Posted 21 October 2007 - 10:31 AM
1. Cf. Armstrong (1978a, 25–26).
2. Rand takes propositions (E), (I), and (C ) to express primary facts and to be fundamental compositions upon three concepts she takes as axiomatic: existence, identity, and consciousness. She takes all concepts to bear implicit propositions that elucidate the concepts (Rand 1966, 48; 1969, 177–81, 228). Propositions (E), (I), and (C ) are immediate elucidations of Rand’s axiomatic concepts (1957, 1015–16). Rand does not present (I) and (C ) as axioms, only as most important elucidations of her three axiomatic concepts; for her order of presentation, she follows what she takes to be the order of cognitive development (1966, 3, 55–56, 59). My order of presentation brings the propositions (E), (I), and (C ) to the fore, and this, I hope, is analytically illuminating.
3. In the case of the concrete that is the universe itself, which is all of existence, the measurable relations are to parts of itself. For example, the total mass-energy of the universe is a measure having relation to each of its constituents having mass-energy.
Rand took (Im) to be axiomatic in that she took it to be entailed by her axiom (I). A thing not measurable in any way “would bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, . . . in short, it would not exist” (1966, 39). Rand is supposing that anything bearing some relationship to the rest of the universe bears some measurable relationship to the rest of the universe. I think that this supposition, which is tantamount to (Im) (all concretes have measurable relations to other concretes), is a postulate additional to the axiomatic postulate (I) (existence is identity). I do not regard the postulate (Im) to be axiomatic; unlike the axiom (I), the postulate (Im) can be denied without self-contradiction and is therefore open to possible restriction by counterexamples. Like Rand I take (Im) to be an unrestrictedly true postulate.
4. This idea is widespread. Antecedents are to be found in Schopenhauer, Goethe, Helmholtz, James, Bergson, and Dewey. For current applications of the idea that perceptual systems measure, see Krantz, Luce, Suppes, and Tversky (1989, 131–53); Churchland and Sejnowski (1992, 183–233).
5. Pale anticipations of this idea of Rand’s may be found in James (1890, 270), Johnson (1921, 173–92), and Heath (1925, 132–33). For relations to Aquinas and Hume, see Boydstun (1990, 24–27).
6. How might the concept existence satisfy this definition of concepts? Might the concept of existence as all existents (Rand 1969, 241) be rendered as all instances of existents, with all measure values of those existents omitted? (1966, 56). See also Armstrong (1997, 194–95).
7. A concept class is at least the sense of class at work for a kind, for mere membership in a kind (Macnamara 1986, 50–53, 152–56). For Rand’s theory of concepts, however, it seems that concept classes might always also be properly regarded as sets. For in Rand’s theory, all concept classes must be measurable. They must afford some appropriate numerical representation, and any such representation can also be expressed in terms of sets.
There are reasons to doubt whether concept classes always satisfy even the extensionality postulate of Zermelo-Fraenkel set theory, the postulate that two classes collecting the same items are the same class (ibid., 152; Bigelow 1988, 102). Concept classes not satisfying that postulate could not qualify as either so-called proper classes nor as sets. Even if those doubts can be put to rest (Bigelow 1988, 101–9), there would remain further doubts about whether absolutely all concept classes satisfy the separation axiom of ZF set theory. Some concepts, such as the concept all items (all things that are either a potential or actual existent or a mere posit), are so comprehensive that they do not themselves stand as substitution units in some superordinate concept. Then concept classes need not always be extensionality-satisfying classes that are also sets. In particular a concept class need not always be itself a member of a larger class. Such concepts are extremely rare; almost always an extensionality-satisfying concept class will qualify as a set. I assume, with trepidation, that concept classes appropriate for Rand’s theory of concepts are not only classes in the sense of a kind, but also are rightly construed as classes that satisfy extensionality and, with rare exception, are rightly construed as classes that are sets (cf. Armstrong 1997, 185–95).
The following are proper classes, extensional classes that are not sets: the class of all items (the universe of class discourse), the class of all sets, the class of all ordinals (all order types of total orders having least members), and the class of all cardinals (all least ordinals for sets such that there is a mapping from least order type to set that is one-to-one and onto). On proper and nonproper classes, see Machover (1996, 10–16); von Neumann (1925, 393–94, 403); Quine (1961, 90–101, 112–17; 1982, 94, 130–31, 302); and Boolos (1998, 35–36, 42–47, 73–87, 223–24, 238–41).
8. Kline (1972, 554–64, 882–86); Churchland and Sejnowski (1992, 183). My measurement analysis of the concept shape supplements Rand’s. Likewise, it could supplement Armstrong’s (1997, 55–56).
9. Cf. Rand (1966, 14); Peikoff (1991, 84). On the intended elementary sense of linear order, see Rosenstein (1982, 3).
10. On ratio scales, see Krantz, Luce, Suppes, and Tversky (1971, 3–5, 9–10, 44–46, 71–87, 518; 1990, 10, 108–13).
11. Simple addition: 153.0 yards joined to 153.0 yards is 306.0 yards. Grade addition: 153.0 yards per mile joined to 153.0 yards per mile is 310.3 yards per mile. Note also that there are valid and specific nonstandard ways of concatenating lengths, and these are faithfully represented mathematically by specific nonsimple additions (Krantz, Luce, Suppes, and Tversky 1971, 87–88, 99–102; 1990, 18–56).
12. Cf. Swoyer (1987, 256–58). Here I take a norm accepted in mathematical physics and adapt it for our broader context (Geroch 1985, 86, 81–84, 119). Physical gets replaced by concrete for our metaphysics. Notice, making that replacement, that to obtain the relation of mathematics to metaphysics, we may look to the relation of mathematics to physics (ibid., 1, 17, 111–13, 183–87, 223, 283–90, 324–40; Geroch 1996).
13. Krantz, Luce, Suppes, and Tversky (1990, 112–25); Martin (1982, 14–17). Structures are characterized by their automorphisms, the set of structure-preserving morphisms of that structure into itself. (Consider the set of rotations and reflections, confined to the plane, that transform a square into itself: 90° rotation about the square’s center, reflection through a diagonal line, and so forth.) The identity morphism is among the set of automorphisms for any structure. The set of automorphisms for a totally disorganized structure (a would-be structure, we might say) has only that one member, the identity morphism. The identity automorphism by itself affords counting, which is a form of measurement known as absolute measurement (Suppes 2002, 110–18). That barest structure is less than the minimal structure required for concept classes under Rand’s measurement-omission analysis of concepts.
14. Contrast Rand’s system in this respect with the systems of Descartes and Kant. Rand’s theory does not entail that there is any 2D or 3D magnitude structure of concretes having the structure of Euclidean geometry. In particular Rand’s theory does not imply that physical space is Euclidean.
Let me also note at least some of what is meant by absolute and affine in the present context. Euclidean geometry contains both absolute geometry and an affine geometry. Absolute geometry consists of those propositions of Euclidean geometry that can be obtained from Euclid’s first four postulates alone, neither affirming nor denying the fifth postulate, which is the parallel postulate. These propositions hold not only in Euclidean geometry but in hyperbolic geometry. Absolute structure permits the comparison of lengths along lines whether or not they are parallel to each other.
Affine geometry consists of those propositions that can be obtained from Euclid’s first two postulates (to draw a straight line from any point to any point and to produce a finite straight line continuously in a straight line) together with the fifth postulate (in one version: for any point P off a line L, there exists a unique line through P that is parallel to L). Affine structure permits the comparison of lengths only along lines that are parallel to each other. See Krantz, Luce, Suppes, and Tversky (1989, 109–11); Coxeter (1980); Martin (1975).
15. Krantz, Luce, Suppes, and Tversky (1989, 31–35).
16. It might be thought that temperature was found to afford ratio scaling once absolute zero was conceived and the “absolute thermodynamic temperature scale” was constructed. That is incorrect. The interval units of the absolute thermodynamic temperature scale (˚K) are the same as the interval units of the Celsius scale (˚C). Like the Celsius and Fahrenheit scales, construction of the absolute thermodynamic temperature scale requires not only that an interval unit be chosen, but that a fundamental fixed point be chosen and assigned a value. The fixed point selected for the absolute thermodynamic temperature scale is the triple point of water (unique temperature and pressure at which water, ice, and vapor coexist). Absolute zero is then defined to be 273.16 ˚K below the triple point exactly.
What if, contrary to my supposition, temperature were found to be a physical quantity that affords ratio measures? That would not change the outcome of my core task in this study. I am to delineate and put aside the richer types of magnitude structures affording measurement until we arrive at the minimal structure required for Rand’s measurement-omission recipe. The physical examples presented need be, for our purpose, only hypothetical illustrations of types of magnitude structures.
On applications of interval-scale measurement in psychophysics, see Krantz, Luce, Suppes, and Tversky (1971, 139, 519–20; 1989, 177–78, 184–85); also, Michell (1999, 20–21, 74–76, 81–87, 147–52, 172–77, 189–90, 198–200, 205–8). On applications of interval-scale measurement in utility theory, see Krantz, Luce, Suppes, and Tversky (1971, 17–21, 139–42); also, Nozick (1985). That there are magnitude structures affording interval-scale measurement in the realms of utility and psychophysics does not mean that every magnitude structure in those realms affords such measures; some may afford merely ordinal measurement. Rand likely supposed that only ordinal measurement is appropriate in utility theory (1966, 32–34), under tutelage from Austrian-school economists (cf. Rothbard 1962, 15–28, 222–31, 276–79).
17. On interval scales, see Krantz, Luce, Suppes, and Tversky (1971, 10, 17–21, 136–48, 170–73, 515–20; 1990, 10, 108–13). Throughout this paper, I use simply concatenation in place of the usual technical expression positive concatenation. That the concatenations are positive means that the resulting, concatenated magnitude is greater than either of the magnitudes entering into the concatenation. So, I say simply that magnitude structures of concretes such as temperature (or chemical potential) do not afford concatenations, rather than say, as would be usual technically, that such structures afford concatenations qualified as intensive in contrast to positive.
18. A body or fluid at 43˚C is at 109.4˚F. If at 45˚C, then at 113.0˚F. If at 56˚C, then at 132.8˚F. The Celsius difference-interval ratio (45 – 43)/(56 – 45) equals the Fahrenheit difference-interval ratio (113.0 – 109.4)/(132.8 – 113.0). The simple ratios of degrees such as 43/45 and 109.4/113.0 are not equal, unlike the character of ratio scales. We should be aware too of an important respect in which magnitude structures affording interval scales are like magnitude structures affording ratio scales. For either type of structure and their scale types, it is the case that whether two intervals in the structure are equal is independent of which measurement scale in the scale type is used. The interval between 43˚C and 45˚C equals the interval between 47˚C and 49˚C. That equality remains when those values are converted to ˚F, though the value of each equal interval changes from 2˚C to 3.6˚F.
19. Ratio scales stand to each other as a metal bar under uniform thermal expansions. A single number characterizes a particular state of expansion, a particular ratio scale. Interval scales stand to each other as an elastic band pinned at some point, then stretched to some degree from that pinned point, for various such pinnings and stretches. Two numbers characterize a particular pinning and stretch, a particular interval scale. On characterization of scale types by degrees of uniqueness and homogeneity, see Krantz, Luce, Suppes, and Tversky (1990, 112–25, 142–50); Suck (2000); Cameron (1989).
20. Krantz, Luce, Suppes, and Tversky (1990, 112–22); Martin (1982, 136–44).
21. Temperature attributes are relational attributes, specifically, difference attributes. When we sense the warmth or coolness of a body by touching it, we are sensing the rate of heat flow into or out of our own body at the contact surface. Rate of heat flow reflects size of temperature difference between the two bodies in contact.
22. Krantz, Luce, Suppes, and Tversky (1989, 107–8); Coxeter (1980); Martin (1975).
23. Krantz, Luce, Suppes, and Tversky (1989, 42–46). The concept color (resolved as hue, saturation, and brightness) is a 3D magnitude structure that is affine, but not also absolute (Krantz, Luce, Suppes, and Tversky 1971, 515–20; 1989, 40, 243–50, 279–85). The concept space-time (flat space-time) is a 4D magnitude structure that is affine, but not also absolute (though it has absolute substructures).
24. Rand did not herself reach a stable understanding of these entailments. See Rand (1966, 31; 1969, 189–90), where she expresses her supposition that our resort to measurements less rich than ratio-scale measurement is a resort to measurements that are less “exact” and reflects our relative ignorance of the thing we are measuring.
25. On linear orders, see Rosenstein (1982). On ordinal measurement, see Krantz, Luce, Suppes, and Tversky (1971, 2–3, 11, 14–15, 38–43; 1989, 83–89) and Droste (1987a; 1987b).
26. The absolute value function here is not taken over the real numbers in their character as a vector space. Then the absolute value function in our merely ordinal context is not a norm (Bartle 1976, 54–55). Our metric is not being derived from a norm; we do not magically convert our merely ordinal scale to an interval one by taking absolute values of numerical differences.
On topological, uniform, and metric spaces, see James (1999) and Geroch (1985). That the topology of a magnitude structure affording ordinal-, interval-, or ratio-scale measurement be a Hausdorff topology seems fitting. In such a topology, any two distinct points have some nonintersecting neighborhoods, and this would seem to be a natural condition for any sort of measurement at all.
27. On ordered geometry, see Krantz, Luce, Suppes, and Tversky (1989, 104–7) and Coxeter (1980). I say a distance geometry rather than a metric geometry because the distance function need be only positive and symmetric. The triangle inequality, an additional requirement for a metric, need not be satisfied (Martin 1975, 68–69; Coxeter 1980, 175–81; Blumenthal 1970, 16; consider also, Krantz, Luce, Suppes, and Tversky 1989, 186–87, 205–8).
A mathematically determinate form from which measure values may be suspended for the concept shape of a curve (in 3D) is a set of curvature and torsion values, one pair of values for each point of the curve. Consider a 2D graph in which curvature values are plotted along one axis and torsion values are plotted along the other axis. Plotting the particular pairs of values for a particular curve in concrete 3D space will form a particular curve in the plane of our 2D graph. Relations among points in this plane satisfy the axioms of a 2D ordered, distance geometry (as well as axioms for richer 2D geometries).
The concept class shape of a curve satisfies my principle, sprung from Rand’s measurement-omission theory of concepts, that all concept classes having a multidimensional magnitude structure have the structure of at least an ordered, distance geometry. Many of our concepts are obviously multidimensional. Consider a general-purpose definition of the concept animal (metazoa): a multicellular living being capable of nervous sensation and muscular locomotion. Surely the mathematically determinate form of the concept class animal is multidimensional (cf. Rand 1966−67, 16, 24−25, 42). My principle alleges that that multidimensional structure will have the structure of at least an ordered, distance geometry.
28. Cf. Armstrong (1978a, 44–50; 1978b, 95–123; 1997, 17–18, 22–23, 47–57); Jetton (1998, 41–42).
29. But consider Rand’s exchange with Leonard Peikoff (Prof. E) in Rand (1969, 275–76).
30. See also Rand’s exchange with Allan Gotthelf (Prof. B ) in Rand (1969, 139–40) as well as Peikoff (1991, 85) and Gotthelf (2000, 59). Further, see Kelley and Krueger (1984, 52–61); Kelley (1984, 336–45); Jetton (1998, 63–72).
31. See Armstrong (1978a, 11–12, 77–87, 108–16; 1997, 14–15, 28–31, 49); Bigelow (1988, 4, 18–27, 40–41, 56–57, 121–78).
32. Cf. Armstrong (1997, 185–95).
33. Rand concluded from research literature as of 1966 that the sensory experience of the infant was apparently entirely “an undifferentiated chaos” and did not contain any percepts (1966, 5, 6). Subsequent research has dispelled that old vision of cognition in neonates. See Bremner (1994); Meltzoff (1993); Clifton (1992); Kellman (1995).
34. The distinction of particular and specific identity is mine and is as follows. Particular identity answers to that, which, where, or when. Specific identity answers to what. Every existent consists of both a particular and a specific identity (Boydstun 1991, 43–46, and 1995, 110).
35. The sense of implicit here is extracted from the relevant cognitive-development research literature (viz., Gelman and Meck 1983, 344). The child is said to have implicit knowledge of the counting principles if she engages in behavior that is systematically governed by those principles, even though she cannot state them. (See Note 40 for the principles.) Gelman and Meck liken this implicitness of the counting principles at this stage of cognitive development to the way in which we are able to conform to certain rules of syntax when speaking correctly without being able to state those rules. That much seems right, but there is a further distinction I want to make. The child’s implicit counting principles are being learned (and taught) as an integral part of learning to properly count aggregations explicitly, expressly. In contrast, we can (or anyway, my preliterate Choctaw ancestors centuries past could) live out our lives, speaking fine in our mother tongue, following right rules of syntax, yet without being able to state those rules; indeed, without even knowing any of the terminology of syntax. Our learning of tacit rules of syntax is not for the sake of becoming able to follow them explicitly, only tacitly.
In the present developmental discussion, I shall reserve the term implicit to indicate that an operative rule is not only tacit, but has become operative as an integral part of becoming explicitly operative. The tacit logical principles, whose acquisition according to Macnamara is traced in the text, are not implicit in my present sense.
There is, of course, another sense of implicit that I am also happy to use. That is the logicomathematical sense, which was pertinent to our analysis section. It is in that sense that we say a certain theorem is implicit in a set of axioms; Hertz’ wave equation for propagation of electromagnetic radiation is implicit in Maxwell’s field equations; an inverse-cube central force law is implicit in a spiral orbit; dimension reductions are implicit in Kolmogorov superposition-based neural networks; certain measure relations are implicit in any similarity discerned in perception; or certain measure relations are implicit in a concept class. Cf. Rand (1969, 159–62); Campbell (2002, 294–96, 300–10); Boydstun (1996, 201–2).
36. Drawn out into our adult expression, here is the logic tacitly put to work by the toddler at this stage: There is a unique kind (class) of which Star is a member, and any object is a ball if and only if it is a member of that kind. For any particular ball, there is a unique member of the kind ball, and as long as that member exists, it is identical (totally same) with that particular ball (Macnamara 1986, 137–39). I should say that such working interpretive principles render one’s perceptual knowledge conceptual. One has conceptual knowledge even at the single-words stage of language development.
My example of proper naming of a special ball Star is contrived for convenience of illustrating the tacit logical resource. Toddlers at this stage are likely to restrict proper names to particular (real or make-believe) animate entities possessing mentality (Bloom 2000, 130–31).
37. By 24 months the child is using two-word utterances such as “Mommy sit!” and “guy there” and “I know [how to do it]” (Bremner 1994, 252–53; Nelson 1996, 112, 124–25). Up to about this time, when grammar begins to develop, “words learned remain tied to their world models and do not form systems of their own” (Nelson 1996, 128). In terms of Deacon’s iconic, indexical, and symbolic levels of representation (1997, 70–83), I should say that concepts at the single-words stage are indexical representations, and these concepts will become symbolic representations with the onset of grammar. Rand’s conceptual level of representation cuts across Deacon’s indexical and symbolic levels.
All three levels of representational cognition—even the iconic level (e.g., drawing a stick man)—are active, deliberate, and constructive. I take the membership relation, which is essential for concepts, classes, and sets, to require this sort of active generation, from our first concept to our last. In this way, the membership relation is unlike perceptual relations of similarity, proximity, or containment (cf. Rand 1964, 20; Maddy 1997, 90–94, 108–9, 152n30, 172–76, 185–88).
38. See further Boydstun (1990, 16–18); Minsky (1974, 111–17); Johnson (1987, 23–30, 102–4); Iverson and Thelen (1999); Nelson (1996, 16–17).
39. Cf. Kelley and Krueger (1984, 47, 52). In saying that this tacit logical principle of application is a surrogate for a concept’s definition, I mean to say only that the tacit principle accomplishes the main function that an explicit definition accomplishes. I do not mean to say that the tacit principle is additionally an implicit definition in the developmental sense of implicit (as in Note 35). Macnamara’s tacit logical principle of application is needed just as much for concepts of things in terms of merely characteristic features as it is for concepts of things in terms of defining features (cf. Bloom 2000, 18–19).
During the first few years of speech, we evidently tend to conceive of things in terms of characteristic features. After about age 5, there is a developmental shift to conceiving of things in terms of defining features. The course of this shift, which occurs at different times in different domains of knowledge, has been partially charted by Frank Keil (1989); see Boydstun (1990, 34–37). The shift need never occur for all our concepts. [In a preliterate culture (my Choctaw ancestors again), is the shift so extensive as in our culture? See Olson (1994).] Acquiring a tacit logical principle of application is not for the sake of becoming able to conceive of things in terms of defining features.
40. The child has gone far beyond learning first words (roughly months 12 to 18) by the time she is learning to count. By 30 months, the basic linguistic system has become established and is fairly stable (Nelson 1996, 106). Not until around 36 months or beyond does the child have an implicit grasp of the elementary principles of counting: assign one-label-for-one-item, keep stable the order of number labels recited, assign final recited number as the number of items in the counted collection, realize that any sort of items can be counted, and realize that the order in which the items are counted is irrelevant (Gelman and Meck 1983; Butterworth 1999, 109–16).
At 22 months, a child in my family could “say his numbers.” This competence is not essentially different than being able to “say his ABC’s” (Bloom 2000, 215). Rand may have mistaken the onset of recitation of count-word sequences with onset of ability to count.
41. Cf. Macnamara (1986, 143); Burgess (1998, 10–11); Boolos (1984, 72).
42. For Ockham on comparative similarity, see Maurer (1994, 387, 389). For more on comparative difference and comparative similarity in theory of concept formation, especially in Rand’s theory, see Kelley and Krueger (1984, 52–61) and Kelley (1984, 336–45). See also Jetton (1998, 63–72) and Livingston (1998, 15–21).
43. Cf. Armstrong (1997, 64–65) for a related extravagance, which he boldly embraces. The extravagant implication I pose is avoided by me in one way; for another way, consider Jetton (1991).
44. Quine (1969, 117–23); Krantz, Luce, Suppes, and Tversky (1989, 207–22); Nosofsky (1992, 38–40).
45. The General Relativity principle that freely falling bodies follow time-like geodesics of space-time is subject to analytical challenges (Torretti 1983, 176–81) and to empirical tests, such as whether Earth and Moon have different accelerations towards the sun (Ciufolini and Wheeler 1995, 14, 88, 113–15). Contrast those methods of evaluating conjectures in natural science with the methods of evaluating various candidate axioms for a formal discipline such as set theory (Maddy 1997). We should expect the forms of evaluation appropriate to measurement conjectures for a theory of concepts and concept classes to lie between forms appropriate to natural science and forms appropriate to the formal disciplines of mathematics, set theory, and logic.
46. This essay was studied at the 2003 Advanced Seminar of The Objectivist Center. The significance of the present work was clearly appreciated. The session indicated that reference to an accessible general overview of modern mathematics would be helpful. I heartily recommend MacLane 1986.