ࡱ>  cbjbj 2HH|S|S````````8 abT`ޠb:%c;c;c;cddd]______$*J`nddnn``;c;ctttn^`;c`;c]tn]ttQ;c `8oz,I0ޠ%ttqtXt` d\rgftikdddbtdddޠnnnntddddddddd|S _: MORE EQUAL THAN OTHERS: A VIEW FROM THE GRASSROOTS John Cable King's College London john.cable @ kcl.ac.uk This article offers some analysis and criticism of the ubiquitous language and symbolism of equality in mathematics. By way of preparation some attention is given to the key processes of idealisation, abstraction, symbolisation and generalisation. The authors standpoint is in mathematics education, but the criticisms mirror those made recently from a more elevated perspective. Introduction I have just discovered a wonderful article entitled Why are some things equal to other things? by the mathematician Barry Mazur ([2008]). This is remarkable, amongst other reasons, because it shows a mathematician drawing attention to some of the chinks in the armour of modern mathematics. The eponymous notion of equality is described as treacherous. Conflicts are revealed over the definition of natural number. In Peanos axioms the entire apparatus of propositional verification is recruited in order to achieve some limited particular end. (To which I would addsee 2.3.2 belowthat that end is to contrive a spurious operation of addition on things that do not naturally possess one.) I do not say that Mazur regards all these blemishes as serious. He is concerned to promote a high level super-theory called category theory, and from this elevated perspective some ambivalence or ambiguity at the lower levels of mathematics can be tolerated, or even welcomed, because it serves to show that the essence of mathematics lies in the structural similarities that (so Mazur believes) lie behind the apparent discrepancies. His main target, therefore, is the rival philosophy of formal systems and foundations, which shows no such tolerance. Mazur acknowledges that this is still the lingua franca of mathematics, and he treats it with tactful respect. But his purpose is evidently to undermine it. I myself live on the lower slopes of mathematics and see things from a different point of view. Despite the grandeur of the peaks above, I am less inclined to dismiss anomalies and obstacles that lie closer to hand. Nor am I convinced that these problems cease to exist just because the observer moves further up the mountain. In the present article, therefore, I should like to send a report from the foothills. I shall not presume to comment on category theory. Nor shall I start by wishing to attack current beliefs in mathematical philosophy. I shall focus on some very elementary mathematics, concluding with the issue of equality. It turns out, however, that by analysing very elementary mathematics one does find oneself challenging many of the current orthodoxies of both mathematics and mathematical philosophy. I like to hope that, on certain points at least, Mazur and I may meet up somewhere in the middle. A note on mathematics education My own background is in mathematics education, but I hasten to add that the present article is about mathematics, not education. It is part of my thesis that the teacher of young children and the mathematical philosopher have a common interest in the foundations of their subject, and that both may have something legitimate to say about it. For the most part, of course, educators are modest people, who see their task as being to transmit knowledge rather than to criticise it, and to some extent this attitude persists when they do research. On our present topic of equality, for example, one will find in the literature of mathematics educationKieran ([1981]), Jones and Pratt ([2012]) etcsome valuable analysis of the different ways in which the = sign can be interpreted. Thus in a formulation like 7 + 4 = it can be taken simply to mean Put the answer here. (And a similar instructional interpretation must be placed on the = key on a pocket calculator.) In a more complex case like 7 + 4 = 6 + 5 = 11 it must be interpreted as some kind of equivalence. There is also what Jones and Pratt call the substituting meaning. The chief aim of this research, however, will be to assist teachers in getting children to master the orthodox mathematics. There is no explicit criticism of the mathematics. Much the same applies to work on the vital processes of abstraction, generalisation and symbolisation, to be considered in section 2 below. I will argue that on these topics the analysis made by the mathematics educator Zoltan Dienes ([1961], [1963]) is more penetrating than anything I have found in the literature of mathematics itself or mathematical philosophy. Yet even Dienes presents his work as an account of childrens thinking rather than an analysis of mathematics. The one person I can think of who has used mathematics education as a platform from which to mount a sustained criticism of mathematical philosophy is Paul Ernestsee particularly Social Constructivism as a Philosophy of Mathematics ([1998]). If I say no more about Ernest in the present article, that is partly because I hope to get away without touching on deep matters like social constructivism, and partly because I shall be as much concerned with mathematics itself as with its philosophy. But I am certainly in debt to Ernest for his general example, and I wish, at some distance perhaps, to follow in his footsteps. Four basic dichotomies or processes I shall argue that the notion of equality is attended by several difficulties. One is that equivalence among objects is always relative to one or more specific attributes and that the omission of a specified attribute can, and often does, create avoidable ambiguity. A second is that equivalence in a domain of concrete elements is not projected into the associated abstract domain because it has been factored outbut this fact is routinely ignored. A third concerns the interpretation of symbols in more formal languages. In order to substantiate these criticisms, however, it will be necessary to make a few preliminary remarks about four basic dichotomies: between ideal and ordinary between abstract and concrete between symbol and concept between generic and specific. These may also be thought of asor at least associated withfour mental processes: idealisation and its inverse modelling abstraction and its inverse concretisation symbolisation and its inverse interpretation generalisation and its inverse particularisation or specification. Up to a point, of course, these processes and distinctions are all familiar enough. They are the stuff of philosophy, and most of them are in the vocabulary of mathematicians. If the mathematician has any misgivings, it will be because mental processes fall under the anathema of psychologismthat is, although they may help to explain how mathematics is learnt, they do not contribute to the analysis of what it actually is. It is my thesis that they are a vital part of mathematics itself, and I will venture to suggest that they are matters where the mathematics educator has a special contribution to make. Space here permits only a most brutally abbreviated treatment, but I will try to bring out those features that bear especially on the notion of equality. Ideal versus ordinary The classic model of idealisation is provided by Euclidean geometry, which postulates the existence of deliberately idealised objects: points, lines, rectangles etc. One immediate benefit for geometry is that geometrical objects, unlike ordinary objects, have relations of exact equality with respect to continuous attributes such as length and area. It is, of course, remarkable, that a theory involving such blatant idealisation should prove to be of such great practical utility. But that is manifestly the case: Euclidean geometry can be used to model a vast range of spatial situations. Euclids example has since been followed in science, most notably in Newtonian mechanics, which likewise postulates ideal objects (infinitesimal particles, rigid bodies, inelastic strings etc), again having relations of exact equality with respect to continuous attributes. Indeed one might sayand it is worth saying in order to combat any residual belief that natural science is a purely inductive subjectthat the chief characteristic of modern science is the postulation of theories or models built up of idealised elements. As all the world knows, the explanatory power and practical utility of these theories have been staggering. Measurement theory A particular example of idealisation occurs in that branch of science known as measurement theory, where the study of length is invariably conducted by reference to ideal rods, which are in effect Euclidean line segments that are free to move. Here the idealisation sometimes seems to embarrass scientists brought up to regard their subject as purely empirical, but in the end they all accept it because otherwise there is no way in which two rods can, when necessary, be exactly equi-linear. (I ignore for the moment the attempt by Krantz et al ([1971], [1989], [1990]) to axiomatise measurement theory in such a way that the ideal rods have only a heuristic role. Even they give some attention to ideal rods.) In my view measurement theory has special importance in current mathematico-scientific thought because it is the one place where the study of continuous quantities like length (which Euclid called magnitudes) is kept alive without trying to subordinate these attributes to real number. It was Norman Campbell ([1928]), one of the founding fathers of measurement theory, who first clearly recognised that degrees-of-length have their own operation of addition, quite independent of numbers, and this is the critical step (so I believe) in developing the whole theory of length on the foundation of ideal rods alone. Much the same can be done for other quantitiesmost readily for other extensive quantities like area, volume, mass and duration. At first sight it is a problem that linear dimensions are found with a great many other objects besides ideal rods. However, the width of a rectangle, for example, can be taken as the length of rod that will fit across it transversely, and so can the diameter of a football. In this way a theory that is developed using ideal rods will find wider application laterjust like geometry. Idealisation and arithmetic In arithmetic the need for idealisation is less immediately apparent because two sets of ordinary objects can usually be declared equi-numerous by simple matching or counting. In consequence arithmetic can seem, ironically, to be more intimately concerned with real things than either geometry or physics. I should say that in other fieldsphilosophy, physics and parts of psychology, to say nothing of educationthe notion of ordinary object has received greater critical scrutiny, and many people have concluded that the bundling of sense-data into objects is part of the human interpretation of sense-data rather something than inherent in the data themselves. Einstein, no less, declared the concept of bodily object to be a free creation of the human mind ([1936]). For much of everyday life, however, such sophistication is unnecessary, and mathematicians, when they condescend to consider ordinary objects, very often follow common sense down the path of nave realism (as indeed did Plato) and assume that ordinary objects are given elements of external reality. The love affair between arithmetic and nave realism reached its apotheosis in Russells proposal that two-ness, for example, should be defined actually to be the set of all pairs-of-things in the universe. But this was also its nemesis, because the ambition to embrace all possible sets-of-things proved to be, as Mazur puts it, over-greedy. In this situation the initiative has been seized by those who see the essence of natural number in rank order rather than how-many-ness, and the result is nowadays enshrined in Peanos axioms. This, however, is a situation in which the teacher of elementary arithmetic has some relevant experience. The teacher cannot afford to ignore how-many-ness. Moreover, he needs to showmore imperatively perhaps than the mathematician or philosopherthat arithmetic applies to sets of ordinary objects. Nevertheless, the teacher will also find it convenient to develop the theory of arithmetic by reference to artificial objects like counters or marks on paper, chosen for ease of manipulation or production rather than for their intrinsic interest as objects. From uniform plastic counters it is a small step to the conception of ideal counters that are entirely featureless and imaginarybut still, of course, sufficiently distinct to be counted. Many thinkers, among them both Dedekind ([1888]) and Cantor ([1895]), have sought to base arithmetic on a domain of objects that might be called ideal featureless counterspeople sometimes call them unitsand I believe that they are right. Unfortunately, there are complications because neither the counters themselves, nor sets-of-counters, have the properties expected of numbers, and it is necessary to think in terms of abstraction as well as idealisationsee immediately below. But the idealisation provides the foundation. A theory of how-many-ness based on a domain of ideal counters can then be applied to whatever phenomena can be adequately modelled by itjust like geometry or natural science or the theory of length in measurement theory. In none of these cases is there any need to embrace all possible applications from the very beginningthat is being over-greedy. Abstract versus concrete I will use the words abstract and concrete to denote the distinction between attributes and the things that bear them. This agrees to some extent with normal practicea table is concrete and its length abstractbut unfortunately normal practice also shows much confusion, and very often the word abstraction is used to mean what I have just called idealisation. Significant steps in the clarification of this rather murky area were made by Frege ([1884]), who focused on the mathematically important attribute of how-many-ness, which, of course, is one meaning of number. Unfortunately, Frege made life difficult for all of us by trying at first to deny that number is an attribute at allor at least an attribute like colourand this has encouraged the belief that it must be treated as a very special case. But this denial was rather contradicted when Frege himself proceeded to draw a telling comparison between how-many-ness and the attribute of direction, which is associated with the equivalence relation of parallelism in such a way that two parallel lines have the same direction. Frege recognised that there was a similar equivalence relation that he called equi-numerosity, whereby two equi-numerous sets-of-objects have the same degree of how-many-ness (or in common language the same number of elements). There has been some reluctance to extend Freges insight to continuous quantities like length and volume, chiefly because of worries over the approximate nature of equality with respect to such quantities (at least if this depends on physical comparison). But this problem disappears once it is realised that theories, even in physics, are based on idealised objects. It is then clearor at least I think it should bethat the equivalence relation of equi-linearity plays the same role in the domain of ideal rods as does parallelism in the domain of fixed geometric lines. In fact the attribute of length is in one respect easier to analyse than how-many-ness, because length is the attribute of individual objects (rods) whereas how-many-ness can be attributed only to sets-of-objects. (As has been well said, in the spectacle of two fat cats the fat-ness is the attribute of each animal, but the two-ness belongs only to the pair as a whole.) This means that the next step in the analysis of how-many-ness is to conceive the power-set of the population of counters (the collection of all sets-of-counters that can be drawn from it), for it is this power-set, rather than the population itself, on which the equivalence relation of equi-numerosity is defined. The relation then partitions this domain so that all pairs-of-counters go into the same class, all triplets into the same class, and so on. Once the concrete domain has been correctly identified, how-many-ness is seen to share many features with other attributes, and I think it desirablefollowing Freges example rather than his preceptto study it in a wide context where different species of attribute may be compared. The classification of attributes The numerous species of attribute may conveniently be grouped into three genera. In one genus you have non-quantitative attributes like shape, colour and direction, which may conveniently be called qualities. In a second genus you have the continuous attributes like length, area, volume, mass and duration. These correspond roughly to what Euclid called magnitudes, but I should like to take up the suggestion made recently by Lucas ([2000]) that they should be called quantities from the Latin quantum, meaning how much. That leaves how-many-ness, which, as I say, is one meaning of number, sometimes distinguished as cardinal number or cardinality. Lucas would rename this quotity from the Latin quot, meaning how many. Quotity constitutes a third genus, and you will notice that it is sui generis, being the only species in its genus. Attributes generally may thus be classified as qualities, quantities and quotity. The important thinglet us call it the abstraction theoremis that each species of attribute is associated with an equivalence relation in the domain of things that bear it. We have illustrated the theorem by the cases of length and quotity, but it applies also to qualities. Shape, for example, is associated with the equivalence relation of geometric similarity in the domain of geometrical objects. The difference between qualities on the one hand and quantities and quotity on the other appears when you consider things that are not equivalent. If two rods are unequal in length, one will be longer and the other shorter. If two sets-of-objects are not equi-numerous, one will be more-numerous and the other less. But, if two things differ in shape or colour, it does not follow that one is more shapely or more colourful than the other. This is to say that in the case of a quantity or quotity the equivalence relation is embedded in an order relation, but in the case of a quality it is not. It is also worth noting that the word equal is not normally used with a quality. Two things will have the same shape, or be similar in shape, but they are not equal in shape. The difference between quotity and quantities, of course, is that the degrees of quotity are discrete and possess rank order, whereas the degrees of a species of quantity are continuous and possess continuum order. Nominalism The nature of abstract attributes presents a certain challenge to common sense, which regards them as more problematic than ordinary objects. Some philosophers have taken the same view, and some of these (the nominalists) have gone so far as to deny that abstract entities exist at all except in name. We shall not take that view here. This is partly because on close inspection concrete objects are just as problematic as their attributes, and partly because abstract attributes sometimes need to be treated as objects, capable of being classified like objects and forming relations among themselves. In modern mathematics nominalism has asserted itself in a new form in the suggestion that a degree of an abstract attribute should be actually identified with the corresponding equivalence class of concrete elements. The classic example was Russells proposal to identify two-ness with the set of all pairs-of-objects, and, although this particular case has encountered philosophic turbulence, the general idea has been widely taken up in other parts of mathematics, where, for example, a geometric vector will be identified with an equivalence class of directed line segments, or an integer with an equivalence class of natural numbers. Applying this principle to our present examples, two-ness would have to be identified with the equivalence class of all pairs-of-ideal-counters and a degree-of-length with an equivalence class of ideal rods. Personally I find it circular to propose that a degree-of-length should be defined to be a class of rods when a rod has been conceived in the first place as a thing having length as its principle attribute. Nor am I convinced that the notion of set-of-objects (in the sense of an unconfigured or unstructured finite set) is entirely free from any premonition of quotity. If the reader feels wedded to this particular incarnation of nominalism, however, I doubt whether any serious problems will arise. It certain has the great heuristic benefit of drawing attention to the equivalence relations. And anyway I believe that many of the problems concerning abstract and concrete can be resolved by simply invoking the abstraction theorem without trying to reduce the one to the other. The confusion between abstraction and idealisation What does need some attention is the regrettable and widespread tendency to confuse abstraction with idealisation. The word abstract means to draw out or take away, and I am using it to suggest that two-ness, for example, is something that can be drawn out of all pairs-of-counters and then treated as a new kind of object, having its own name, its own attributes and capable of forming its own relations with other concepts. Likewise a degree-of-length is something drawn out of equi-linear rods. I believe that this kind of drawing out is sometimes known as hypostatic abstraction. The process supposes that one has in view from the beginning a whole domain of relevant concrete objects (or sets-of-objects) from which to abstract. Unfortunately, Aristotle uses abstraction to mean a rather different process. (See, for example, the glossary in Irwin and Fine [1995] p 564.) He seems to start, not with a whole domain of objects, but with an individual object, like say a wooden rod, and from this he takes away the woodiness and the width and the colour and the mass, to be left with what we are calling an ideal rod, possessing length and rigidity and very little else. Hypostatic abstraction is like gold mining, where what is taken away is the gold, but Aristotles is more like wood carving, where what is taken away is discarded. To my mind, of course, what Aristotle is describing is more like idealisation. And Aristotle seems to lack, or at least to avoid, the holistic vision by which a whole domain of objects is conceived as a whole. The result, I fear, is a legacy of confusion between the abstract and the ideal that has affected many fields, including science and mathematics. It presumably lies behind the faux pas committed by Euclid on the very first page of the Elements, where he declares that a line is length without breadth. Euclids aim was evidently to explain how the ideal line segments of geometry differ from the quasi-linear objects of ordinary lifeand I have no quarrel with thatbut a normal person would say that a line HAS length, whereas Euclid writes IS, thus confusing the ideal object with its principal attribute. In arithmetic both Dedekind ([1888] para 1) and Cantor ([1895] section 1) describe the transition from ordinary objects to ideal objects as a process of abstraction rather than idealisation, and, although one can hardly quarrel purely on the ground of vocabulary, this pre-emption of the word abstract appears to inhibit both of them from identifying clearly the truly abstract entities that are attributes of their ideal objects (or sets-of-objects). Symbol versus concept Symbolisation is the act of giving names to things. None of us does much of it because most things already have names, especially in mathematics. Indeed learning mathematics consists largely of accepting traditional words and symbols and then trying to work out what concepts they are intended to refer to. Since this is often difficult, there is a temptation at all levels of the subject to rest ones attention on the symbols themselves and abandon the search for any more nebulous entities that might lie beyond. As Frege put itand time has done nothing to diminish the force of his criticism a frequent fault of mathematicians is their mistaking symbols for the objects of their investigations. ([1984] p 229) I will suggest that two of the most radical and unsettling conclusions about symbols emerge from a simple-minded but resolute look at the language and symbolism of elementary number. Empirical Note firstly that the basic number-words in different languages one, two three un, deux, trois eins, zwei, drei which are almost perfect synonyms, prove by their very plurality that what you might call the numbers themselvesone-ness, two-ness, three-ness etcmust be different from any of the words that represent them. This goes some way to redress Freges complaint. On the other hand, the standard numerals 1, 2, 3 because of their adoption as the official symbolism of modern arithmetic, have a remarkable universality across different languages. It is hardly surprising that they are often referred to simply as numbers, a word that is therefore used ambiguously to refer to both the symbols and the concepts they denote. It is to be noticed, however, that the standard numerals are still pronounced differently in different languagesin fact they are pronounced the same as the local number-wordsand this reminds us that symbols can be spoken as well as written. The sound two is just as much an oral symbol as the written two or the written 2 is a visual symbol. In mathematics the stress is overwhelmingly on written symbols, but in human language generally the spoken word is much more primitive. There are also tactile symbols, as in languages for the blind. The only essential requirement for a symbol is that it should be accessible to at least one of the senses, for otherwise it will not serve the purpose of communication. It seems fair to conclude that the symbols are the one part of any subject that are unequivocally empirical, being accessed by the senses. I find it ironic that mathematical philosophers of the formalist school, seeking escape from their problems in the construction of formal languages, should in this way make it a principle to reduce their subject to sensory experience, as if it were ultimately a matter of grunts and scratches like those of a wild animal. Ambiguity The other conclusion is so shocking that I shall merely raise it and leave it in the air for future consideration. Consider the following: A: I have 3 children B: Section 3 of a report C: I have 3 times as many children as you D: The length is 3 times the width. The person in the street is likely to be struck above all by the common occurrence of the numeral 3, and will instinctively assume that its meaning remains constant. But closer inspection reveals that the context imposes significant changes in meaning. In A the numeral denotes a degree of how-many-ness (which I am calling quotity).In B it serves primarily to indicate position in a sequence (referring to the third section). In C and D it provides a comparison, denoting a ratio. There are also differences that arise if you start to generalise. Expressions of the form of A require that the numeral remain integral, but in C it could be a fraction, and in D it could even be irrational. In B the sections of the report could as well be labelled A, B, C as 1, 2, 3, but this substitution would be unnatural in the other cases. Wittgenstein has taught us that the meaning of a word or other symbol depends on the way in which it is used, and in the present case it seems fair to conclude that the numeral 3 is being used ambiguously to denote four distinct concepts. In A it denotes, as I say, a degree of quotity. In B it denotes a degree of rank order. In C it denotes a degree of quotity-ratio, and in D a degree of quantity-ratio. Using these terms, there is no need to employ the ambiguous word number at all, except perhaps as part of the compound number-symbol. The mathematician, of course, is alert to the distinctions we have noted, but he believes the situation can be managed by postulating different types of number. What I have called quotity-ratio, he will call rational number. What I have called quantity-ratio, he will call real number. Quotity and rank order he will class together under the name of natural number, and, if pressed, will concede that the difference between A and B anticipates the distinction between cardinal number and ordinal number, which only normally becomes important to him with transfinite numbers. The question arises, however, as to whether this continued use of the term number adequately reflects the conceptual realities or whether it is merely a reflection of the popular prejudice that a symbol like 3, which is itself sometimes called a number, must have, in some deep or perhaps mystical sense, an underlying essential core of common meaning. Fortunately there is no need to make a complete answer this question herewhere in any case we shall make no further reference to ratios. But it will be desirable to maintain at least the distinction between quotity and rank order. These two attributes are surprisingly different in structure, mostly obviously in the fact that quotity has an operation of addition and rank order does not. This is evident to the innocent observer, who can see that, if you combine a set of three counters with a (disjoint) set of four counters, you get a set of seven counters, whereas there is no way in which the third person in a queue can be combined with the fourth person to give the seventhor Section C combined with Section D to give Section G. This is a case, I believe, where the innocent are vouchsafed a truth that eludes the more sophisticated. Those who wish to bundle quotity and rank order together as natural numberand, above all, those who take rank order as the dominant aspectrequire that rank order should have an operation of addition, even thought it does not naturally possess one. Dedekind tried to achieve this for his domain of rank-ordered ideal counters by bringing in transformations of the domain (such as move 3 places to the right), which is cheating of a very basic kind, because the domain of transformations of a domain is a quite different thing from the domain itself. Others appeal to the principle of mathematical induction, which Mazur rightly castigates in the quotation of my opening paragraph. In these circumstancesand without prejudice to any future adjudication on the wider issueyou will, I hope, permit me in the present article to avoid the word number as far as possible, and to continue to refer to how-many-ness as quotity. Generic versus specific The terms species and genus, which of course have their origins in biology, may be used metaphorically in many ways. We have already applied them to the classification of abstract attributes. They may also be applied at the concrete level, and this produces a further set of distinctions among attributes. Consider the power-set of the population of ideal counters. When this domain is partitioned by the equivalence relation of equi-numerosity, the equivalence classes may be likened to species and the whole power-set to a genus. Then two-ness, for example, may be called a specific attribute (because it is the common property of elements in the same species), while quotity itself is the generic attribute (being the common property of the whole genus). When the domain of rods is partitioned by the equivalence relation of equi-linearity, each equivalence class is a species of rod and the whole domain the genus. Then each degree-of-length is a specific attribute and length itself generic. It may be noted that in the case of length only a few of the specific attributes have proper names (metre, kilometre etc), but the distinction between specific and generic is not thereby diminished. These two applications of the biological metaphor are not in conflict but fit together. Length, for example, is simultaneously generic (because it is the common property of a genus of concrete elements) and a species of attribute (within the genus of quantities). Likewise, quotity is simultaneously generic (because it corresponds to a genus of concrete elements) and a species of abstract attributes. To go further, two-ness is specific (because it corresponds to a species of concrete elements), but, considered as a member of the abstract domain of attributes, it has the character of an individual specimen. (The abstract domain of attributes breaks down into genera; each genus breaks down into species; and each species has its individual specimens.) There is no conflict, provided that species of and specific are not confused. This is all fairly straight forward, provided that abstract entities are assigned sufficient substantiality to be classified at alljust like objects. But there are, I think, one or two implications that are worth mentioning. The bureau-of-standards definition The distinction between specific and generic attributes illuminates the various presentations of natural number suggested by Mazur, and particularly the bureau of standards approach, whereby the degree-of-quotity of a set-of-objects is determined by matching its elements to those of a standard set kept in a safe place for this very purpose. To my mind the bureau of standards approach is a legitimate wayindeed the only wayto define a particular degree-of-quotity like five-ness, while Freges definition in terms of equi-numerosity is a way of defining the generic attribute of quotity itself. These are certainly not the same, but nor are they alternatives, because both are necessary. The bureau-of-standards definition enables you to answer a question of the form How many objects in this set? But in order to understand the question in the first place you must have some idea of quotity itselfthat is, the generic concept. Incidentally, I think that Frege, whom Mazur associates with the generic definition, himself appreciated the need to define the specific attributes as well, for it was in pursuit of this that he constructed his remarkable if insubstantial gallery of spectral specimen sets built entirely out of the empty-set. General names for specific attributes We may now make a few further remarks about symbols (i.e. names) in the light of the distinction between specific and generic concepts. A word like two-ness is the proper name of one of the specific attributes of quotity. Likewise quotity is the proper name for the generic attribute. Both of these, like all proper names, are necessarily singular. But there is also a term degree-of-quotity, which can take a plural, This is a general name for the specific attributes. I do not think that the distinction between a general name for a set of specific attributes and the proper name for the related generic attribute is particularly difficult, provided that one has established two firm distinctions: between generic and specific concepts and between concepts and symbols. (It is more difficult to achieve in nominalism because there the abstract concepts only exist in name anyway.) In traditional language, however, the word number is made to serve in both capacities, and, although one could introduce the term degree-of-number for the general name, it is a relief if number has to be jettisoned on other grounds, for then the new vocabulary can establish the two terms quotity and degree-of-quotity from the beginning. Note in passing that the words one, two, three etc, as nouns, are a considerable embarrassment in serious analysis. As adjectives they are easily acceptable, for three oranges obviously refers to a triplet of objects (which is concrete). But the specific attributes of quotity are naturally called one-ness, two-ness and three-ness rather than one, two, three. (Note the language that was used naturally in our earlier brief discussion of the two fat catspara 2.2.) In the same way, the specific degrees of rank order can naturally be called first-ness, second-ness, and third-ness. I suggest that in any serious analysis of natural number the nouns one, two, three should be jettisoned at the same time as number, except possibly as names for the numerals (that is, as symbols for symbols). Euclid and magnitudes We must also note that the whole distinction between generic and specific attributes is seriously neglected in Greek geometry. Euclid, the master of idealisation, had a much less confident grasp of abstraction, and in consequence he showed little interest in the classification of abstract attributes. The problem manifests itself most obviously in the poverty of his vocabulary. Even though his home ground is geometry, he has no clear terms for the geometric attributes that we call length, area and volume. The word for length only occurs once in the Elements, in the ill-fated definition of line, already considered. The word usually translated area is only ever used to refer to an object having area. There is no word at all that I can find that might be translated volume. Euclid does recognise the genus to which length, area and volume belong and he calls it magnitudes, but he seems never clearly to recognise that this genus breaks down into the species that we call length, area and volume (let alone non-geometrical species like mass and duration). This has implications for Euclids use of the word equal, as discussed below. (It also has important implications at a higher level, concerning the super-abstract concept of quantity-ratio. Given the diverse species of quantity, we can nowadays appreciate that two degrees of length may be proportional to two degrees of volume or to two degrees of mass etc, so that, instead of there being separate concepts of length-ratio, volume-ratio and mass-ratio, there is a single thing that may be called quantity-ratio. This universality gives to quantity-ratio much of the character that later mathematicians have come to look for in real number, but Euclid failed to perceive the significance of this unity at the super-abstract level because he had earlier failed to appreciate the contrasting diversity at the first level of abstraction.) Equality After these preliminaries we may turn to the language and symbolism of equality. Equality in Euclid Euclid uses equality as a general name for equivalence relations embedded in order relations. It is not a general name for any equivalence relation whateverit does not cover parallelism, for example, or geometric similaritybut it certainly covers what we are calling equi-linearity and equi-numerosity, and would appear intended to cover the equivalence relation in the domain of any species of quantity or quotity, but not a quality. This is also, more or less, the way the word is used today in everyday life, where, if two things are unequal, it is expected that one will be bigger and the other smaller. Unfortunately, there is at least one problem with Euclids usage. Euclid has no proper names (like equi-linearity) for the specific equivalence relations. Nor is he able to qualify the general term equal to given compound terms like equal-in-length. He just uses equal each time. When he says The straight lines joining equal and parallel straight lines are themselves also equal and parallel. (Bk I Prop 33) he evidently means equal-in-length or equi-linear. When he says construct a square equal to a given figure. (Bk II Prop 14) he means equal-in-area. When he says Parallelepipedal solids which are on equal bases and of the same height are equal to one another. (Bk XI Prop 31) the first equal refers to equality of area and the second to equality of volume. Up to a point this does not matter because in simple cases the meaning is clear from the content, and Euclid never puts the reader into a position where she is tempted to equate, say, a line segment to a rectangle or a rectangle to a sphere. However, when he makes a context-free statement like Things which are equal to the same thing are also equal to one another. (Bk I Common notion 1) he gives the unfortunate impression that equality is a relation in its own right over and above the various specific relations. That is, the general name for the specific relations is easily misinterpreted as the proper name of a spurious generic relation. People come to think of equality as a new kind of relation, somehow intermediate between equivalence and identity. Factoring out Euclids lack of detailed interest in the abstract domain has at least this advantage, that all his relations of equality are firmly located at the concrete level. The same feature characterises our own analysis, for it is rods that are equi-linear, and sets-of-counters that are equi-numerous. With this in mind it may be noted that, if two rods are equi-linear, they belong to the same equivalence class or species (not to equal species), and it follows that they have the same degree-of-length, not equal lengths. In the same way, if two sets-of-counters are equi-numerous, they belong to the same equivalence class or species (not to equal species), and it follows that they have the same degree-of-quotity (not equal numbers of elements). In neither case does the equivalence relation penetrate into the abstract domain because it has been factored out. That, at least, is the theory. In practice, as we all know, people do talk about equal numbers and equal lengthsin fact the literature of abstract arithmetic (to say nothing of algebra and every other branch of higher mathematics) is littered with the word equality and the ubiquitous = sign. This creates serious problems of interpretation. Equality in arithmetic and algebra Although the = sign is almost the hallmark of arithmetic, it is worth noticing that much arithmetic could be written without using it at all. Addition, for example, could be symbolised more naturally as  EMBED Equation.3  With this in mind the conventional notation 3 + 5 = 8 can be regarded as a surviving historic alternative, where the + and the = are parts of a single compound symbol (comparable to the big S and the dx of the integral calculus). However, there are genuine equivalence relations to be found (or created) in the abstract domain. For example, having-the-same-sum is an equivalence relation on sets of degrees-of-quotity. Under this relation one has (1, 7) = (1, 6) = (3, 5) = or in traditional notation 1 + 7 = 2 + 6 = 3 + 5 = This interpretation will serve to justify 3 + 5 = 8 if you include singleton sets along with the others. (Note in passing, however, that this equivalence relation is defined in terms of addition. It is not itself a definition of addition.) Synonymy Dedekind tried to account for the intrusion of equality into abstract domains by saying that a=b means that a and b are symbols or names for one and the same thing. ([1888] para 1) This is say that the = sign denotes synonymy, which is indeed an equivalence relation, but on symbols not concepts. The distinction might perhaps be brought out by the use of quotation marks: two = deux or two = 2 or a = b More subtlety is required in a case like The length of AB is equal to the length of CD or The number of men is equal to the number of women. Here the length of AB and the number of men are symbolic expressions rather than simple symbolsdescriptions rather than proper namesand such an expression has a meaning of its own (what Frege [1892] calls its sense), which is lacking in a simple label. In general the length of AB will be different from the length of CD, and the number of men different from the number of women. It is only in a special case that the two things will be the same, and then one may say (in Freges term) that the the length of AB has the same reference as the length of CD. With Freges analysis in mind one may take equality in these cases to mean having-the-same-reference-despite-a-difference-in-sense. It is now an equivalence relation on symbolic expressions (compound symbols) rather than on simple symbols. But it is still not an equivalence relation on the things that are symbolised. Paul Ernest has pointed out that the Fregean interpretation can be applied to arithmetic if 3+5 and 8 are regarded as two expressions having different senses but the same reference. I do agree, although, if this is the intended meaning, a more faithful symbolisation might be 3 + 8 = 8 Equality in mathematical logic Consider finally, as an illustration from higher mathematics, the axiom of extensionality in set theory. I think myself that, properly understood, this so-called axiom is part of the definition of the term set. The author is declaring that she will use set to denote, not a configured set (like a constellation of stars, where spatial disposition is relevant), nor an ordered set (like a list of aristocrats in order of precedence or the letters of the alphabet in their conventional order), nor a set-with-repetition (like the set of five letters of the word SWEET), but what might be called a free or unstructured set, determined solely by the identities of its constituent members. There is nothing especially difficult here. One might want to argue over whether there is any deep difference between an axiom and a definition anywaybut that is not the present point. The point is that in contemporary mathematics the axiom will be presented as a string of symbols such as  EMBED Equation.3  The intended meaning is clear enough: If each member of set y is also a member of set z, and vice versa, then y and z are merely alternative names for the same set. But this paraphrase manages without the word equal, and it suggests that the = in the symbolic formulation must be interpreted as Dedekinds relation of synonymy. It is here that we join forces with Mazur in wanting to challenge the current hegemony of formal systems. On my view synonymy is a relation on symbols, so that the present example might more accurately be written  EMBED Equation.3  But this is not catered for in the formal language of logic. Indeed the very phrase first order logic with equality (alias first order logic with identity) seems designed to pre-empt any discussion of symbols versus concepts. To my mind this extreme form of formalism is a slap in the face for conceptual analysis altogether, asserting that the problems of mathematical philosophy can be settled by laying down a few rules for the manipulation of symbolsof scratches and grunts. Conclusions I think our discussion amply confirms Mazurs contention that the notion of equality is treacherous, and I have tried to argue that the treachery lies mainly in three failures: a failure to distinguish among specific kinds of equivalence, a failure to acknowledge that equivalence is absent from abstract domains because it has been factored out, and a failure, especially in formal systems, to distinguish symbols from the things they symbolise. In popular language the lack of specificity has often been exploited in slogans like All men are equal which does not claim they are equal in height or weight or intelligence or any other specific attribute, but rather in something vaguer like equality in the sight of God or being equally worthy of respect. In Orwells Animal Farm even this vague equality has to be qualified when the original premise All animals are equal is amended by adding but some are more equal than others. It seems to me remarkable that such an ambiguous and potentially troublesome term as equality should have been taken up so wholeheartedly in mathematics, a discipline which on other occasions makes claims to precision, rigour and lack of ambiguity. The lack of specificity can be blamed on Euclid, but the failure to acknowledge factoring out lies rather in the department of arithmetic and symbolic representation. It is interesting to note that in the famous passage in which Robert Recorde first introduces the = sign to a receptive world he explains his choice of a paire of paralleles on the ground that noe 2. thynges, can be moare equalle. He means no doubt that equi-linearity is intuitively the most obvious kind of equivalence, but his phrase more equal seems designed to cloud the issue and pave the way for Orwell. This issue, however, is still one of generality. The matter of factoring out relates to the distinction between abstract and concrete, which, as I maintain, is still seriously bedevilled by Aristotles confusion between abstraction and idealisation. And the problems of equality or identity in mathematical logic relate to the processes of symbolisation and interpretation. The four processes of idealisation, abstraction, symbolisation and generalisation underlie everything in mathematics. Bibliography Campbell N. R., An Account of the Principles of Measurement and Calculation ([1928]) London: Longmans Green. Cantor G., Contributions to the Founding of the Theory of Transfinite Numbers ([1895] and [1897]) trans Jourdain P. B. ([1915]) New York: Dover Dedekind R., Was sind und Was sollen die Zahlen? ([1888]) translated as The Nature and Meaning of Numbers in Essays on the Theory of Numbers ([1901]) Chicago: Open Court Dienes Z. P., On abstraction and generalisation Harvard Educational Review Summer [1961] Dienes Z. P., An experimental study of mathematics learning ([1963]) London: Hutchinson Einstein A., Physics and Reality ([1936]) in Einstein A. Ideas and Opinions ([1954]) New York: Random House Ernest P., Social Constructivism as a Philosophy of Mathematics ([1998]) New York: State University of New York Press Frege G. (tr Austin), The Foundations of Arithmetic ([1884/1959]:) Oxford: Basil Blackwell Frege G., On Sense and Reference ([1892]) in Geach P. and Black M. Translations from the Philosophical Writings of Gottlob Frege ([1952]) Oxford: Blackwell Frege G., Gottlob Frege: Collected Papers ([1984]) Oxford: Basil Blackwell Jones I. and Pratt D., A Substituting Meaning for the Equals Sign in Arithmetic Journal for Research in Mathematics Education Vol 43(1) ([2012]), 2-33 Kieran C., Concepts associated with the equality symbol Educational Studies in Mathematics Vol 12(3) ([1981]), 317-326 Krantz D. H., Luce R. D., Suppes P. and Tversky A. Foundations of Measurement Vols 1-3 ([1971], [1989], [1990]) San Diego: Academic Press Lucas J. R., The conceptual roots of mathematics ([2000]) London: Routledge Mazur B., When is one thing equal to another thing? in Gold B. and Simons R. 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