept of length involves as much as and nothing more than the set of operations by which length is determined.56

^®Homel Hart, "Operationism Analyzed Operationally," Philosophy of Science. VII {July 1940), pp. 292, 311, «57 Williams and Williams, Plane Geometry, p. 64.

^®P. W. Bridgman, The Logic of Modem Physics, p. 5. 169 An example of an operational definition in psyohology is the following: The I, Q,* is the result obtained by dividing a person's mental age by his ohronologioal age and then multiplying by one hundred. d. Explioit definitions and definitions in use. Ayer makes the distinotlon between these two types of definition : In a diotionary we look mainly for what may be called explicit definitions; in philosophy, for définit ions in"use. A brief explanation should suffice to make the nature of this distinction clear. -Ve define a symbol explicitly when we put forward another symbol, or symbolic expression which is synonymous with it. . . . In particular, it is worth remarking that the process of defining per genus et differentlam, to which Aristotelian logicians devote so much attention, always yields definitions which are explicit in the foregoing sense, . . . ,/e define a symbol ^ use, not by saying that it is synonymous with some other symbol, but by showing how the sentences in which it signi­ ficantly occurs can be translated into equivalent sentences, which contain neither the definiendum itself, nor any of its synonyms.^0 All of the definitions of plane geometry seem to be of the explicit type. Definitions in use, while not appearing In plane geometry, are very important in advanced mathe­ matics, in such works, for example, as the Prlncipia Mathe­ matics by V/h it ahead and Russell. However, there, are some

60 Alfred Ayer, Language. Truth and Logic. p. 60. 170 who olalA that définitions In use do ooour In plane geometry In the fozm of the postulates because they serre as "defini­ tions In use" of the undefined tews# e. artenslre or denotative definition and oonnotatlve definition. Robinson makes the traditional distinction between denotative and oonnotatlve definition: One obvious name fo r th is method Is the 'exem­ plifying* or 'ezempllfleatory* method or 'exemplifi­ cation. * But I shall usually call It the 'denota­ tive* method. In order to have the benefit of the well known doctrine of denotation and connotation. The word 'ocean* denotes the Atlantic and the Pacific and only two or tnree other things, because there are only these few things that have the connotation which the word * ocean* namely, a huge body of water not enclosed by land7" Thus by 'denotation* of a given word I usually mean roughly either or a ll the classes which Include all and only the particular things to which it Is applied. And by the 'connota­ tion* of a word I usually mean roughly the common characteristic or sort or class In virtue of which _ the word Is applied to these and only these things. It Is evident that young children learn many geometric concepts by observing examples of them. A child might define "perpendicular" denotatively as follows: "Per­ pendicular means the position of the edges of my book pages; perpendicular Is the position of the edges at the comer of a square ; perpendicular Is how I should stand on the line In gym class, etc." Many people use the word "extension" synonymously with "denotation" and we shall not attempt to

Ô1 Robinson, op. c lt. . pp. 108-109. 171 inak» a distinotlon hers. The Aristotelian method of defin­ ing provided oonnotatlve definitions, hut as It has been pointed out In this seotlon, this Is not the only way to provide oonnotatlve definitions* Ostenslve definition. From one point of view this oould be oonsldered a sub-oase of denotative definition* Ostenslve definition oonslsts In using the term being defined and then pointing to examples or Illustrations of the term without further use of words* This Is not a form of definition to those who confine definition to statements about words. However, If definition applies to any process by which the meaning of a word Is conveyed, then It Is evi­ dent that babies must make use of ostenslve definition In the process of learning to talk* We might say that an **ostenslve" process Is used by geometry students vrhen they point out examples of angles In the room* One of these examples might be the act o f a student turning* When "angle" Is defined by means of a large recorded lis t of these examples, such a definition Is called a denotative one. After the students have analyzed this lis t, deter­ mined an appropriate characteristic coomon to a ll the exam­ ples, and have formed a definition In terms of this common characteristic, such a definition Is called a connotatlve one* f * Implicative definition. Robinson provides the following explanation of this kind of definition: 172 Oonsid«r the eentenoe, 'A square has two diag­ onals, and each of them divides the square into two rightangled isosoeles triangles.* It does not pro­ fess to be about words at all. It is not ezplieitly a nominal definition; for it does not say that 'the word means so and so.* Yet a person who oomes to i t knowing the meaning of a l l the words in i t ezeept 'diagonal* oan learn from it what the word * diagonal* means. Robinson claims th a t th is method d iffe rs from a l l the others in two respects: 1. It uses the word being defined and does not mention i t , whereas the previous methods men­ tion the word defined and do not use it. . . . 2. Here for the firs t time we come to a method of definition that does not depend on equivalence.^^ The follow ing comments by Robinson in d icate th at his "implicative” definition is really the same as "definition in use" described earlier: We might also call it the 'contezual' method, be­ cause i t puts the word in a oontezt which d eter­ mines its sense. Gergonne,6 * who probably was the first to describe it, called it 'implicit* defini­ tion in his ezcellent article on definition. The implicative method is closely connected with the synthetic method. Probably most implica­ tive definitions can be transformed into synthetic d e fin itio n s ; i f so, though they do not give an equivalence in their implicative form, they imply one. When mathematicians say that a system dispenses

^^Ibid.. p. 106.

^^Ibid.. p. 107.

D. Gergonne, "Rssai sur la theorie des defini­ tions' , in Ann^am ds Mathématiques Pures et Appliques. IX (1818), pp. 1^0. 173 with definitions hsoauss the terms are defined by the postulates in which they ooour, the postulates are Implicative definitions of the terms* It is not clear to what extent implicative definitions are possible outside mathematics, nor to what extent we actually use this method*^* g* Coordinating definition. Since this type of defi< nition does not appear to occur in high-school plane geometry, only a b rief explanatory paragraph from Robinson w ill be included here: Some definitions are said to be "prescriptions for the interpretation of calculi, by means of which formulae or parts of formulae belonging to a calculus are coupled with objects that are to be investigated through the calculus* (Dubislav), Or, as Sigmund Koch puts what appears to be the same idea, 'coordinating definition correlates empirical constructs to the formal terms of the postulate set, and thus transforms an abstract system into an empirical one’ ,®® h. Descriptive definition. This form of definition is attributed to Pepper and defined by him as follows: For the facts are that in empirical enquiry observers desire expressions which ascribe mean­ ings to symbols with the definite proviso that these meanings shall be as nearly true to fact as the available evidence makes possible* The investigator's desire is not for a nominal defi­ nition but for what I shall define as a descrip­ tive definition. A descriptive definition cannot be reduced to a combination of nominal defini­ tions plus a description. It is a Gestalt-like triadic relation in which at one and the same

^^Robinson, op. cit.. p. 107,

^^Ibid., p. 199. 174 time a symbol is given a meaning and the meaning is given a truth reference. The slgiifloanoe of a descriptive definition is that the syiid)ol defined Is reponslble to facts meant by the s y m b o l . ^7 The situation which Pepper describes in the case of descriptive definition, might occur In plane geometry under the following circumstances: the word **clrcle" Is to be defined and the figures, which the word "circle** la agreed to denote, are being analyzed for a defining property which will be taken as descriptive of the meaning of circle. The defining property may be said to have a truth reference In terms of the examples of circle. The defining property Is based on the existing extension of figures called "circle. ** This type of definition has not been recognized In plane geometry, but It seems to be covered adequately by Pepper's "descriptive definition." 1. Definitions In mathematical systems. It should be emphasized that, of the two groups of definitions in geometry, namely, those which are used In proofs (parallel­ ogram, perpendicular lines) and those which are not (postu­ late, deduction, conclusion), the former group la being dis­ cussed here, In so far as geometry Is mentioned. Sxcept for the real and nominal types, most of the above types, according to Robinson, are based on distinctions of method.

^^Stephen Pepper, "The Descriptive Definition," The Journal of Philosophy. 3CLIII (January 1946), p. 29. 175 However, It seeniB that dlatinotlons of purpose are also Implied in many cases* following is a list of six kinds of definitions in mathematical systems, presented by Robinson, which are apparently based on distinctions of purpose: 1* Abbreviations, i#e. the introduction of a new term to mean the same as a certain complex of old terms* 2* The analysis of concepts* 3* The analysis of concepts into specified con­ cepts of the system* 4* The improvement of concepts* 5* The nominal definition of the symbols of the system* qq 6 * Co-ordinating definitions* As has been stated and illustrated before in this section, all of these kinds of definition occur in plane geometry, except coordinating definition* No discussion of definitions in mathematical systems would be complete without reference to the article by Karl Schmidt, "The Nature and Function of Definitions in a 69 Deductive System*" Schmidt questions two attitudes

toward the nominal theory of definition: (1 ) that nomineü. definitions are mere typographical conveniences,

and (2 ) all nominal definitions must be in the form of an equation* Regarding the first point, Schmidt quotes Russell:

66 Robinson, op* oit*. p* 200* 69 Karl Schmidt, "Studies in the Structure of Systems: No* 5* Oie Nature and Function of Definitions in a Deductive System," The Journal of Philosophy* XXX (November 1933), pp* 64S-6S^* 176

*It 18 8 ourloua paradozi* lÆr. Ruaaell wrote in the Principlea of Mathemat1os. p. 65, * puzzling to the aymoollo mind, tiiat définit Iona, theoret­ ically. are nothing but statementa of aynbolio abbreriationa, irrelevant to the reaaoning and inserted only for practical convenience, while yet, in the development of a subject, they always require a large amount of thought, and often embody some of the greatest achievements of analysis * ^ 0

With regard to the second point, Schmidt disagreed with Feano that definitions must be propositions in the form of an equation. To support his argument, Schmidt cited the famous definition which Dedeklnd gave of a "simply infinite system." Since this example (in German) consisted of four separate statements, it appeared to be a synthetic, con­ textual type.

Schmidt went on to list what he considers to be the essential features of a mathematical definition:

(1 ) it states a system of conditions, which are satisfied by those and only those "things" which are recognized as "examples" of the defined; (2 ) this system of conditions suffices for the deduction of the other propertiesof the defined. The same remarks apply to the definitions of geometry.^! Chapter This chapter took up each of the following difficulties regarding definition found in textbooks and methods books.

'^°Ibld.. p. 647,

^4 b l d .. p. 649, 177 and diaouaaad them in the light of modern logle and mathe- matlos books and periodioala: 1. Diffioulties arising out of the Aristotelian distinction between essence and property. £• Difficulties arising out of the Aristotelian notion that classes and rank of intelligibility are absolute. 3. Definitions are not the exclusive conveyors of meaning. 4. Definitions can be true or false. 5. Definitions establish existence. 6 . Difficulties relating to the reversibility of definitions. Under the general heading "Other Theoretical Points," eight major types of definition were discussed: (1 ) real and nominal definition, (2 ) stipulative and lexical defi­ nition, (3) analytic and synthetic definition, (4) explicit definitions and definitions in use, (5) denotative defini­ tion and oonnotatlve definition, (6 ) implicative definition (shown to be the same as definition in use), (7) coordina­ ting definition, and (8 ) descriptive definition. Genetic and operational definitions were discussed under synthetic definition, while ostenslve definition was discussed under denotative definition. As far as possible, the extent and manner in which these types of definition occur in mathe­ matical systems, and in particular, geometry, was pointed out. The chapter terminated with a discussion of definitions in mathematical systems. CHAPTER Y

SUmASJ, CONCLUSIONS, AND RECOMMENDATIONS

Dlff#r#no#8 between defining praotioes and statements about definitions in contemporary plane geometry textbooks on the one hand and the treatment of definitions recom­ mended by modem mathematics and logic books on the other hand have indicated an important problem in present-day geometry teaching. The difficulties and misconceptions regarding definition which have been shown to occur in vary­ ing numbers of contemporary plane geometry textbooks are the following: 1. The Aristotelian method of defining by class and difference is the only way to define in geometry. S. There is only one purpose of defining in geometry— to state the essence and therefore the essential meaning of a term. 3. There is only one possible definition of a term in geometry. The definition must show the essence, but not the properties of a term. 4. Vie do not know the meaning of a term until we define it. Undefined terms are terms left without meaning. 5. Existence is established by definition.

6 . Reversibility of definitions is not mentioned in most of the contemporary geometry text­ books examined. Only sixteen mention reversi­ bility in any way, while five of these are

178 179

content with obscure and conflicting ezplana­ tions justifying reversibility. 7. The rule against overloading or redundancy is not consistently followed in most contemporary geometry textbooks. 8 . There is an absolute order of terms such that some are prior and more intelligible than others. This results in the fact that there can be only one set of undefined terms and but one sequence of definitions# 9. A distinction oan be made between an ideal and a material figure in geometry. 10. Definitions within geometry are justified exclusively by illustrating that definitions settle arguments In some disputes outside of geometry. 11# Definitions oan be said to be true or false. 12. Undefined terms are too simple to be defined. Chapter II of this study analyzed in oonsiderable detail the extent to whioh six defining practices, originat­ ing in the works of Aristotle, occurred in contemporary plane geometry textbooks# The six practices were as follcsst 1. Definitions should be in the form of "genus et differentia" or class and difference# 2# Definitions should not use obscure language and should use terms which are prior and more intelligible.

5 # Definitions should not contain redundant or superfluous information. 4. Definitions must be reversible. 5. There is only one true definition of a thing#

6 # Definitions state what a thing is but not the fact that the thing exists# 180 For oaoh of thoso practloom, supporting passages from Aristotle's writings were first presented and disoussed* The manner and extent to whioh contemporary geometry text­ books followed the praotioe under consideration was next presented in the form of appropriate quotations from each textbook whioh adTOoated the practice or referred to it in any way. These quotations were then summarized end dis­ cussed. This manner of treatment resulted in a rather com­ plete survey of defining practices in contemporary geometry textbooks. The principal conclusion which might be drawn from this survey is that Aristotelian defining practices are deeply imbedded in contemporary geometry textbooks to the almost total exclusion of modem pedagogical and logical recom­ mendations. This chapter concluded with a survey of the purposes of definitions indicated in textbooks. Chapter III undertook to review the treatment of definitions recommended in each of the modem works dealing with mathematics methods. In general, as was to have been expected, these works were more acceptable than the text­ books with regard to matters of teaching definitions. lYith regard to Aristotelian methods of defining, the larger number of methods books did not commit themselves, but a few specifically recommended definition per genus et dif­ ferent i am. A number of the difficulties listed previously 181 In this ohapter were attaoked by the methods books, but only a few at a time. There is no evidenoe in this 1 i terature of a recognition of different types of definition in geometry with regard to purpose and method.

Chapter 1 7 considered each of the ten difficulties previously listed in this chapter and presented the modern point of view regarding it by documentation from the current mathematical and logical literature. These views concerned chiefly the logical nature of definition and the defining process. This chapter also described the major types of definition discussed in the above literature and pointed out the relation of the appropriate definitions to geometry. The points made in Chapter IV, in addition to the peda­ gogical points in Chapter III constitute the findings and recommendations made by this study with regard to definition in plane geometry. These findings, along with the recommendations which they imply, are now summarized and listed as follows: Finding 1. The Aristotelian class-difference method of defining seems to be the only method specifically recommended in contemporary geometry textbooks. There are other logically and mathematically acceptable ways of defining in geometry besides the Aristotelian method.

Recommendation This does not mean that teachers and textbooks should reject the Aristotelian class-difference 182 method of defining* It means that they should not convey the impression that Aristotelian definition is the only acceptable way to define# The genetic, operational, and descriptive methods are some of the other ways whioh might be explored*

Finding 2. The Aristotelian distinction between essence and property, which is found explicitly or in dis­ guised form in so many textbooks, is no longer valid according to modern logic and mathematics* The per­ sistence of this distinction has contributed to the retention of the wide-spread belief that there is only one really correct definition for each term.

Recommendation 2. a* Any one of several different combinations of defining properties oan often be chosen to make an acceptable definition* V/hich combination should be chosen for a formal definition is a matter of choice, convenience, or convention* For example, it might prove worthwhile to have the students vary the definition of "parallelogram" from "opposite sides are parallel" to "opposite sides are equal" and then determine what changes v;ould be necessary in the proofs establishing the other properties of this figure. Classification by students of figures, such as quadrilaterals, in terms of angles, equality of sides, and parallelism of sides would be useful in showing the varieties of ways in whioh the different quadrilaterals may be defined*

Recommendation Z, b * The linguistic form of a geometric name, such as "isosoeles" and "parallelogram," should not be allowed to control the selection of defining properties in a definition* These forms merely sug­ gest what the traditional definitions of these terms have been, but should not prevent the adoption of alternative definitions when desirable* A notable example of a current defining phrase being incon­ sistent with the linguistic form of the term occurs in the case of triangle. The defining phrase of "triangle" in terms of three sides is not consistent with the linguistic form "triangle," which suggests three angles. Many textbooks and methods books, when Invoking the linguistic form of the term in 183 determining a definition, call up the ghost of the essenoe->property distinotlon. They olaim, for example, that since "isosoeles" is etymologloally derived from "isos" (equal) and "akeloa" (legs), this must be the essence of isosoeles triangle and consequently muat be used in the definition. From the m o d e m mathematical or logical point of view there is nothing wrong with any of the following definitions; "An isosceles triangle la a triangle with two equal angles," or "An Isosceles triangle is a triangle which has two equal angle bisectors," or "An isosoeles triangle Is a triangle where a median and an angle bisector coincide."

Finding 3. Definitions of the names of figures serve the purpose of providing a minimum beginning for deducing the other properties of the figures. Instead of presenting the above purpose many textbooks attempt to justify definitions and their importance within geometry by citing their argument-settling function outside of geometry.

Recommendat1on 3, The mathematical purpose stated above should not be overlooked or shunted aside in favor of the questionable argument-settling reason. Many disputes outside of geometry, such as between management and labor and between Russia and the Uhited States, are based on differences more fundamental than the mere definitions of terms. There are many other more valid kinds of illustrations of the purpose of definitions outside of geometry, besides that of settling arguments, such as the original and revised definition of "axle" in the Ohio law establishing an "axle-mile tax" for trucks. Situations inside and outside of geometry should be selected which help pupils develop a deeper understanding of the role of definitions. Finding 4. Existence is not established by definition but by existence postulates, by existence theorems, or by constructions. Sometimes In mathematics the formation of a definition amounts to concept- construction. Since existence is independent of 164 definition, the exemplifioetion of any concept might be a null-olaes,

Recommendation 4. In plane geometry this point becomes important when an attempt is made to propose a definition of a figure with a complex set of properties whose internal consistency has not been established. The question of the existence of such a figure is always subject to a special investigation. Finding 5• Textbooks offer many conflicting explanations of the indispensability of undefined terms in a deductive system. According to modern mathematics undefined terms are necessary because of the rule requiring the use of previously defined terms, the related rule against circularity, and the finiteness of words in the language. It is the modem view that membership of terms in the set of undefined terms is relative to the postulates and to the other definitions. The idea that terms can be ranked in an absolute order of simplicity and that undefined terms are absolutely the most simple is an Aristotelian concept whioh is no longer acceptable. Recommendation Undefined terms are not terms whioh are left without meaning. Meanings of the undefined terms are developed both by informal, intuitive experi­ ences and also by their use in the system itself. Textbooks should not convey the impression that in a system there must be undefined terms because they are simpler than all the others. Also textbooks should not claim that the undefined terms consti­ tute an unchangeable set of the simplest terms.

Finding 6 . According to modern logic, definitions are not used to convey the meaning of a term. Many textbooks, either directly or by implication, give the impres­ sion that the purpose of definition in geometry is to give the meaning of a term. Recommendation 185 Contrary to the mistaken view above, it is more oorreot to consider meaning a prerequisite to defini­ tion. Definitions should be formulated by pupils after they have investigated examples of the term and have developed a group of possible defining properties.

Finding 7. Reversibility of definitions is inadequately treated in most contemporary geometry textbooks. Modern logical literature provides satisfactory explanations of reversibility. Recommendation %. If reversibility of definitions is justified, it should be done either on the basis of the class equivalence of term and defining phrase or on the basis of nominal equivalence, or both. Nominal equivalence is indicated in the definition by the important mathematical words, "if, and only if." Another way to describe nominal equivalence is that the defining phrase presents a necessary and suffi­ cient set of conditions for the use of the term being defined. Since these conditions are neces­ sary, the inverse and therefore the converse of the definition is acceptable. Finding 8. Most of the textbooks whioh present a rule against overloading or redundancy violate the rule themselves in such cases as the definitions of rectangle, square, similar triangles, and congruent triangles. Recommendation 8. It might well prove that the redundant defini­ tions referred to above are better pedagogically than more rigorous definitions with minimum condi­ tions. Certainly no serious logical or mathematical difficulties will result from the above type of redundancy. The redundant definition, "A rectangle is a parallelogram with four right angles," will merely require more steps to prove a figure is a rectangle. The rule against redundancy is mainly a rule of economy. In some cases of complex defini­ tions overloading might increase the danger of 186

having inconsistent properties in a definition. In any case the rule against redundancy should not be stated and then violated without some word of explanation to the student.

Recommendations for other studies. I. A study m i ^ t well investigate the varieties of acceptable definitions of each of the geometric terms used in proofs. For example, define "convex polygon";

1. Per genus et differentiam definition

A convex polygon is a polygon wherein each interior vertex angle is less than 180°.

2. Operational definition

If, whenever any side of a polygon is extended, the extension does not enter the polygon, the polygon is convex.

3. Generalized operational definition

If, whenever we join two points on the perimeter of a figure, the segment fozned lies within the figure, the figure is convex,

II. Another study might be to investigate other ways of classifying quadrilaterals for the purpose of exhibiting varieties of definitions involving the special quadrilat­ erals.

For example, almost all books define a square in terms of its being a rectangle with a specific property. A rectangle is in turn defined as a parallelogram with a special property. It is obviously possible to define a square as an equiangular, equilateral quadrilateral, from which the properties of parallelism and reotangularity are deduoible. One method of conducting this analysis of the kinds of quadrilaterals would be the classical method of successive dichotony, known as the "Tree of Porphyry." III, Still another possible study would be to analyze the classification of geometric figures of any one class, 187

not by the two-valued approaoh of the Tree of Porphyry, but by a multivalued approach. For example, instead of dichotomizing quadrilaterals into parallelograms and non-parallelograms, the analysis might begin with a tripartite division, as follows: a quadrilateral with two pairs of parallel sides, one with only one pair of parallel sides, and one with no pair of parallel sides, similarly, a still different analysis of quadrilat­ erals might be made which is based upon a four-fold divi- 8 i on involving the equality of sides, as follows: begin with a quadrilateral with four sides equal, one with three and only three sides equal, one with two and only two sides equal, and one with no sides equal. Obviously the category of "two and only two sides equal" can be broken down into adjacent and opposite sides. BIBLIOGRAPHY

188 BIBLIOGRAPHY

PART I, HIGH SOHOOL GEOMETRY TEXTBOOKS EXAMINED

Auerbach and Walsh, Pleme Geometry. Philadelphia; J. B« Lippinoott, 19BO. Austin, William, A Laboratory Plane Geometry. Chicago: Scott, Foresman end do., 1986. Avery, Royal, Plane Geometry. Boston: Allyn and Bacon, 1947. Barber and Hendrix, Plane Geometry. New York: Haroourt, Brace and Co., 19^7, Barnard and Child, A New Geometry for Schools. London: Macmillan, 1903. Bartoo and Osbom, Plane Geometry. St. Louis: Webster Publishing Co,,1939. Bernard, D. Meade, Plane Geometry. Richmond: Johnson Publishing Co., 1927. Birkhoff and Beatley, Basic Geometry. Chicago: Scott, Foresman and Co., 1940. Blackhurst, J. Herbert, Humanized Geometry. Des Moines: University Press, 1935. Breslich, Ernst, Plane Geometry. Chicago: Laidlaw Brothers, 1938. Carson and Smith, Plane Geometry. Boston: Ginn and Co., 1914. Chauvenet, William, A Treatise on Elementary Geometry. Philadelphia; T. B. Llppinoott and Co., 1884. Cook, Alexander, Geometry for Today. Toronto: The Macmillan Co., 1940.

189 190 Dupuis, N« F«, Synthstlo Gsometry. London: Maomlllan and Co•, 1Ô89• Durai1 and Arnold, New Plane Geometry. New York: Charles Merrill Co., 1924. Edwards, George C., Elements of Geometry. New York: Maomlllan and Co., 1695. Fallor, Isaac, Plane Geometry. New York: The Century Co., 1910. Farnsworth, Ray D., Plane Geometry. New York: MoGraw- Hlll Book Co., 19^3. Godfrey and Slddons, Elementary Geometry. Cambridge: University Press, 190é. Haertter, Leonard, Plane Geometry. New York: The Century Co., 1928. Hall and Stevens, A School Geometry. London: Macmillan and Co., 1918. Halsted, George Bruce, Elements of Geometry. New York: John Wiley and Sons, 1866. , Rational Geometry. New York: John Wiley and Sons, 1904. Hart and Feldman, Plane and Solid Geometry. New York: American Book Co., 19ll.

Hawke8 , Luby, and Touton, New Plane Geometry. Boston: Ginn and Co., 1930. Hayn, Julius, A Geometry Reader. Milwaukee: Bruce Pub­ lishing Co., 1 6 2 6 . Henrlcl, Olaus, Elementary Geometry. London: Longmans, Green, and Co., 1891. Herberg and Orleans, A New Geometry. Boston: D. C. Heath and Co., 1940. Kenlston and Tully, Plane Geometry. Boston: Ginn and Co., 1946. 191 Leary and Shuster, Plane Geometry. New York: Oharles Scribner*8 and Sons, 1956. Legendre, Adrien, Elements of Geometry and Triaonomet^. Edited by Dayies and Amringe, New York: A# S. bames and Go,, 1885, Leonhardy, Joseph, MoLeary, New Trend Geometry. New York: Charles Merrill Co., 19357 Lougheed and rforkman, Geo^try for High Schools. Toronto: The Maomlllan Co., 1^46. Major, George T., Plane Geometry. New York: Charles Scribner's Sons, 1938, Mallory and Oakley, Plane Geometry. Chicago: Benjamin H, Sanborn and Co., 1953. cCornack, Joseph, Plane Geometry. New York: D, Appleton and Co,, 1928, Milne, VITilliam, Plane Geometry. New York: American Book Co., 1899, Mlrick, Newell, and Harper, Plane Geometry. New York: Row, Peterson and Co,, I559T Morgan, Eoberg, Breokenridge, Plane Geometry. Boston: Houghton Mifflin Co., 1933. Nyberg, Joseph, Plane Geometry. New York: American Book Co., 1944. Palmer and Taylor, Plyie Geometry. Chicago : Scott, Fores­ man and Co., 1915, Palmer, Taylor, Famum, Plane Geometry Revised. Chicago : Scott, Foresman and do,, 1924. Parkinson and Pressland, A Primer of Geometry. Oxford: Clarendon Press, 192?. Reiohgott and Splller, Today's Geometry. New York: Prentioe-Hall, Inc., 1938. Rosskopf, Aten, and Reeve, Mathematios A Second Course. New York: McGraw-Hill hook Co.,1?52, 19£

Sohnell and Oravford, Plana Gaomatry. A Clear Thinking Approaoh. New York: MoGrow-Hlll Cook Bo., 1953. Soho ling, Clark, and Smith, Modarn-Sohool Qaomatry. Yonkara -on-Hud son : VTorlÀ 5ook Co . , 1938. Sohultza, Savenoak, and Sohuylar, Plana Gaomatry. Now York: The Maomlllan Co., 1930. Saymoar, Bugana, Plana Geometry. New York: American Book Co., 1925. Slgley and Stratton, Plane Geometry. New York: The Drydan Praam, 1948. Skolnlk and Hartley, Dynamlo Plana Geometry. New York: D. Van Noatrand Co., Ino., 1950. Slaught and Lannes, Plane Geometry. Boaton: Allyn and Baoon, 1918. Smith, Eugene R., Plana Geometry Davalopad by the Syllabua Method. New York: American Book Co., 19Ô3T Smith and Marino, Plane Geometry. Columbus : Charles £. Merrill Co., 1948. Smith, Reave and Moras, Text yad Teats In Plane Geometry. Boaton: Ginn and Co., 1943. Solomon and «/right. Plane Geometry. New York: Charles Scribner's and Sons, 1929. Stone and Mallory, Modern Plane Geometry. Chicago: Benjamin ?. Sanborn and Co., 1940. Strader and ^hoada. Plane Geometry. Chicago: The John C. vrinaton Co., 1927. Swenson, John A., Integrated Mathematics with Special Application to Plane Geometry. Ann Arbor: Edwards Bothers, 193Ô. Sykes and Comstock, Plane Geometry. Chicago: Rand McNally and Co., 1918. Symon, Alexander, The New Geometry Complete. Glasgow: Robert Gibson and sons, 19287 193 Trump, Paul, Gaometry - A First Course. New York: Henry Holt and Co., 1949. Welohons and Kriokenberger, Plane Geometry. Boston: Ginn and Co., 1943 and 1952. /elkowitz, ^itomer, Snader, Geometry. Meaning and Mastery. Philadelphia: John G. Winston Co., 1950. Jells and Hart, M o d e m Plane Geometry. Boston: D. C. Heath and Co., 1926. , Progressive Plane Geometry. Boston: D, C. Heath and Co.,19 4 3. Jentworth, G. A., Plane Geometry. Boston: Ginn and Co., 1902. Williams, John H. and Williams, Kenneth P., Plane Geometry. Chioago: Lyons and Carnahan, 1915. Jorkman and Craoknell, Geometry. Theoretical and Praetioal. London: Uhlverslty tutorial Press, 1909.

PART II. METHODS IvIANUALS IN MATHEMATICS AND RELATED WORKS

Beatley, Ralph, "Third Report of the Committee on Geometry." The Mathematios Teacher. XXYIII (November, 1935), 401-45o T Breslioh, Emst, Proh^wmw in Teaohing Seoondary Mathe­ matios. Chioago: ' bhTversity of cJhioago Press, 1931. Brown,, Claude, The Teaohing of SeooiSeoondary Mathesiatios. Now York: Harper and Brothers, 19! Butler and Wren, The Teaohing of Seoondary Mathematios. New York: MoGraw-kill Book Co., 19^1. Cajori, Florian, The Teaohing and History of Mathematios in the United States. Bureau of aduoatlon. Oiroular of information Ko. 3. Washington: Government Printing Offioe, 1890. Christofferson, H. C., Geometry Professionalized for Teaohers. Oxford, Ohio: The Author, 1^35. 104 DavlB, David R., The Teaohimc of Mhthematiee. Cambridge: Addison-Weeley Frees, ino•, iO^lV DeMorgan, Augustus, 0^ the Study yid ptrfloulties of Mtoth- matios. LaSalle : Open Court Fublishing Co.,1043. Fawoett, Harold P., The Nature of Proof. Thirteenth Year­ book of the National CounoTT of Taaohere of Me the- matios. New York: Bureau of Publications, Teachers College, Columbia University, 1938. Godfrey and Siddons, The Teaching of Blwnantary Mathe­ matics. Cambridge; the Uhiversity Press, 1031. Hassler and Smith, The Teaching of Secondary Mathematics. New York: The Maoiuilian Co., 1037. Kinney and Purdy, Teaching Mathematics in the Secondary School. New York: Rinehart and Co., 195£. Minniok, J. H., Teaching Mathematics in the Secondary School. New York: Prentice-Hall, Inc., 1930. Heave, William D., Mathematics for the Secondary School. New York: Henry Holt u d do., 1)54. The Reorganization of Mathematics in Secondary Education. Report of the ^tional Committee on Mathematical Requirements under the auspices of the Mathematical Association of America. The Mathematical Association of America, Inc., 1983. Schorling, Raleigh, The Teaching of Mathematics. Ann Arbor; The Ann Arbor Press, lb^6 . Schultzs, Arthur, The Teaching of Mathematics. New York: The Macmillan do., 1931. Shibli, J., Recent Developments in the Teaching of Geome­ try. State College, Pennsylvania: y. sh i b ^ Rub- liaher, 1932. Smith, David £., The Teaching of 1S1 amentarv Mathematics. New York: The Macmillan So.,1901. Smith and Reeve, The Teaching of Junior High School Mathe­ matics. BostonMnn and Co. , lëiST/. 190 Stamper, Alva, A History of the Teaohlng o^ Blementary ometrv. "iîew York: Teachers College Gontributions to 2&&uoatlon No. 23, Columbia Uhiversity# 1909. The Teaohlng of Geometry. Fifth Yearbook of the Rational Counoil of Teaokers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia U., 1930. A First Report of the TeaoMng of Geometry In Schools. " Prepared for the ÉrltIsh MathZcatloal Association, London: G. Bell and Sons, Ltd., 1923. Jestaway, F. W., Craftsmanship In the Teaching of Slamentary Mathematics. London: Blaokle and Son, 1937.

PART III. LOGIC AND PHILOSOPHY On MAOHEMATICS BOOKS

The

, editor. Philosophical Analysis. Ithaca: Cornell T]niTeP8lty I « 8 8 , 15TSÜ.------196 Blaokhurst, J . Herbert, BuolideaLii QeoBietry. Its Nature end Its née. Dee Moinee: Gamer Publishing Co., 1947. Boole, George, ^ Invaattgatlon of tby Leiry of T^ougbt# Newiw York: DoTer PublloatlPubllo ations • (First A^rloan Print - Ing of the 1954 edition).

Bridgman, P. VI., % e Nature of Some of Our Physio^ Con- eeots. New York: PbilosopbioallAbrary, 1952. Burke, Kenneth, A Grammar of Motives. New York; Prentioe- Hall, Ino.,"i9Ü2. Burnham, James and Wheelwright, Philip, Introduction to PhilosoDhloal Analysis. New York; Henry Holt and Co., 1938. Burtt, Edwin, Prjficiples and Problems of Right Thinking. New York: Harper and brothers, ITS b . Camap, Rudolph, The Logical Syntax of Language, New York: Haroourt, Brace and Co., 1937. Chatterjee, S. C., The Nyava Theory of Knowledge. Calcutta; University of Calcutta, T555V Churchman, C. West, Elements of Logic and Formal Science. Chicago: J. B. Lippincott Co., 1940. Cohen, Morris and Nagel, Ernest, Introduction to Logic and Scientific Method. New York: Hareourt,“Brace and Co., 1934. Conger, George, A Course in Philosophy. New York: Haroourt, Brace and Co., 1924. Cooley, John, A Primer of Formal Logic. New York: The Maomillan*"Co., 1949. Copi, Irving, Introduction to Logic. New York: The Mac* millan Co,, l95&. Cotter, A. C., ^gic and Epistemology. Boston: The Stratford CoT,19&0, Couturat, Louis, The Algebra of Logic, Chicago: The Open Court Publishing Co,, 1914. Creighton, James, ^ Introductory Logic. Revised by Harold Smart, New York: The iMiomlllan Co., 1932. 197 Cunningham» H* £•» Tartbook of Logio. New York: The Maomillan Co., 1926. Dewey, John, How We Think. Boston: D. C. Heath and Co., 1910. Logio. The Theory of Inquiry. New York: Henry Holt and Co., 1938. , The a.uest for Certainty. New York: Minton, Balch and Co., 1929. Dodgson, Charles, Buolid and His Modern Rivals. London: Maomillan and Co., 1885. Dotterer, R. H,, Beginners* Logio. New York: The Maomillan Co., 1929. . Philosophy by *<^ay of the Soienoes. New York: The Maomillan Co., 1930. DubiSlav, iYalter, Die Definition. Leipzig: Felix Meiner, 3 rd. ed., 1931. Eaton, Ralph, General Logio. New York: Charles Soribner’s Sons, 1931. , Symbolism and Truth. Cambridge: Harvard Univer­ sity Press, 1925. Snoyolopedia of the Philosophioal Soienoes. 7ol I: Logio. London; Maomillan and Co., 1913. Evans, D. Luther and Gamertsfelder, V/alter S., Logio. Theoretioal and Applied. New York: Doubleday, Doran •and Co., Ino., 1937. Feigl, Herbert, and Sellars, Wilfrid, editors. Readings in Philosophioal Analysis. New York: Appleton-Century- Croft8, ino., 1949. Heath, T. L., Buolid. The Elements. Annapolis: St. John's College Press, 19497 History of Greek Mathematios. Oxford: The Ôiarendon Press, 1921. Jeffreys, Harold, Soientlfio Inferenoe. Cambridge: The University Press, 1937. 198 Joad, 0. E, M . , A Orltlque of Loidoal Positlvlam. Chicago: The UhlveraTty of onloago ^resa, Jones, Adam Leroy, Logio. Induotive and Dednotire. New York: Henry Holt and Co,, 1909. Kant, Immanuel, Critioue of Pure Reason. New York: Willey Book Co., 1945. (ïntroduotîion by the translator, J. M. D. Meikeljohn) Kattsoff, Louis, A Philosophy of Mathematios. Ames, Iowa: The Iowa Sta^e College Press, 1948. Keyser, Cassius J,, The Pastures of Wonder. New York: Columbia University Press, 1929. Klein, Felix, Elementary Mathematios from an Advanqed Standpoint: Geometry, tfew York: Dover PiAlioations. 1939. (First M i t ion 1908) Langer, Susanna K., An Introduction to Symbollo Logic. London: George"Zllen and tnwin, Ltd., l9o7, Lazar, Nathan, The ^porteaice of Certain Conoepts and Laws of Logio for the Study anT"Teaotiing o i Oeometyr. Mehasna, Wisconsin: George Banta Puolishing Co., 1932. Lewis, Clarenoe Irving, ^ Analysis of Knowledge and Valuation. LaSalle, Illinois: “îhe Open Court Pub- lishing do., 1946. MScKaye, James, The Logio of Language. Hanover, New Hampshire: Dartmouth“^ollege Publications, 1939. Mill, John Stuart, A System of Logio. London: Longaans, Green, and Co,, 1919. Miller, C. Sherwood, A Theory of Definition and Meaning Applied to the Study and Teaobing of Plane Geometry. Unpublished Master*s thesis, university of Buffalo, 1940. Montague, Wm. Pepperell, The Ways of Knowing. London: George Allen and Uhwin, ltd., 1925, Nagel, Ernest, On the Logic of Measurement. New York: The Author, 1930. 199

Nlood, Jean, Foundation» of Oeometry and Induction. New York; The Humanitiea Preaa, 1950. Ogden, C. K. and Richards, I. A., The Meaning of Meaning. New York; Haroourt, Brace anTl^o,, 1930• Pap, Arthur, Elements of Analytic Philosophy. New York: Maomillan Co., 1949. Poincare, Henri, Science and Hypothesis. New York: Dover Publications, Inc., 1952 (Original translation 1905). ______, Science and Method. New York: Dover Publica­ tions, Inc., 1952. %uine, ,71 Hard, Matheyitical Logic. New York: Norton and Co., Ino,, 1940. Robinson, Daniel S., The Principles of Reasoning. New York: D. Appleton and Company,"T929, Robinson, Richard, Definition. Oxford, Clarendon Press, 1950. Rosenbloom, Paul C., The Elements of Mathematical Logic. New York: Dover Publications, Ino., 1950. Runes, Dagobert, editor, IVentleth Century Philosophy. New York: Philosophical Library, 19^7. Russell, Bertrand, ^ Inquiry Into Meaning and Truth. New York: «7. V7. Norton and do., Inc., 1940. , Introduction to Matoematleal Philosophy. London: The Macmillan do., 1919. Searles, Herbert, Logic and Scientific Methods. New York: The Ronald Press Co., 1948. Sellars, R. IV., The Essentials of Logic. Boston: Houghton Mifflin Co., 1925. Stabler, E. R., An Introduction to Mathematical Thought. Cambridge: Tddison-,/esley Publishing Co., Inc., 1953. Stebbing, L. S., A Modem Introduction to Logic. New York: The Humanities Press, 1930. 200 Stebbing, L* S«, Logie in Praotioe. London: Methuen end Co., 1948. Tarski, Alfred, Introduction to Logio and to the Methodol- ogy of Deduotlve Solenoea. Neur York: Oxford Uhlver- sity Press., 1941. y/aloott, Gregory, ^ Zl«mentary Logic. New York: Haroourt, Brace and Co., 1951. ./erthelmer. Max, Productive Thinking. New York: Harper and Brothers, l64&. V/eyl, Hermann, Philo so phy of Mathematics and Natural Science. Princeton; Princeton Uhiversity Press, 1949. »Vhltehead, A, N. and Russell, B., Principle Mathematloa. Cambridge: Cambridge University, 1910-1913. •/Indelband, Mf., A History of Philosophy. New York: The Macmillan Co., 1985.

H o l t f A., The Correspondence of Spinoza, New York: Lincoln Mac Yeagh, The Dial Press, 1927, Young. J. A., Lectures on the Fundamental Concepts of Algebra and Geometry. ïTew York: TEe kacmlllan Co., 1911.

PART 17. PERIODICAL LITERATURE

Abraham, Leo, "IVhat Is the Theory of Meaning About,” Monlst. XLVI (July, 1936), 228-258. Bentley, Arthur, "Kennetlc Inquiry,” Science. CXII, (December, 1950), 775-783, Black, Max, "Definition, Presupposition, and Assertion,” Philosophical Review. IXI (October, 1952), 532-550.

Butler, John P. , ”0n Definition," Monlst. XLVI (January, 1936), 1-12. Dewey, John and Bentley, Arthur, "Definition,” Journal of Philosophy. XLI7 (May, 1947), 281-306. £01 D#wey, John and Bent lay, Arthur, **Speolfloatlon, ** Journal of Philosophy. ÏLIII (November, 1946), 645—663# "A Terminology for Khowlngs and Zhowns,** Journal of Philosophy# XLII (April, 1945), 285-E47. Fouoh, Robert 3», "Another View of the Prooesa of Defini­ tion," Teaoher, 2XVIII (March, 1955), 178, ISTT" Hart, Hornel, "Operationism Analyzed Operationally,” Philosophy of Science. VII (July, 1940), £88-513. Kaplan, A#, "Definition and Specification of Meaning,” Journal of Philosophy, ILIII (May, 1946), £81-286#

Kattsof, L# 0#, "Dhdefined Concepts in Postulate Sets," PhilOTOPhioal Review. XLVII (May, 1938), 293-300# Keesler, E. R., "Vocabulary in Plane Geometry," The Mathe­ matics Teacher. 300CV (November, 1942), 351# Lazar, Nathan, "The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry, " The Mathematics Teaoher. XXXI (1938), 99-113, 156-174# LeBlanc, Hughes, "On Definitions," Philosophy of Science. XVII (October, 1950), 302-309. Lewis, C# I#, "Some Logical Considerations Concerning the Mental." Journal of Philosophy. 3DQCVIII (April, 1941), 225-233. Moore, Omar Khayyam, "Nominal Definitions of 'Culture,*" Philosophy of Science. XIX (October, 1952), 245-256# Nagel, Ernest, "The Formation of M o d e m Conceptions of Formal Logic in the Development of Geometry," Osiris. VII (1959), 14fi-884# Nichols, Eugene, "How I Teach Understanding of Definition," The Mathematics Teacher, XLVII (April, 1954), 274-276# Peano, G#, "Le Definizioni in Matematica," Perlodlco Di Matematiche. Serie IV, Vol. I, No# 5, (1^21), £79^288# Pepper, Stephen, "Descriptive Definition," Journal of Philosophy. XLIII (January, 1946), 29-33% 202 Reid, J. R., "Definitional Rules: their Nature, Status, and Normative Function," Journal of Philosophy* XX (April, 1943), 108-192, . "The Dilemma of Definition," Joornal of Philosophy. HB cyI (September, 1939), 505-517, . "What Are Definitions?" Philosophy of Soienoe. STlI (March, 1946), 170-175, Schmidt, Karl, "Studies in the Structure of Systems," Journal of Philosophy. XXX (November, 1933), 645-659, Smith, T, W,, "Definition in Geometry,** School Science and Mathematics. XI (December, 1911), 79i-001, AOTOBIOORAPHY

I, Sh«ldon Stephen Myere, warn b o m In Norwood» Ohio, April 13» 1917. I received my elementary and aeoondary school education in the public schools of Norwood» Ohio. My undergraduate training was obtained at the University of Cincinnati where I received the Bachelor of Science degree in 1941 and the Bachelor of éducation degree in 1942. I received the degree of Master of Education from the Uhiver­ sity of Cincinnati in 1946 when I was awarded the Phi Delta Kappa Prize. Prom 1941 to 1946 I taught mathematios and occasionally science in several Ohio high schools. In 1946 I became a supervising teacher in mathematics in State High School» V/estem Michigan College» Kalamazoo» with duties involving the training of student teachers. In 1946 I became a mathematics instructor in University High School» Ohio State University» and began studying towards the Ph.D.» majoring in Mathematics Education. In 1951 I became an instructor in the Department of Education» Ohio State Uhi­ versity» where I taught orientation courses and supervised student teachers. From 1950 to 1951 I edited the department entitled "Applications" in The Mathematics Teaoher and wrote a section of the Twenty-Second Yearbook of the National Council of Teachers of Mathematics entitled "Emerging Practices in Mathematics Education." Between 1946 and 1955 I participated as a consultant or speaker in thirty work­ shops and professional meetings of mathematics teachers.

203