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This essay of mine was first published in V5N2 of The Journal of Ayn Rand Studies – 2004. Universals and Measurement I. Orientation Rand spoke of universals as abstractions that are concepts (1966, 1, 13). Quine spoke in the same vein of "conceptual integration—the integrating of particulars into a universal" (1961, 70). Those uses of universal engage one standard meaning of the term. Another standard meaning is the potential collection to which a concept refers. This is the collection of class members consisting of all the instances falling under the concept[1]. In the present study, the character of universals in the latter sense will be brought into fuller articulation and relief. That vantage will be attained by amplifying the mathematical-metaphysical requirements of Rand's theory of universals as conceptual abstractions. To begin I situate the topic of the present study within Rand's larger system of metaphysics and epistemology. My core task for the present study will then emerge fully specified. Rand's system relies on three propositions taken as axioms[2]. (E) Existence exists. (I) Existence is identity. (C ) Consciousness is identification. Rand's set of axioms conveys the fundamental dependence of consciousness on existence. Existence is and is as it is independently of consciousness, whereas consciousness is dependent on existence and characters of existence (Rand 1957, 1015–16; 1966, 29, 55–59; 1969, 228, 240–41, 249–50). As part of the meaning of (I), Rand contended (Im): All concretes, whether physical or mental, have measurable relations to other concretes (1966, 7–8, 29–33, 39; 1969, 139–40, 189, 199–200, 277–79)[3]. Every concrete thing—whether an entity, attribute, relation, event, motion, locomotion, action, or activity of consciousness—is measurable (Rand 1966, 7, 11–17, 25, 29–33; 1969, 184–87, 223–25). As part of the meaning of (C ), Rand made the point (Cm): Cognitive systems are measurement systems (1966, 11–15, 21–24; 1969, 140–41, 223–25). Perceptual systems measure[4], and the conceptual faculty measures. Concepts can be analyzed, according to Rand's theory, as a suspension of particular measure values of possible concretes falling under the concept. Items falling under a concept share some same characteristic(s) in variable particular measure or degree. The items in that concept class possess that classing characteristic in some measurable degree, but may possess that characteristic in any degree within a range of measure delimiting the class (Rand 1966, 11–12, 25, 31–32)[5]. This is Rand's "measurements-omitted" theory of concepts and concept classes. All concretes can be placed within some concept class(es). All concretes can be placed under concepts. Supposing those concepts are of the Randian form, then all concretes must stand in some magnitude relation(s) such that conceptual rendition of them is possible. What is the minimal magnitude structure (minimal ordered relational structure) that all concretes must have for them to be susceptible to being comprehended conceptually under Rand's measurement-omission formula? That is to say, What magnitude structure is implied for metaphysics, for all existence, by the measurement-omission theory of concepts in Rand's epistemology? My core task in the present study is to find and articulate that minimal mathematical structure. With that structure in hand, we shall have as well the fuller articulation of the class character of universals implied by Rand's theory of concepts. Such mathematical structure obtaining in all concrete reality is metaphysical structure. It is structure beyond logical structure; constraint on possibility beyond logical constraint. Yet it is structure ranging as widely as logical structure through all the sciences and common experience. The minimum measurement and suspension powers required of the conceptual faculty by Rand's theory of concepts calls for neuronal computational implementation. Is such implementation possible, plausible, actual? This is a topic for the future, bounty beyond the present study. We must keep perfectly distinct our theoretical analysis of concepts and universals on the one hand and our theory of the developmental genesis of concepts on the other. Analytical questions will be treated in the next section, and it is there that I shall discharge the core task for this study. The logicomathematical analysis of concepts characterizes concepts per se. It characterizes concepts and universals at any stage of our conceptual development, somewhat as time-like geodesics of space-time characterize planetary orbits about the sun throughout their history. The analysis of concepts and universals offered in the next section constrains the theory of conceptual development, as exhibited in §III.

II. Analysis (cont.) Affordance of Ordinal Measures Recall again Rand's characterization of measurement: identification of "a quantitative relationship established by means of a standard that serves as a unit" (1966, 7). The phrase "a standard that serves as a unit" suggests that Rand's conception of measurement for her measurement-omission analysis of concepts was ratio-scale or interval-scale measurement. These two types possess interval units that can serve as interval standards. They possess interval units that can be meaningfully summed to make measurements. The quantitative relationship established in measurements equipped with interval units entails summation of elementary units. The summation might be simple addition or a more elaborate mathematical combination, and the basis for the summation in concrete reality might be susceptibility to concatenation (for ratio scales) or to composition of ordered difference-intervals (for interval scales). The measure values required for Rand's theory need not be interval units. As Rand realized, merely ordinal measurement suffices for her measurement-omission scheme (1966, 33). I say that the magnitude structure captured by ordinal measurement is the minimal structure implied for metaphysics if, as I supposed at the outset, all concretes fall under one or more concepts for which Rand's measurement-omission analysis holds. What is the magnitude structure captured by ordinal measurements? All magnitude structures captured by ratio- or interval-scale measurements contain a linear order relation. A magnitude structure consisting only of such a linear order relation is a structure for which merely ordinal measurement is appropriate. An example is the hardness of a solid. I mean specifically the scratch-hardness, which is measurable using the Mohs hardness scale. Calcite scratches gypsum, but not vice versa; quartz scratches calcite, but not vice versa; therefore, yes, quartz scratches gypsum, but not vice versa. Degrees of hardness have an order that is anti-symmetric and transitive. Mohs scale assigns the numbers (2, 3, 7) to the degrees of hardness for (gypsum, calcite, quartz). All that is intended by the scale is to be true to the order of the degrees of hardness. That Mohs has chosen these three numbers to be integers is of no significance. They could as well be the rational triple (14.7, 55.3, 56.9) or the irrational triple (√2, π, 1.1π). Unlike the numbers on interval scales, the ratios of difference-intervals between the numbers on these scales are not meant to be of any significance. The hardness degrees (2, 3, 7) are not intended to imply that the hardness of calcite is closer to the hardness of gypsum than it is to the hardness of quartz. For all we know, and for all our ordinal measurements signify, there simply may be no fact of the matter whether the scratch-hardness of calcite is closer to that of gypsum than to that of quartz. The magnitude structure of hardness (scratch-hardness, not dent-hardness) evidently does not warrant summations or equal subdivisions of some sort of interval unit of hardness. This particular hardness concept is founded analytically on merely ordinal measure. To fall under this concept hardness, an occasion need only present the quality at some measure value on the merely ordinal scale, and that may be any measure value on that scale. Affordance of ordinal measurement is all that Rand's measurement-omission recipe entails for the magnitude structure of all concretes. Her theory does not entail that every attribute of concretes—hardness, for example—must in principle afford ratio- or interval-scale measurements. Her theory does not imply that, were only our knowledge improved enough, it would be possible to make ratio- or interval-scale measurements of scratch-hardness[24]. The magnitude structure affording merely ordinal measurement is a linear order whose automorphisms are the order-automorphisms of (same-order subsets of) the real numbers in their natural order. Such a magnitude structure affords characterization by a lattice (a type of partially ordered set) formed of sets and subsets of possible Dedekind-cuts of its linear order. This linear order might be scattered or dense; ordinal measurement is possible in either case[25]. The magnitude structure affording merely ordinal-scale measurement affords metrics. Each of the three scales adduced above to capture degrees of hardness bears a metric defined by the absolute values of those scales' numerical differences. A magnitude structure affording a (separable) metric belongs to the topological category known as a (separable) uniformity. Topologies that are uniformities in this sense are Hausdorff topologies, but they need not be compact nor (topologically) connected[26]. The topological character of the magnitude structure entailed for all concretes by Rand's measurement-omission theory of concepts is the character of a uniformity. The magnitude structure entailed by Rand's theory has the algebraic character of a lattice, which has more structure than a partially ordered set (or a directed set) and less than a group (or a semi-group). In terms of the mathematical categories, Rand's magnitude structure for metaphysics is a hybrid of two: the algebraic category of a lattice and the topological category of a uniformity. Rand's structure belongs to the hybrid we should designate as a uniform topological lattice. Concerning multidimensional magnitude structures of concept classes, I concluded in the preceding subsection that Rand's theory entails neither affine nor absolute structure. What is entailed: concept classes with a 2D or 3D magnitude structure will have the structure of at least an ordered, distance geometry[27]. Significantly, it is implied that planes and spaces concretely realizable will have at least that much structure. Superordinates and Similarity Classes Hardness, fatigue cycle limit, critical buckling stress, shear and bulk moduli, and tensile strength all fall under the superordinate concept strength of a solid. The conceptual common denominator (Rand 1966, 15, 22–25; 1969, 143–45) of these various strengths of solids is that they are all forms of resistance to degradations under stresses. What is the common measure of this resistance the different species of strength have in common? What is the common measure of strength that all specific forms of the strength of solids have in common? The magnitude structure of hardness affords only ordinal-scale measurement. The magnitude structure of tensile strength affords ratio-scale measurement. Only the ordinal aspect of the tensile-strength measure could be common with the hardness measure. Is the ordinal aspect of each specific form of strength a same, single, common measure? No, the ordinal measure of hardness is not the same as the ordinal measure of tensile strength. Resistance to being scratched is not the same as resistance to being pulled apart under tensile stress. The way in which an ordinal measure of hardness and an ordinal measure of tensile strength can form a common ordinal measure for the general concept strength of a solid is only as substitution units, not as distinct measure values along some common ordinal measurement scale. The some-any locution can be applied to substitution units (e.g., Rand 1966, 25). We sensibly say that strength of a solid in general must have some type of ordinal strength measure but may have any such type. That sort of use of some-any pertains to units as substitution units: there must be some specific form of strength to instantiate the general concept strength of a solid, but it may be any of the specific forms. The substitution-unit standing of concepts under their superordinate concepts is a constant and necessary part of Rand's measurement-omission recipe as applied to the superordinate-subordinate relationship. But this part is not peculiar to Rand's scheme for that relationship. Here is what is novel in Rand's measurement-omission theory for superordinate constitution, as I have dissected it: Whichever concept is considered as an instance of the superordinate concept, not only will that subordinate concept and its instances stand as a substitution instance of the superordinate, each instance of the subordinate will have some particular measure value along a specific dimension. And that particular value is suspended for the concept, thence for the superordinate concept. Analytically, identity precedes similarity[28]. For purposes of her theory of concepts and concept classes, Rand defined similarity to be "the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree" (1966, 13). I concur. Occasions of scratch-hardness are similar to each other because they are all occasions of scratch-hardness, exhibiting that hardness in various measurable degrees. This much accords with Rand's definition and use of similarity in the theory of concepts. Occasions of scratch-hardness are also more like each other than they are like occasions of tensile strength. This is a perfectly idle invocation of comparative similarity (comparative likeness). The work that comparative similarity pretends to be doing here can be accomplished fully by simple identity (sameness) without any help from similarity: scratch-hardness is itself and not something else, such as tensile strength. The shapes of balls are similar to each other because they have principal-curvature measures at various values within certain ranges. Likewise for the shapes of cups (to keep the illustration simple, consider a Chinese teacup, not a cup with a handle). Moreover, ball shapes are more like one another than they are like cup shapes because ball values of principal curvatures are closer to each other than they are to cup values of principal curvatures. Here the invocation of comparative similarity is not idle. To say that ball shapes are more like one another than they are like cup shapes is to say something beyond what is claimed in saying: Shapes that balls have are themselves and not something else, such as shapes that cups have. The strengths of a solid are of various kinds that are not simply of various values along some common dimension(s). The shapes of a solid are of various kinds, and unlike kinds of strengths, these kinds are of various values along some common dimension(s). Rand's conception of similarity as sameness of some characteristic, but difference in measure, can be put squarely to work in analyzing comparative similarities of shapes of solids with each other. Then this conception of similarity is a genuine worker, too, in the analysis of the concept shape of a solid, superordinate for the concepts ball-shape and cup-shape. This employment of Rand's conception of similarity in the analysis of comparative similarity, thence in the analysis of superordinates, is just as Rand would have it (1966, 14). But such an employment of Rand's conception of similarity as sameness of some characteristic, but difference in measure, is incorrect in application to the comparative similarities of the various strengths of solids, thence to their superordinate concept strength of a solid. What will be the proper analysis as we move on up the superordinates? Strengths of a solid are more like strengths of a solid than they are like shapes of a solid. Let us suppose, as Rand supposed, that the reason we can say that strengths of a solid are more like each other than they are like shapes of a solid is because there is some common dimension, the dimension of the conceptual common denominator, between strengths and shapes of a solid. Property of a solid fits the bill for conceptual common denominator. Strengths and shapes of a solid are both properties of a solid. What is the measurable dimension of the concept property of a solid that is common to both strength and shape of a solid? Like the common dimension for strength, it is a dimension consisting of nothing more than various substitution dimensions. The measurable dimension of property of a solid will be the hardness dimension or the tensile-strength dimension or the principal-curvature dimensions or . . . . There is no single, common measure of property of a solid that all specific properties of solids have in common. Rand supposed in error that there were, for she supposed it always the case that there is some same, common measurable dimension supporting the conceptual common denominator for any superordinate concept (1966, 23)[29]. That supposition is here rejected, and measurement-omission analysis of superordinate concepts is here corrected in this respect. Suppose for a moment, though it be false, that there were some common measurable dimension of property of a solid that was singular, not common merely by substitutions. Then in saying that strengths of a solid are more like each other than they are like shapes of a solid, we could reasonably contend that the values of strengths are closer to each other on the hypothetical common property-of-a-solid dimension than they are to the values of shapes on that common dimension (Rand 1966, 14)[30]. Then the magnitude structure of the common dimension for property of a solid could not be one that affords only ordinal measures. On such measures, there is no telling whether a value between two others is closer to the one than to the other. (Then in an order of values ABCD, one has no measure-basis for clustering B or C with A or D : B might cluster with A and C cluster with D; or B and C might both cluster with A; or . . . .) The magnitude structure of the common dimension for property of a solid would have to afford additional measurement structure. It would need to afford ratio- or interval-scale measurements. But it is not at all plausible that a measurable dimension common to each instance of property of a solid should have not only ordinal-scale structure, but ratio- or interval-scale structure, when hardness, for instance, has only ordinal structure. Amended Measure-Definitions of Similarity and Concepts With possible exception for the most general concepts such as property, Rand supposed that concept classes are always similarity classes (1969, 275–76). This is immediately apparent from comparison of her definition of similarity with her definition of concepts. In the present study, I likewise make Rand's supposition. Now I have said that a solid's resistance to being scratched is not the same as its resistance to being pulled apart under tensile stress. Nonetheless, these two sorts of strength of a solid are similar. Occasions of hardness are similar to occasions of tensile strength because the same characteristic, limit of resistance to some sort of stress, is possessed by both in different measurable forms. These measurable forms could be merely ordinal, yet in this way be a basis of similarity. Moreover, hardness and tensile strength are more similar to each other than to shape because hardness and tensile strength are two different measurable forms of the same characteristic that is different from the measurable characteristic (pair of principal curvatures) shared by shapes in different degrees. So I should amend Rand's definition of similarity as follows: Similarity is the relationship between two or more existents possessing the same characteristic(s), but in different measurable degree or in different measurable form. The corresponding definition of concepts would be: Concepts are mental integrations of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted or with the particular measurable forms of their common distinguishing characteristic(s) omitted. Every concrete falls under both sorts of concept. Both sorts of conceptual description have application to every concrete. Occasions of hardness fall under the hardness concept by sameness of characteristic in various measures. Those very same occasions of hardness fall under the strength concept by sameness of characteristic varying in measurable form. Occasions of cup-shape fall under the cup-shape concept by sameness of (pairs of) characteristics in various measures. Those very same occasions of cup-shape fall under the spatial property concept by sameness of a characteristic, spatial extension, that has various measurable forms. Conclusion of Core Task My amendments to Rand's definitions of concept and concept class (similarity class) do not implicate metaphysical structure beyond what is already implied by Rand's definitions. Where I have spoken of various measurable forms of a characteristic, all of those forms are the same measurable dimensions that are also at work in concept classes based on variation of measure values along a dimension. What is the magnitude relation under which all concretes must stand such that conceptual rendition of them is possible? They must stand in the relation of a uniform topological lattice, at least one-dimensional. This is the magnitude structure implied for metaphysics, for all existence, by the theory of concepts in Rand's epistemology. The same magnitude structure is implied by that theory with my friendly amendment. What is the mathematical character of universals, of the collection of potential concept-class members, implicit in Rand's theory of concepts? In Rand's theory, universals are recurrences, repeatable ways that things are or might be. Properties, such as having shape or having hardness, are examples of such ways. That universals are recurrences is a traditional and current mainstay in the theory of universals[31]. In Rand's theory, however, universals are not only recurrences, they are recurrences susceptible to placement on a linear order or they are superordinate-subordinate organizations of recurrences susceptible to placement on such linear orders. Universals as (abstractions that are) concepts are concept classes with their linear measure values omitted. If the concept is a superordinate, then the linear measurable form might also be omitted, that is, be allowed to vary across acceptable forms. Universals as collections of potential concept-class members are recurrences on a linear order with their measurement values in place[32]. For either sense of the term universals, they are an objective relation between an identifying subject and particulars spanned by those universals (Rand 1966, 7, 29–30, 53-54; 1965, 18; 1957, 1041). (Continued below.)

II. Analysis Rand gave three definitions of concept. I shall tie them all together in the next section, but for the present section, we need this one alone: Concepts are mental integrations of "two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted" (Rand 1966, 13)[6]. The units spoken of in this definition are items appropriately construed as units by the conceiving mind. They are items construed as units in two senses, as substitution units and as measure values (Rand 1969, 184, 186–88). As substitution units, the items in the concept class are regarded as indifferently interchangeable, all of them standing as members of the class and as instances of the concept. Applied to concept units in their substitution sense, measurement omission means release of the particular identities of the class members so they may be treated indifferently for further conceptual cognitive purposes[7]. This is the same indifference at work in the order-indifference principle of counting. The number of items in a collection may be ascertained by counting them in any order. Comprehension of counting and count number requires comprehension of that indifference. The release of particular identity for making items into concept-class substitution units is a constant and necessary part of Rand's measurement-omission recipe. But this part is not peculiar to Rand's scheme. What is novel in Rand's theory is the idea that in the release of particular identity, the release of which-particular-one, there is also a suspension of particular measure values along a common dimension. Before entering argumentation for the minimal mathematical structure implied for the metaphysical structure of the world, let us check that we have our proper bearings on objective structure and intrinsic structure. I have ten fingers, eight spaces between those fingers, and two of my fingers are thumbs. That's how many I have of those items. Period. Those numerosities are out there in the world, ready to be counted, and they are what they are whether I count them or not. In our positional notation for expressing and calculating numbers, we choose the number base, but the different base systems designate the same things, the numbers. In base ten, my (fingers, spaces, thumbs) are (10, 8, 2); in base eight (12, 10, 2); and in base two (1010, 1000, 10). The three numbers referred to in all these bases are the same three numbers. In Rand's terminology, the various bases are objective schemes; they are appropriate tools for getting to the intrinsic structure of numbers. But the numbers have intrinsic character—even or odd, whole or fraction, rational or irrational, analytic or transcendental—quite independently of our choices, such as choice of number base. In asking for the minimal magnitude structure that all concretes must possess if all concretes can be subsumed under concepts for which Rand's measurement-omission analysis holds, we are seeking intrinsic structure, obtaining under every adequate objective expression of that structure. Now we are ready. Affordance of Ratio or Interval Measures I have said that the units suspended in the formula "two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted" are units in a double sense: substitution units and measure values. We focus now on units in the latter sense. Rand spoke of measurement as "identification of a relationship in numerical terms" (1966, 39) and as "identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit" (1966, 7; also 33; see further 1969, 188, 199–200). The measure-value sense of unit is the one at work here. By the expression "a standard that serves as a unit" and by some of her examples of concepts and their measurement bases, one might suppose that Rand's theory of concepts entails that all concretes stand under magnitude relations affording some sort of concatenation measurement. That supposition would be incorrect. Rand illustrates her theory with the concept length. The pertinent magnitudes of items possessing length are magnitudes of spatial extent in one dimension. Another illustration of Rand's is the concept shape (1966, 11–14; 1969, 184–87). The pertinent magnitudes of items possessing shape, in 3D space, are pairs of linear, spatial magnitudes such as curvature and torsion for shapes of curves or the two principal curvatures for shapes of surfaces[8]. Shapes must possess such pairs of magnitudes in some measure but may possess them in any measure. Observe that Rand's measurement-omission theory does not entail what number of dimensions for the magnitude relations among concretes is appropriate for the concept. Length requires 1D, shape requires 2D. Rand's theory works for any dimensionality and does not entail what the dimensionality must be, except to say that it must be at least 1D. Observe also that the conception of linearity to be applied here to each dimension is not the more particular linearity familiar from analytic coordinate geometry or from abstract vector spaces. It is merely the linearity of a linear order[9]. The magnitude structure of the concretes falling under the concept length affords concatenations. Take as unit of length a sixteenth of an inch. Copies of this unit can be placed end-to-end, in principle, to form any greater length, such as foot, mile, or light-year. This standard concatenation of lengths is properly represented mathematically by simple addition. That is a numerical rule of combination appropriate to concatenations of the concrete magnitude structure in the case of length. The magnitude structure of the concretes falling under the concept length also affords ratios that are independent of our choice of elementary unit. The ratio of the span of my left hand, thumb-to-pinky, to my height is simply the number it is, regardless of whether we make those two measurements using sixteenths of an inch as elementary unit or millimeters as elementary unit. Mass is another concept whose concept-class magnitude structure affords simple-addition concatenations and affords ratios of its values that are independent of choice of elementary unit. Because of the latter feature, conversion of pounds to kilograms requires only multiplication by a constant. Such measurement scales are called ratio scales[10]. The mathematical combinations reflecting the concatenations need not be simple addition. This category of scales is somewhat more inclusive than that. It would include the scale for the concept grade (grades of roads, say). Grades can be concatenated, although the proper mathematical reflection of this concatenation is not simple addition[11]. Finest objectivity requires measurement scales appropriate to the magnitude structures to which they are applied. What does appropriate mean in this context? It means that all of the mathematical structure of the measurement scale is needed to capture the concept-class magnitude structure of concretes under consideration. It means as well that all the magnitude structure pertinent to the concept class is describable in terms of the mathematical structure of the measurement scale[12]. What is the magnitude structure of concretes that is appropriately reflected by ratio-scale characterization? It is a magnitude structure whose automorphisms are translations[13]. Translations are transformations of value-points (i.e., points, which may be assigned numerical values) of the magnitude structure (the ordered relational structure of the concept-class concretes) that shift them all by the same amount, altering no intervals between them. Rand's measurement-omission analysis of concepts and concept classes applies perfectly well to cases in which the measurement scale appropriate to the pertinent magnitude structure of concretes is ratio scale. But Rand's theory does not entail that all concretes afford ratio-scale measures. For Rand's theory does not necessitate that the scale type from which measurements be omitted be ratio scale. Her analysis also works perfectly well for scales having less structure. The magnitude structure entailed for all concretes by Rand's theory is less than the considerable structure that ratio scales reflect. An analogous conclusion obtains for multidimensional magnitude structures of concept classes. Rand's theory does not entail that all 2D or 3D magnitude structures have both affine structure and absolute structure, as Euclidean geometry has[14]. That is, Rand's theory does not entail that multidimensional magnitude structures of concept classes afford a metric (a measure of the interval between two value-points) definable from a scalar product (a measure of perpendicularity of value-lines)[15]. Physical temperature, certain aspects of sensory qualities, and certain aspects of utility rankings are examples of concretes whose magnitude structures afford what are now called interval measures, but evidently do not afford ratio measures[16]. The magnitude structure underlying the concept class temperature affords only an interval scale of measure. Such magnitude structures do not afford concatenations, unlike the natures of length or mass, but they do afford ordering of differences of degree, and they afford composition of adjacent difference-intervals[17]. Such magnitude structures do not afford ratios of degrees that are independent of choice of unit, but they afford ratios of difference-intervals that are independent of choice of unit and choice of zero-point[18]. Ratio scales have one free parameter, requiring we select the unit, such as yard or meter. These scales are said to be 1-point unique. Interval scales have two free parameters, requiring we select the unit, such as ˚F or ˚C, and requiring we select the zero-point, such as the freezing point of an equally portioned mixture of salt and ice or the freezing point of pure ice. These scales are said to be 2-point unique[19]. The magnitude structure of concretes affording interval-scale characterization is one whose automorphisms are fixed-point collineations, preeminently stretches[20]. Stretches are transformations of the value-points of a magnitude structure such that one point remains fixed and the intervals from that point to all others are altered by a single ratio. Rand's measurement-omission analysis of concepts and concept classes applies perfectly well to cases in which the measurement scale appropriate to the pertinent magnitude structure of concretes is interval scale. The temperature attribute of a solid or fluid must exist in some measure, but may exist in any measure[21]. But Rand's theory does not entail that all concretes afford interval-scale measures. For Rand's theory does not necessitate that the scale type from which measurements be omitted be interval scale. Her analysis also works perfectly well for a kind of scale having less structure. The magnitude structure entailed for all concretes by Rand's theory is still less than the considerable structure that interval scales reflect. An analogous conclusion obtains for multidimensional magnitude structures of concept classes. Rand's theory does not entail that all 2D or 3D magnitude structures have not only order structure, but affine structure, as Euclidean and Minkowskian geometry have[22]. That is, Rand's theory does not entail that multidimensional magnitude structures of concept classes afford a metric definable from a norm (a measure on vector structure)[23]. (II. Analysis – continued below)