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Boydstun

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  1. Publications Ayn Rand Society Philosophical Studies (Currently in process, a volume on Aristotle and Rand) Darwinism 2. Darwin and Darwinism 2.1 Darwin’s Life 2.2 Darwin’s Darwinism 2.3 Philosophical Problems with Darwin’s Darwinism 3. The Five Core Philosophical Problems Today 3.1 The Roles of Chance in Evolutionary Theory 3.2 The Nature, Power and Scope of Selection 3.3 Selection, Adaptation and Teleology 3.4 Species and the Concept of ‘Species’
  2. As it has turned out, the US reached this mortality number in only two months, rather than the predicted four months. ~~~~~~~~~~~~~~~~ On US public action, there is some good historical perspective here. Excerpts
  3. Some Comments on Rand’s Theory and Some Historical Notes Merlin Jetton would not want to count ordinal scales as measurement scales (“Omissions and Measurement” JARS Spring 2006). Similarly, on his view, ordered geometry and affine geometry should not pass muster as multidimensional measurement systems. I do count ordinal and other scales on up to ratio scale as measurement scales, I count ordered geometry and affine geometry as measurement systems, and in all of that I’m in league with the principal measurement theorists of the last few decades. Even if one did not think of ordinal ranking as measurement, it would remain that it takes the set structures the theorists have found for it, going beyond the structure for counting (absolute scaling). This makes Rand’s conjecture (her analysis conjecture presupposed by her formation conjecture) and mine (weaker than hers) an addition to the simple substitution-unit standing of instances under a concept that is common to pretty much all theories of concepts or universals. There are, I say, some indispensable concepts we should not expect to be susceptible to being cast under a measurement-omission form of concepts. Among these would be the logical constants such as negation, conjunction, or disjunction. The different occasions of these concepts are substitution units under them, but the occasions under these concepts are not with any measure values along dimensions, not with any measure values on any measure scale having the structure of ordinal scale or above. Similarly, it would seem that logical concepts on which the fundamental concepts of set theory and mathematical category theory rely have substitution units, but not measure-value units at ordinal or above. The membership concept, back of substitution units and sets, hence back of concepts, is also a concept whose units are only substitution units. Indeed, all of the logical concepts required as presupposition of arithmetic and measurement have only substitution units. Still, to claim that all concretes can be subsumed under some concept(s) other than those said concept(s) having not only substitution units, but measure values at ordinal or above, is a very substantial claim about all concrete particulars. ~~~~~~~~~~~~~~~~ These notes below do not go to the truth or importance of Rand’s theory (and its presuppositions), only to its originality or uniqueness and its relations to other theories in the history of philosophy. From my essay: In the years after composing this paper (2002–03), I learned of a “pale anticipation” of Rand’s measurement-omission perspective on concepts way back in the fifth or sixth century. My studies of Roger Bacon, a contemporary of Aquinas, led me to study Bacon’s mentor and model Robert Grosseteste (c. 1168–1253). The latter mentioned that Pseudo-Dionysus (an influential Neoplatonic Christian of the fifth or sixth century) had held a certain idea about the signification of names. From James McEvoy’s The Philosophy of Robert Grosseteste (1982): “[Grosseteste] reminds us that Pseudo-Dionysius himself at one point introduced the hypothesis that the names signify properties held in common, but subject to gradation in the order of intensity. Thus the seraphim, for instance, are named from their burning love; but it goes without saying that love is a universal activity of spirit” (141–42). Angels were thought to exist and to have ranks, I should say. Some kinds have burning love; others do not have that kind of love. The thought of Pseudo-Dionysius and of Grosseteste was that angels in the different ranks, angels of different kinds, all shared some properties (e.g. their participation in being, their knowledge, or their love) that the various types possessed in various degrees. I have located the pertinent text of Pseudo-Dionysius. It is in chapter 5 of his work The Celestial Hierarchy. The heading of that chapter is “Why the Heavenly Beings Are All Called ‘Angel’ in Common.” Dionysius writes: “If scripture gives a shared name to all the angels, the reason is that all the heavenly powers hold as a common possession an inferior or superior capacity to conform to the divine and to enter into communion with the light coming from God” (translation of Colm Luibheid 1987). To the preceding compilation of historical anticipations of Rand’s analysis of concepts in terms of measurement omission, I should add the case of John Duns Scotus. Continuing from Aristotle and Porphyry, medieval thinkers reflecting on universals and individuation held specific differentia added to a genus make a species what it is and essentially different from other species under the genus. Similarly, individual differentia added to a species make an individual what it is and different from other individuals in the species. Scotus held individuals in a species to have a common nature. That nature makes the individuals the kind they are. It is formally distinct from the individual differentia, a principle that accounts for the individual being the very thing it is. The individual differentia, in Scotus’ conception, will not be found among Aristotle’s categories. Individual differentia are the ultimate different ways in which a common nature can be. Individual differentia are modes of, particular contractions of that uncontracted common nature. “The contracted nature is just as much a mode of an uncontracted nature as a given intensity of whiteness is a mode of whiteness, or a given amount of heat is a mode of heat. It is no accident that Scotus regularly speaks of an ‘individual degree’ (gradus individualis)” (Peter King 2000—The Problem of Individuation in the Middle Ages. Theoria 66:159–84).
  4. References Armstrong, D.M. 1978a. Nominalism and Realism (Vol. 1 of Universals and Scientific Realism). Cambridge: Cambridge University Press. ――. 1978b. A Theory of Universals (Vol. 2 of Universals and Scientific Realism). Cambridge: Cambridge University Press. ――. 1997. A World of States of Affairs. Cambridge: Cambridge University Press. Bartle, Robert. 1976. The Elements of Real Analysis. 2nd edition. New York: John Wiley & Sons. Bigelow, John. 1988. The Reality of Numbers. Oxford. Clarendon Press. Bloom, Paul. 2000. How Children Learn the Meanings of Words. Cambridge: MIT Press. Blumenthal, Leonard. 1970. Lattice geometry. In Studies in Geometry, 3–129. San Francisco: Freeman. Boolos, George. 1998. Logic, Logic, and Logic, edited by R. Jeffrey. Cambridge: Harvard University Press. Boydstun, Stephen. 1990. Capturing concepts. Objectivity 1(1):13–41. ――. 1991. Induction on identity (Part 2). Objectivity 1(3):1–56. ――. 1995. Volitional synapses (Part 2). 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Mathematical Physics. Chicago: University of Chicago Press. ――. 1996. Partial differential equations of physics. In General Relativity. Edited by G.S. Hall and J.R. Pulham, 19–60. Bristol: Scottish Universities and Institute of Physics. Gotthelf, Allan. 2000. On Ayn Rand. Belmont, CA: Wadsworth. Heath, Thomas. [1925] 1956. Euclid’s Elements (Vol. 1). New York: Dover. Iverson, Jana, and Esther Thelen. 1999. Hand, mouth, and brain: The dynamic emergence of speech and gesture. Journal of Consciousness Studies 6(11–12):19–40. James, I.M. 1999. Topologies and Uniformities. London: Springer. James, William. [1890] 1950. The Principles of Psychology (Vol. 1). New York: Dover. Jetton, Merlin. 1991. Formation of concepts. Objectivity 1(2):95–97. ――. 1998. Pursuing similarity. Objectivity 2(6):41–130. Johnson, Mark. 1987. The Body in the Mind. Chicago: University of Chicago Press. Johnson, Mark H. 1990. Cortical maturation and the development of visual attention in early infancy. Journal of Cognitive Neuroscience 2(2):81–95. Johnson, W.E. [1921] 1964. Logic (Part 1). New York: Dover. Keil, Frank. 1989. Concepts, Kinds, and Cognitive Development. Cambridge: MIT Press. Kellman, Philip. 1995. Ontogenesis of space and motion perception. In Perception of Space and Motion. 2nd edition. Edited by W. Epstein and S. Rogers, 327–64. New York: Academic Press. Kelley, David. 1984. A theory of abstraction. Cognition and Brain Theory 7:329–57. Kelley, David, and Janet Krueger. 1984. The psychology of abstraction. Journal for the Theory of Social Behaviour 14(1):43–67. Kline, Morris. 1972. Mathematical Thought. New York: Oxford University Press. Krantz, D.H., Luce, R.D., Suppes, P., and A. Tversky. 1971. Foundations of Measurement (Vol. 1). New York: Academic Press. ――. 1989. Foundations of Measurement (Vol. 2). New York: Academic Press. ――. 1990. Foundations of Measurement (Vol.3). New York: Academic Press. Livingston, Kenneth. 1998. Rationality and the psychology of abstraction. Objectivist Studies, no. 1. Poughkeepsie: Institute for Objectivist Studies. Machover, Moshé. 1996. Set Theory, Logic and their Limitations. Cambridge: Cambridge University Press. MacLane, Saunders. 1986. Mathematics: Form and Function. New York: Springer-Verlag. Macnamara, John. 1986. A Border Dispute: The Place of Logic in Psychology. Cambridge: MIT Press. Maddy, Penelope. 1997. Naturalism in Mathematics. New York: Oxford University Press. Mandler, Jean, and Patricia Bauer. 1988. The cradle of categorization: Is the basic level basic? Cognitive Development 3:247–64. Martin, George. [1975] 1998. The Foundations of Geometry and the Non-Euclidean Plane. New York: Springer. ――. 1982. Transformation Geometry. New York: Springer. Maurer, Armand. 1994. William of Ockham. In Individuation in Scholasticism, edited by J.J.E. Gracia, 371–96. Albany: State University of New York Press. Meltzoff, Andrew. 1993. Molyneux’s babies: Cross-modal perception, imitation, and the mind of the preverbal infant. In Spatial Representation, edited by N. Eilan, R. McCarthy, and B. Brewer, 219–35. Oxford: Blackwell. Michell, Joel. 1999. Measurement in Psychology. Cambridge: Cambridge University Press. Minsky, Marvin [1974] 1997. A framework for representing knowledge. In Mind Design II, edited by J. Haugeland, 111–42. Cambridge: MIT Press. Needham, Amy, and Renée Baillargeon. 1993. Intuitions about support in 4.5-month-old infants. Cognition 47:121–48. Nelson, Katherine. 1996. Language in Cognitive Development. Cambridge: Cambridge University Press. Nosofsky, Robert. 1992. Similarity scaling and cognitive process models. Annual Review of Psychology 43:25–53. Nozick, Robert. [1985] 1997. Interpersonal utility theory. In Socratic Puzzles, 85–109. Cambridge: Harvard University Press. Olson, David. 1994. The World on Paper: The Conceptual and Cognitive Implications of Writing and Reading. Cambridge: Cambridge University Press. Peikoff, Leonard. 1991. Objectivism: The Philosophy of Ayn Rand. New York: Dutton. Quinn, Paul. 1987. The categorical representation of visual pattern information by young infants. Cognition 27(2):145–79. Quine, Willard van Orman. [1961] 1980. From a Logical Point of View. 2nd edition. Cambridge: Harvard University Press. ――. 1969. Natural kinds. In Ontological Relativity and Other Essays, 114–38. New York: Columbia University Press. ――. 1982. Methods of Logic. 4th edition. Cambridge: Harvard University Press. Rand, Ayn. 1957. Atlas Shrugged. New York: Random House. ――. [1961] 1964. The Objectivist ethics. In The Virtue of Selfishness, 13–35. New York: New American Library. ――. [1965] 1989. Who is the final authority in ethics? In The Voice of Reason, edited by L. Peikoff, 17–22. New York: Meridian. ――. [1966] 1990. Introduction to Objectivist Epistemology. Expanded 2nd edition. New York: Meridian. ――. [1969] 1990. Transcripts from Ayn Rand’s epistemology seminar. Edited by L. Peikoff and H. Binswanger. Appendix to Introduction to Objectivist Epistemology. Expanded 2nd edition. New York: Meridian. ――. [1970] 1982. Kant versus Sullivan. In Philosophy: Who Needs It, 83–94. New York: New American Library. Rosenstein, Joseph. 1982. Linear Orderings. New York: Academic Press. Rothbard, Murray N. [1962] 1970. Man, Economy, and State: A Treatise on Economic Principles. Los Angeles: Nash. Spelke, Elizabeth, and Gretchen Van de Walle 1993. Perceiving and reasoning about objects: Insights from infants. In Spatial Representation, edited by N. Eilan, R. McCarthy, and B. Brewer, 132–61. Oxford: Blackwell. Starkey, Prentice, Spelke, Elizabeth, and Rochel Gelman. 1990. Numerical abstraction by human infants. Cognition 36:97–127. Streri, Arlette, and Elizabeth Spelke. 1988. Haptic perception of objects in infancy. Cognitive Psychology 20:1–23. Suck, Reinhard. 2000. Scale type (n,n) and an order-based topology induced on the automorphism group. Journal of Mathematical Psychology 44:582–99. Suppes, Patrick. 2002. Representation and Invariance of Scientific Structures. Stanford: CSLI Publications. Swoyer, Chris. 1987. The metaphysics of measurement. In Measurement, Realism and Objectivity, edited by J. Forge, 235–90. Dordrecht: D. Reidel. Torretti, Roberto. [1983] 1996. Relativity and Geometry. New York: Dover. Von Neumann, John. [1925] 1967. An axiomatization of set theory. In From Frege to Gödel, edited by J. van Heijenoort, 393–413. Cambridge: Harvard University Press.
  5. Notes 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. 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.
  6. III. Genesis Rand takes concepts to be mental products of a mental process "that integrates and organizes the evidence provided by man's senses" (1970, 90). She gives three definitions of concepts. (1) Concepts are mental integrations of "two or more perceptual concretes, which are isolated by a process of abstraction and united by means of a specific definition" (1961, 20). More generally in terms of the data processed, (2) concepts are mental integrations of "two or more units which are isolated according to a specific characteristic(s) and united by a specific definition" (1966, 10). Finally and most deeply, (3) concepts are mental integrations of "two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted" (ibid., 12). The "two or more perceptual concretes" spoken of in definition (1) are the elementary type of "two or more units" spoken of in (2) and (3). Rand proposes, in a general way, a developmental intellectual ascent from apprehending the world only in terms of perceptual concretes and actions they afford to apprehending that same world in terms of units in classes. That ascent is a refinement and sophistication in our apprehensions of existents: an ascent from apprehending existents as entities to apprehending them as identities to apprehending them as units (1966, 6–7; 1969, 180–81). Rand's measurement-omission analysis of concepts could be correct even if her account of their genesis were incorrect. In particular, her analysis could be correct even if her proposed developmental intellectual ascent were incorrect. I contend that her general proposed ascent is correct. I shall give a thumbnail sketch of the developments I think should be seen as tracing an entity-identity-unit ascent in the apprehension of existents. Elaboration of Identity For the first day or two after birth, existents for us are plausibly only entities. Such would be the occasions of Mother's face or voice[33]. Very soon existents become for us not mere entities, but identities, particular and specific[34]. At 20 days, there is expectation of the reappearance of a visual object gradually occluded by a moving screen; here is rudimentary particular identity of visual objects (Bremner 1994). At 4 weeks, there is some oral tactile-to-visual transfer of object features, without opportunity for associative learning; here is rudimentary specific identity (Meltzoff 1993). At 5 weeks, there is recognition memory of color and form; here is growth of specific identity. At 8 weeks, there is onset of attention toward internal features of patterns and onset of smooth visual tracking, and there is hand tactile-to-visual transfer of object features; here is growth of particular and specific identity. At 10 weeks, there is expectation that one visual solid object cannot move through another (Bremner 1994). By 3 months, visual tracking is becoming anticipatory (Johnson 1990); there is visual fill-in of invisible parts of objects (Bremner 1994); visual objects are being identified as separate using various static-separation and motion traits (Spelke and Van de Walle 1993); there is categorical perception of objects and events (Quinn 1987). At 4 months, haptic apprehensions of shapes can be transferred to the visual mode (Streri and Spelke 1988); visual solid objects are expected to endure and retain size when occluded for a brief period (Bremner 1994); objects are expected to fall if not supported (Needham and Baillargeon 1993). The infant's world of entities-identities will continue to elaborate. Units are not yet. At 6 months, the infant will have some sensitivity to numerosity; will be able to detect numerical correspondences between disparate collections of items, even correspondences between visible objects and audible events; and will be able to detect the equivalence or nonequivalence of numerical magnitudes of collections (Starkey, Spelke, and Gelman 1990). At 7 months, still without words, the infant distinguishes global categories (e.g., animals v. vehicles) which will later become superordinates of so-called basic-level categories (e.g., dog v. car) yet to be formed (Mandler and Bauer 1988; cf. Rand 1969, 213–15). By 12 months, the infant reliably interprets adult pointing, looking from hand to target (Butterworth and Grover 1988). First Words, First Universals At around 12 months, the infant puts first words, single-word utterances, into her play. Words at this stage are used only in play, not for communication, which is still accomplished with cries, gestures, and gazes (Bremner 1994, 249–51; Nelson 1996, 105, 112). An infant in my family, just past his first birthday, uses the word ba. He says it quietly to himself whenever he sees or is handed a spherical ball; he does not say his word when the ball is a football. We should not suppose too hastily, I should note, that his word ba refers simply to the spherical ball with which he is engaged. At this first-words stage, his utterance may designate the object as component of his whole activities that go with those objects (activities like training the adults to fetch) (Bremner 1994, 251–52; Nelson 1996, 97, 109–10, 115, 227–29; Bloom 2000, 35–39). By 14 months, the toddler points to indicate items (Butterworth and Grover 1988). By 16 months she spontaneously groups objects of a single category (Bremner 1994, 173). In another month or two comes the naming explosion, naming of objects especially (Nelson 1996, 111–15; Macnamara 1986, 144–45; Bloom 2000, 91–100). (That there really is such a dramatic burst in the rate of word acquisition at this time is disputed; Bloom 2000, 39–43.) By that time, at 17 or 18 months, the toddler is using single words to refer (Macnamara 1986, 56–57). These words (50 to 100 words) include demonstratives such as that, common nouns such as ball, and proper names such as Star, say, to refer to a particular ball. The use of common nouns and proper names in single-word reference indicates certain competencies of identification, certain representational comprehensions of identities specific and particular. The representational comprehensions of specific and particular identity that are evidently coming into operation at this stage are class-membership relation, individuation within a class, and particular identity over time. Skillful reference for the utterance ball indicates that the beginning speaker has some working principles for deciding whether a given item qualifies as being in the category ball. Then such a speaker has some operational sense of class-membership relation (ibid., 61–62, 72–74, 124–28, 148–49, 152–56). Ball is a count noun. Although the beginning speaker does not yet possess the principles of counting, not even implicitly, she has some working principles of individuation within a class, some principles for holding in mind individual balls as distinct from one another (ibid., 128–30)[35]. Moreover, ball refers to any individual ball as a distinct individual over time (ibid., 59–60, 130–36, 141–42, 152). Finally, the name Star is attached to a particular one of those individual balls over time (ibid., 55–62, 71–83)[36]. I suggest that even at the single-words stage of language development, the toddler has entered the conceptual level of consciousness in Rand's sense of that level. The utterance ball refers, and marks a concept, already at this stage. One problem for that conjecture is the following. Rand required that the items falling under a concept be united with a specific definition. [see (1) and (2) above and Rand 1966, 48–50.] But at the single-words stage of development, the toddler cannot yet form two-word expressions. That competence will not be attained for another six months or so, at around 24 months of age[37]. Not yet having two-word expressions, she cannot yet form a sentence, cannot yet use words in assertive sentences. Without propositions one is without defining propositions, hence, without definitions. Then at the single-words stage of development, the items falling under a "concept" cannot be united by a specific definition. Then it would seem one does not yet possess a concept in Rand's sense. I think that conclusion would be an overstatement. For an older child or an adult, of course, "a concept identifying perceptual concretes stands for some implicit propositions" (Rand 1966, 48, 21). For a single-words toddler, no propositions can be adduced. Actions can be adduced. A ball is something that can be handled and thrown down. It bounces and rolls. These things are clearly known of balls even by the one-year-old whose first and only word is ba. The concept ball is likely held in mind in the form of image and action schemata as well as by the term ball (Rand 1966, 13, 20, 43; 1969, 167–70)[38]. There is something else, something profoundly conceptual, at hand in linguistic competence at least by the time of the naming explosion. John Macnamara concludes that having a word such as ball at this stage means having a logical principle of application. That is a surrogate for definition at this single-words stage. A principle of application is the working principle, spoken of above, for determining whether an item is or is not a ball (Macnamara 1986, 124–28)[39]. A principle of application determines class membership. That is the basic function a definition accomplishes for more advanced language users (Rand 1966, 40; 1969, 231–32). To have an operational grasp of the class-membership relation is to have a tacit grasp of the notion of unit in the sense of a substitution unit, which is the unit for counting. That does not mean that one has yet grasped the elementary principles of counting (nor that one can put the notion of a substitution unit to work in counting). At 18 months, one has evidently gotten some working hold on the notion of a substitution unit, the notion of a simple member-of-a-class, without yet having the principles of counting. But at this single-words stage, one has taken the first step into the dual realms of the conceptual and the mathematical. With a tacit grasp of the notion of unit in the substitutional sense, "man reaches the conceptual level of cognition, which consists of two interrelated fields: the Conceptual and the Mathematical" (Rand 1966, 7). Rand was correct in thinking that "man's mathematical and conceptual abilities develop simultaneously," even though she was incorrect in thinking that "a child learns to count when he is learning his first words" (1966, 9; 1969, 200)[40]. Having ball, one is getting hold of "ball, any one." That is the membership relation and its requisite principle of application. Having ball, one is also getting hold of "some things of a class, the balls" (Rand 1966, 17–18). That is the individuation-within-class relation and principle (Macnamara 1986, 128–30). Then already at the stage of first concepts, one has beginning working principles of universal quantification (any) and existential quantification (some)[41]. Analytic Constraint As we have seen, in Rand's view, in the analysis of any concept there can be found a double application for some and any: with respect to substitution units and with respect to measure values. To form our concepts, however, Rand supposes that we do not need to grasp, expressly nor tacitly, the notion of units as measure values. We discern similarities. Where there is similarity, there can be found various measure values along a common dimension, in Rand's view, but we need not know anything about such measure bases. When we pick up a ball, our sensory systems measure it in several ways. When we perceive a similarity between two items, according to Rand's account, we are perceiving some same characteristic(s) they both possess in different measure or degree (1966, 13–14; 1969, 139–40, 143). They both possess that characteristic in some measure or degree. Items of their class possess that characteristic in some degree, but may possess it in any degree within a range of measure delimiting the class (Rand 1966, 11–12, 25, 31–32). On which characteristic(s) does the similarity class, thence the concept class, rest? Like Ockham, Rand observed that items in a similarity class are more similar to (and less different from) one another than they are to things not in the class. A ball is more similar in various ways to other balls than it is to sticks, hands, and so forth. As we know, Rand analyzed similarity in terms of measurable dimensions, in terms of measures of dimensional characteristics. The characteristic(s) on which the similarity class and its concept ball rests, analytically and genetically, in Rand's theory, is whichever measurable characteristic(s) makes a ball measurably closer to other balls than to sticks, hands, and so forth (1966, 13–14, 21–23, 41–42; 1969, 144–47, 217, 274–76)[42]. I have addressed the defect and remedy of this measure-theoretic analysis of similarity classes and concepts in the preceding section. It remains to address the genetic aspect, which I cast as: in forming a similarity class and its concept, one is relying on (tacitly using) whichever measurable characteristic(s) makes items in that class and under that concept measurably closer to one another than to opponent items. Rand thought, rightly I should say, that formation of any concept whatever requires differentiating two or more existents from other existents. She thought also that such differentiation requires comparative degrees of difference, measurable as such on a dimension(s) common to existents in the class and existents outside the class (Rand 1966, 13). What if Rand were right in this second doctrine? What if, in order to form any concept whatever, there had to be a dimension common to the concept class and its opponents and this had to be a dimension along which comparative closeness measurement is possible? What would that imply for metaphysics? It would imply that every concrete can be placed in concept classes whose linear measures are not only ordinal-scale, but interval- or ratio-scale as well[43]. I avoid that extravagant implication as follows: I retain Rand's assumption that formation of any concept requires differentiating two or more existents from other existents and her assumption that all concept classes are similarity classes and her measure-definitions, as amended above, of concepts and similarity. I reject the assumption that differentiation between existents included in and existents excluded from a concept class require comparative degrees of difference (beyond the comparative-difference-degree pretender that merely says a thing is less different from itself than it is different from things not itself). Such differentiation may sometimes be based at least partly on fairly blunt sameness and difference. Spherical balls are the same with one another in that they roll regularly, and in this they are baldly different from floors. A dimension along which items in a concept class have various measure values need not be a dimension common with items in an opponent concept class. Differentiation of existents included in or excluded from a concept class may enlist nontrivial comparative degrees of difference (or likeness). I see three forms of these. In one the comparative degrees are along dimensions common to both included and excluded existents, and those dimensions afford either ratio- or interval-scale measures. Along the dimensions of shape, a spherical ball can be distinguished from a football in that way. The sets of pairs of principal curvatures (ratio scaling) over the surfaces of spherical balls are less different from each other, from one ball's set of pairs to another ball's set of pairs, than they are from the sets of pairs of principal curvatures over the surfaces of footballs. In light of my amendment to the measure-definition of similarity, we should allow also for a second nontrivial variety of comparative similarity. I observed earlier that hardness and tensile strength are two different measurable forms of a same characteristic (resistance to degradation under some sort of stress) that is different from the measurable characteristic (pairs of principal curvatures, which are spatial extension properties) shared by shapes. This second manner of decomposing a comparative similarity permits concepts based on comparative degrees of similarity without requiring that linear measures of the concept dimensions be anything beyond ordinal measures. A third decomposition of nontrivial comparative similarity does not rely on shared and unshared dimensions of the relata. It relies simply on numbers of shared and unshared features[44]. Perhaps any concept based on this sort of comparative similarity can be recaptured in a more sophisticated way by ascertaining measurable dimensions on which to base the concept (Boydstun 1990, 31–33). I expect that is so. Notice, however, that the metaphysical implication drawn in the present study (uniform topological lattice structure) need not suppose that all concepts can be analyzed in terms of Rand's measurement-omission formula; only that all concretes can be placed under one or more concepts analyzable in those terms. I have exhibited a way in which a measurement analysis of concepts can constrain theorizing about the genesis of concepts. I do not want to create the impression, however, that theory of the genesis of concepts based on observations and empirical testing cannot rightly constrain one's analysis of concepts. The analytical principles stating that all concretes can be placed in concept classes having a measurement structure and that these structures are of such-and-such characters are conjectures open to restriction through counterexamples. These conjectures of analysis are subject to reform or replacement in the face of contrary analytical and empirical results, somewhat as the General Relativity principle that freely falling bodies follow time-like geodesics of space-time is subject to reform or replacement[45]. One of the avenues for empirical confrontation of our analytical conjectures concerning concepts and concept classes is research on conceptual development. The core task I have undertaken in the present study has been a certain extension of Rand's metaphysics arising from her analysis of concepts (not her theory of their formation). I have not undertaken here a survey of the various ways in which empirical research on conceptual development may challenge Rand's analysis of concepts. But there is one form of challenge that is invalid, and I want to draw attention to this fallacy, which has required a long struggle for me to overcome. That is a fallacy I insinuated in Boydstun (1990, 33–34). It says that because preschoolers do not possess—not even tacitly—mathematical understanding sufficient to be forming their concepts using a principle of measurement-omission, their concepts do not bear analysis in terms of measurement-omission. That is the fallacy of confusing genesis with analysis[46].
  7. 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.)
  8. 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)
  9. 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.
  10. Welcome to Objectivism Online, Carl Leduc. I was wondering, given your university, whether you are bilingual French/English. Also, if you read both well, would you say there has been a good translation of Atlas Shrugged into French? ~~~~~~~~~~~~~~~~ This is only a sidebar to your question, Carl, but I do not agree with the idea that understanding Objectivism completely takes years. I know that the philosophy can go on and on, effectively endlessly, in the different traditional and new philosophical questions it can be developed to tackle. And on and on in detailed scholarly comparisons with other philosophies. And on and on in the ‘philosophy of x’, where x stands for the various special areas of knowledge such as mathematics and the various sciences. Objectivism itself—considering Rand’s writings she chose to publish as well as subsequent works by competent expositors in this close period beyond Rand’s life—can be thought to be of various sizes it seems to me. The first size would be simply what all is in the novel Atlas Shrugged (mainly Galt’s Speech, with its organized conceptual progression). In my own estimation, anyone fully understanding what is said in that book alone understands Objectivism. Everything further, fine and fascinating as it is concerning the philosophy set out there, is inessential to Objectivism insofar as the further work delineates the philosophy at all beyond what was said in that book. It has been my experience that people interested in learning more of the philosophy beyond what they could or did find in Atlas are somewhat above average general intelligence, usually at least one standard deviation above. Seekers of more, in my encounters with them, were seldom genuinely seeking to get something clarified they had found in Atlas nor figure out what good applications the book and its philosophy might have for making their own life. Rather, they were reaching for additional intellectual adventures and realms stemmed from aspects of the Atlas one. There are two books beyond Atlas that present the philosophy, in its larger, more luxurious size, in an organized way. So to a great extent, these present the philosophy with the integration needed for integrated understanding of it. Those are Objectivism: The Philosophy of Ayn Rand and The Blackwell Companion to Ayn Rand. Stephen
  11. PNC Ground Shifts to the Side of the Subject – Kant IV-c Let’s test that analogy Kant floated (see last quotes from KrV in this post) between his distinction Pure Logic v. Applied Logic and his distinction Pure Morality v. Proper Virtue. Among the things he writes concerning the latter distinction are the following. “[Moral laws] hold as laws only insofar as they can be seen to have an a priori basis and be necessary” (1797 6:215). Kant does not here mean necessary as in necessary for some purpose, though such laws are required for some purpose; he means just plain necessary as when we say something is necessarily so. By the proposed analogy, we have: Pure general logical laws hold only insofar as they can be seen to have an a priori basis and be necessary. By its moral laws, “reason commands how we are to act even though no example of this could be found, and it takes no account of the advantages we thereby gain, which only experience could teach us. For although reason allows us to seek our advantage in every way possible to us . . . still the authority of its precepts as commands is not based on these considerations. Instead it uses them (as counsels) only as a counterweight against inducements to the contrary, to offset in advance the error of biased scales in practical appraisal [for what best to do], and only then to insure that the weight of practical reason’s a priori grounds will turn the scales in favor of the authority of its precepts.” (1797 6:216) Rather like F. Bacon and Locke, Kant thought of errors we make in applied logic as largely from prejudices (Jäsche 1800, §§75–81; Lu-Adler 2018, 67, 102, 114), and this tunes pretty well with the role of prejudice for advantage that Kant sees in the origin of moral errors. Learning applied logic—which like pure logic, applies to any topic whatever—can free one from prejudices that come onto the scene in the workings of empirical psychology as logic is put to concrete occasions of thinking. Transcendence of prejudices in the arena of applied logic frees one to think for oneself (Lu-Adler 2018, 114). All analogies break down in some ways, but I take seriously for Kant’s account of logic the conception of commanding by principles from pure logic as analogous with commanding by pure morality. Kant characterized pure general logic as well as pure morality as objective, in contrast to subjectivity in the arena of applied general logic and in the arena of proper virtue. In pure morality, there is found a necessary law for all rational beings connected a priori with the concept of a rational will as such. Kant does not apply the sort of argument in that last paragraph to support his contention that pure general logic is objective. Kant uses objective in various senses. He does not reach the sense of objectivity to be recognized in pure general logic by recognizing merely that this logic is common to all agents of human cognition. Indeed, applied logic and the errors it diagnoses and remedies are also at hand in all such agents. Applied logic is subjective, in one of Kant’s senses of the term, simply because it treats not purely a priori, necessary norms for how we ought to think, but partly how we actually think. In my estimate, however, Kant’s contention that principles of pure logic have normative character at all is nothing known independently of concrete operations of thought. It was not unreasonable to consider a prior, necessary logical norms as objective, but I submit that their objective character we discern comes not at all from their purportedly a priori normative element, from their character as “commanding,” but from the a priori standing and necessity in the formal structures of logic, as in those two features of the formal structures of pure mathematics. They are objective in the sense that you cannot get around them. They are stubborn, as Whitehead would say. Pure geometry of Kant’s era, preeminently Euclid’s geometry, provides norms for all sorts of practical constructions and discernments, but normative import of pure geometry is incidental to it. In contrast to the character of pure geometry, normative character, in the view of Kant, is essential to pure general logic and, as well, to applied general logic. The distinction Kant drew under those titles of general logic do not hold so sharply as he had thought in his general pronouncements about the distinction. He drew other sorts of divisions of what may reasonably be called logic, but this particular distinction is a failure. In his lectures on logic, Kant would count the prescriptions that we now call formal fallacies of inference as belonging to pure general logic, for they are norms purely of form, and they treat no psychological impediments; the informal fallacies of inference would be glossed as applied general logic, for they concern partly how we actually think, not only how we ought to think, and they deal with contingent, psychological impediments. Among logical prescriptions for concepts, Kant has rules for definitions of concepts. Examples would be that (i) species must parse the members under the concept such that each member belongs exclusively under one of the concept’s species or another and (ii) definitions should be in terms of internal characteristics of things under the concept being defined. These rules are guides at work unconsciously in untutored concrete reason, according to Kant’s lectures, and logic only makes them abstract, explicit formulas. The standing Kant conceived for the logical rules of judgment and rules of inference is quite different from the standing he gives to those logical rules for concepts. All these rules of general logic are for perfection of cognition as to form, but the premier rules for judgments and inferences are as commands coming externally to concrete operations of reason, tutored or untutored. It will be noticed that a definition of a concept is a judgment, so theory of the perfection of concepts as to form is brought to completion by logical rules for judgments. This entire scheme of externality and command-character with respect to concrete judgment and deduction is a forced contrivance, not, I say, a sound characterization of logic that is in our elementary texts and in Kant’s own logic lectures. “The proposition, No thing can have a predicate that contradicts it, is called the principle of contradiction” (A151 B190). Kant rightly held that the premier fundamental principle of formal rightness in judgments is the principle of (non)contradiction (further, Pluhar 1987, lxxxixn90). He rightly held that the fundamental principle in deductive inference is not PNC, but this: whatever holds of the genus or species holds also of all things under the genus or species (Kant 1762, §2; 1792, 773; cf. Peikoff 1964, 134–35*). “A judgment is the way that concepts belong to one consciousness universally, objectively” (Kant c.1780, 928). To say “God is moral and is not moral” is a contradiction. Is avoidance of such perfectly plain error of contradiction the great command from pure general logic concerning judgments? I hardly think so. What I see Kant wielding in his logic lectures is the likes of “God is moral and evil is not punished.” That is what was traditionally called contradictio in adjecto. In the example, one is forgetting the elementary traits of God and contradicts that concept of God in contending that evil is not punished. I want to stress that not forgetting elementary aspects of what one is talking about is touching on contingent, empirical psychological processes and hardly a purely a priori rule. In the next installment, I intend to end the long study that is this thread. So we’ll end with the finish of these Kant issues, and I’ll not then be doubling back for more on conventionalist theories of logic. In that next, final installment, I’ll show the untenability of Kant’s sharp divide of pure general logic and applied general logic as it pertains to deductive inference and the failure of Kant’s attempted escape from the criticism (e.g. from Peikoff 1964) that Kant cannot have both his way of norm-setting by general logic yet its necessity of law straightly descriptive of human thought (as Kant appeared to maintain, on the plain face of it, in the Jäsche Logic [and in the Wiener Logic, unavailable for Peikoff 1964]). References Gregor, M. J., trans. 1996. Immanuel Kant – Practical Philosophy. Cambridge. Jäsche, G. B. 1800. Kant’s Logic. In Young 1992. Kant, I. 1762. The False Subtlety of the Four Syllogistic Figures. In Walford and Meerbote 1992. ——. c.1780. The Heschel Logic. In Young 1992. ——. 1781. Critique of Pure Reason. W. S. Pluhar, trans. 1996. Hackett. ——. 1785. Groundwork of the Metaphysics of Morals. In Gregor 1996. ——. 1788. Critique of Practical Reason. In Gregor 1996. ——. 1790. Critique of Judgment. W. S. Pluhar, trans. 1987. Hackett. ——. 1792. The Dohna-Wundlacken Logic. In Young 1992. ——. 1797. The Metaphysics of Morals. In Gregor 1996. Lu-Adler, H. 2018. Kant and the Science of Logic. Oxford. Peikoff, L. 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University. Pluhar, W. S. 1987. Introduction to Kant 1790. Walford, D. and R. Meerbote. 1992. Immanuel Kant – Theoretical Philosophy, 1755 – 1770. Young, J. M., trans. 1992. Immanuel Kant – Lectures on Logic. Cambridge.
  12. Thanks, Richard for the poem 'Evans'. I'd not known of it or its author, and I like it. Thanks also for the feedback on my poem 'The Song'. Your interpretation seems a natural one, now I think of it. My own meaning in that one was not concerning a romantic close relationship. It is rather about what I take for right about the human psyche and standpoint generally and the 'you' in the poem is only a pronominal one, a placeholder of any other human. I had begun writing a philosophy book in January 2014 because all of my years of independent studies of philosophy had come round into some sort of critical point of a formation of my own basic metaphysics and other theoretical areas, all gelling together. It is indebted to Rand's work in metaphysics and is a transfiguration of hers. The first and last verses of this poem were early-on in the book writing and were simply as first and last paragraphs for that book. I worked every morning on the book for six and three-quarters years, by which point (last fall) so much had stabilized and yet so many ramification remained to be worked out for the philosophy that I asked the editor of The Journal of Ayn Rand Studies if the basic, settled part might be appropriate scholarly material for that journal. He said yes indeed, and it's under review there now, most likely will appear this coming winter. That work, that paper, is the most important thing I have ever written by far. I forgot to mention in it that there is a fitting name for my system, and I'll say it here, noting that the first and third verses of the poem are some facets of what that name means: the philosophy of Resonant Existence. (The middle verse of the poem, by the way, refers [poetically, inexactly] to most of the fundamental categories in my system, category division of existence being one of four fundamental ways in which existence is naturally divided. Rand also had four ways of dividing existence, some of the elements in mine differ from hers, and this will have important ramifications for theoretical philosophy downstream.)
  13. In my case, just some of my poetry. The Song The world and you are with me. It and you make and move me. I make and move in the world and you. Now was will be it is. It is pass we pace. It is field we trace. It is trait we lace. We it we live, of it, of we. The world and I are with you. It and I make and move you. You make and move in the world and me.
  14. I got to attend this lecture Certainty in person on 7 July 1992. That was part of a series of lectures he delivered at a conference in Williamsburg, VA. In the early 70’s, I had attended a recorded series of lectures he had made on history of modern philosophy. In ’77 I had attended the recording-series of his ’76 lectures “The Philosophy of Objectivism.” I learned from and was profitably stimulated by all those lectures. I’m overwhelmingly a reading-and-writing sort of person myself. I gather that much of this recorded lecture material of Peikoff’s was never worked up into papers or books, and anyway for many other people, recorded lectures are a better way for them to learn than by reading, especially given their life circumstances. So it’s nice so many of these compositions were recorded and preserved and are becoming widely available. Nice too, on part of the personal level, that a trace of Peikoff’s personality stays on in these recordings. But of course, for Peikoff, as for Rand or me, the most deeply personal are the ideas, and these are our communion with minds deceased long before our start.
  15. All recorded lectures of Leonard Peikoff are to become available here.
  16. Lawrence Edward Richard, firstly, welcome. I wondered if you are related to the Lawrence Edward Richard who died in 2011, because a Facebook man of that name stopped posting there at that time and recently that page has started again having posts under that name. I wondered if perhaps you were his son or other relation. Anyway, welcome to Objectivism Online. I enjoy your posts, as so many others here. ~~~~~~~~~~~~~~~~~~~ I think Rand, as any person in a sensible moment, would squarely object to the statement of Feynman’s as stated, which William Hobba rightly disputed, at the root post of this thread. In its context, which is unknown to me, we might see some better sense to Feynman’s remark. To the remark as it stands here, I would add to Mr. Hobba’s remark that Newton’s definition of Force, as well as its expanded formula by Einstein/Planck, is precise. They are both precise. That the later one is wider in correct application and contains the earlier one in the appropriate physical limit, does not make the later one more precise, but more widely correct. On and on, there is precise definition in physics. The definition of what are canonically conjugate pairs of dynamical variables is precise. The indeterminacy of their precise joint values in the quantum regime is precise. The definition of what is a Feynman Diagram is precise. Rand praised modern science a lot, but had criticisms of a number of general things being said about science by ’57, quoted from the fictitious book Why Do You Think You Think? (AS 340-41). Also in Atlas Shrugged, she made a couple of criticisms of some particular modern science. Most famously, she criticized Behaviorist psychology, which critique she extend in a later essay concerning Skinner. She indicated what was by her lights a wise attitude towards QM, with its “Uncertainty Principle” so salient with the educated public at the time, through words of the fictional character Dr. Stadler (346). She never returned to QM physics stuff herself, but she put her stamp of approval on all the contents of Peikoff’s 1976 lecture series “The Philosophy of Objectivism” which included his understanding and critique of the “measurement problem” in QM. Rand’s rejection of Behaviorism and (with Branden) of human instincts (under some prominent meanings) and the subconscious (under some prominent meanings) was under her view in what is usually called philosophical psychology. Her conception of What is a human being? was at odds with those quasi- or pseudo-scientific psychology schematics. Rand carried in The Objectivist a serial article on epistemological issues in biology that was authored by Robert Efron, a distinguished neuroscientist (Christoff Koch was a student of his). The title was “Biology without Consciousness” (1968). Rand savaged a paper by philosopher of science Feyerabend in her 1970 essay “Kant v. Sullivan.” Rand’s philosophy has also had some interface with science in her conceptions of what sort of thing could or could not be a cause anything.
  17. Clive, being a “kindergarten for communism” is not the same thing as communism or even conceiving of communism. Christianity in the New Testament does not advocate a kind of communism in any verse I recall. Still, regardless of Christianity, your question of whether Objectivism would class voluntary communism as immoral is a good one. I think Rand should say Yes to that question based on her writings. The character Prometheus in Anthem is learning to climb out of the communist society of his birth and youth, climb out of psychologically, not only physically by escape. He comes to recognize a natural organic unified function of his mind, desires, and action that is better to choose over the model of self and society he had been raised in. Similarly, I think on Objectivist ethics that the self-effacement that goes on in the American Amish communities should be graded as immoral, notwithstanding the voluntariness of membership (and leaving aside the immorality of fideism).
  18. Part 3 – Kant, Precritical We have seen that Locke had a keen appreciation of the profound place of construction in Euclid’s geometry. We have seen also his alignment with Aristotle in conceiving as aspects these geometric figures and relationships in the physical world; they are not only in the mind. Leibniz denied the first point. Thought alone, free of sensory perception, reveals geometry. Euclid’s constructions are not essential to justifying the truth of the geometric propositions, according to Leibniz. He opposed Aristotle and Locke on the second point as well. Nature conforms to principles of geometry because, like logical principles, those principles are necessary principles of possibility, coeternal with the author of nature. We have seen Leibniz arguing that truths of geometry could not be reached by induction from experience, for then they could not have the formality, necessity, and universality we know them to have. That charge is not fair to Locke or Aristotle. If their idea was that geometric figures and their natures are arrived at by induction from experience, it was surely by the sort of induction known as abstractive or intuitive, not by the sort one first thinks of, which is called incomplete, problematic, or ampliative. The latter sort is “a passage from the individuals to universals” (Topics 105a12) and a passage “from the known to the unknown” (Top. 156a5). The former sort is induction as “exhibiting the universal as implicit in the clearly known particular” (Posterior Analytics 71a8; Boydstun 1991, 36; further, Peikoff 1985). So we can agree with Leibniz’ point against ampliative induction to geometric figures and their natures, while leaving open the possibility of a role for abstractive induction antecedent the process of thought that is Euclid’s Elements (further, Grosholtz 2007, 53–55; Stekeler-Weithofer 1992). In Kant’s mature philosophy of transcendental idealism (which is also known as critical or formal idealism), he was rightly sensitive, like Locke, to the profound role of construction in Euclidean geometry. Contrary to Leibniz, discursive thought is not enough in the cognition that is geometry. True, as Leibniz stated, the results of geometry are not attained by (ampliative) induction. Contrary to Locke and Hume, the necessity in starting points and conclusions in Euclid is not a kind of feeling arising from a comparison of our ideas. It is not, as Locke had it, from an intuitive sense of agreement or disagreement in our ideas. According to Kant, it is from a kind of intuition, a kind of content for concepts, one that tells us some constraints of form on sensory experience, namely the constraint of spatial relations. We begin before Kant had arrived at those positions. We begin in Kant’s precritical period with his 1764 essay “Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality.” Kant observes that in geometry definitions are arrived at synthetically. One can define clearly a category of figures, and thereby they are so. This is reminiscent of Leibniz saying that lines and defined figures of geometry do not exist (outside the divine understanding) before we define and draw them. Kant sees it this way: “Whatever the concept of a cone may ordinarily signify, in mathematics the concept is the product of the arbitrary representation of a right-angled triangle which is rotated on one of its sides. In this and in all other cases the definition [Erklärung] obviously comes into being as a result of synthesis” (2:276; cf. Spinoza c. 1662, 2.27.15–25). By synthesis, Kant here means a free combination of elements into a concept. Recall the attempts of Aristotle, Locke, and Rand to account for exactitude and certainty of mathematical concepts in comparison to ordinary and philosophical concepts. Here is how Kant contrasted mathematical definitions with philosophical definitions. Mathematical definitions are synthetic; philosophical ones are analytic. Compare with Rand’s analysis of justice or reason (ITOE 51). According to Kant in this essay, had we defined a philosophic concept synthetically, “it would have been a happy coincidence indeed if the concept, thus reached synthetically, had been exactly the same as that which completely expresses the idea . . . which is given to us” (2:277). Granted some philosophers have “defined” philosophic concepts synthetically. Consider synthetic proposals concerning the concept substance. In the case of Leibniz’ concept of monad, I should take some issue with Kant. It was partly synthetic, but it was partly analytic of the concept substance, which was given ordinarily and was given to Leibniz in the philosophic formulas prior to his own. Still, Kant’s point is a very good one. Fundamentally and pervasively, philosophical concepts are analytic in the present sense. In Inquiry Kant argues against the programs of Leibniz and Wolff to rewrite Euclidean geometry using philosophical definitions of similarity in place of Euclid’s; philosophical concepts do none of the work distinctive to geometry. He argues also against their efforts to assimilate geometry into an overarching philosophical, logical formalism. Leibniz proposed that “the theory of similarities or of forms lies beyond mathematics and must be sought in metaphysics” (1679, 254–55). Geometers, in his view, could make greater use of similarity, but heretofore philosophers had not produced a definition of the concept clear, distinct, and adapted to mathematical investigation. Leibniz offered a philosophical definition of similarity and employed it to derive some of Euclid’s theorems. “We call two presented figures similar if nothing can be observed in one viewed in itself, which cannot be equally observed in the other” (1679, 255). With this definition, a related lemma, and an axiom, Leibniz composed a direct proof for Elements XII.2, which says the ratio of the areas of two circles equals the ratio of the squares of their diameters. In Euclid the result had been obtained only by reductio ad absurdum, not directly. The concept of similarity is defined in Elements, which is what one likely learned in high school: Two rectilinear figures are similar if they have corresponding angles equal between the figures and the ratio between corresponding sides of the two figures are the same for all those pairs of sides. Daniel Sutherland explains why Leibniz’ definition of similarity is what it is. Prof. Sutherland points to the tradition of Aristotle on that last point. Aristotle had defined similarity and equality in terms of quality and quantity. Those things “are like whose quality is one; those are equal whose quantity is one” (Metaph. 1021a11–12). Leibniz observed a further distinction of quantity and quality: With this understanding of quantity and quality, Leibniz’ metaphysical definition of similarity follows, and Euclid’s definition is seen to be an instance of that metaphysical definition. Contrary his hopes, however, Leibniz never delivered a presentation of Euclidean geometry with similarity ascendant over congruence nor geometry subordinate metaphysics. Early in the eighteenth century, Christian Wolff, preeminent follower of Leibniz, had written two geometry texts. One was in German, and it came to be the standard introductory student text in Germany, replacing Euclid’s Elements. It was a practical text, permitting use of scaled rulers and protractors. The second text was in Latin and was intended for a more scholarly audience. Its title is Elements of Universal Mathematics (Elementa Matheseos Universae). Wolff thought that all human knowledge would be derivable if concepts were properly defined and organized according to their logical relations. He thought the axioms of geometry could in principle be replaced with definitions and derivations from them. In Elementa Wolff dropped Euclid’s reliance on congruence. Inspired by Leibniz’ proposal to put a philosophical definition of similarity to work directly in geometry, he attempted just that, though with his own, related, but more elaborate, definition of similarity (Sutherland 2010, 163–69). I note in passing that part of one of Wolff’s two corollaries of his definition of similarity was quite close to Rand’s definition of similarity. Wolff’s second corollary includes the clause “similar things, without loss of similarity, are able to differ in quantity” (quoted in Sutherland 2010, 165). In Elementa Wolff radically rewrote Euclid’s geometry, resting the new deductions on his philosophical definition of similarity, joined with a definition of things “determined in the same manner.” The deductions of the theorems of geometry he sets forth are in fact fallacious. The results do not follow from the premises. Sutherland thinks Wolff’s lack of genuine rigor stems from Wolff’s preconceptions about the way human knowledge should be organized. We have noted Aristotle’s mistaken idea that the theorems of Euclidean geometry, with their necessity and universality, can be deduced purely by syllogistic demonstration. However, unlike Wolff, Aristotle did not suffer the further misconception that mathematics and demonstrative science more generally requires no axioms among its starting points, only definitions. We have seen that in Kant’s “Inquiry” philosophy’s distinctive method and task is to render complete and determinate certain concepts that are given in a confused manner in general usage. This task, which is called analysis, is in contrast to that of mathematics: “combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establish what can be inferred from them” (2.278). Wolff’s attempt to bring the concept similarity in an analytic version into geometry and make it a base of geometry was a big mistake. Euclid’s conception of similarity is sufficient for geometry, and similarity under more general philosophic definitions fills no gap and does no real work in geometry. As a matter of fact, not trading in concepts as analytic is why geometry is not in “the same wretched discord as philosophy itself” (2: 277; further, Sutherland 2010, 177–88). Kant observes, furthermore, in geometry: Proposition 35 of Book III would be an example.* This would seem to be an act of abstractive induction, though not for analytic concepts. I say it is abstractive induction for synthetic concepts and their interrelations. Bear in mind that abstractive induction is also called intuitive induction. Kant begins to correct the error of Wolff, Leibniz, and Aristotle, who had not recognized that proofs in Euclid have additional legitimate resources beyond deductions with words marking general concepts having not concrete, perceptual, combinatorial indication of the interrelations of these concepts (2:278–79). Kant at this stage, like Locke before him, has not smoothly embedded his theory of mathematical cognition into a general theory of cognition. Like Locke, his account of cognition peculiar to proofs of Euclid does little more than say we have such intellectual capabilities. Philosophers before Kant, as we have seen, had things to say about the greater exactitude and certainty of mathematical knowledge over philosophical knowledge, such as knowledge of the category of substance. Rand and Gotthelf touched on the issue too. Kant joins that chorus, mentioning factors along the lines of their factors, in addition to his original notice that the mathematical concepts are synthetic, not analytic, and mathematical proof includes handling mathematical universal concepts in concreto. The basic concepts and starting propositions of geometry are few in comparison to such concepts and propositions in philosophy (2:279–82). The object of mathematics is magnitude, which is easy in comparison to the object of philosophy. The latter object is difficult and involved. Nevertheless, in the area of philosophy that is metaphysics, as much certainty is possible as in geometry. The object of mathematics is magnitude, the object of metaphysics is encountered in a thicket of various, numerous qualities. Grasp of the object is in principle attainable in either case. In all disciplines, the formal elements in judgments rely on the indubitable “laws of agreement and contradiction” (2:296). In metaphysics, as in mathematics, there are material concepts and principles that are indemonstrable and foundational. The number of these is greater in metaphysics than in mathematics. Metaphysics is more difficult than Euclidean geometry, though not in principle less secure in its truths. The grounds of metaphysical truths are objective. They are not subjective criteria of conceivability or feeling of certainty (2:294; also 285–86). (To be continued.) References Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton. Boydstun, S. 1991. Induction on Identity. Objectivity 1(2):33–46. Grosholtz, E. R. 2007. Representation and Productive Ambiguity in Mathematics and Science. Oxford. Jetton, M. 1991. Philosophy of Mathematics. Objectivity 1(2):1–32. Kant, I. 1764. Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality. In Theoretical Philosophy 1755–1770. D. Walford, translator. Cambridge. Leibniz, G. W. 1679. On Analysis Situs. In Loemker 1969. Loemker, L. E. 1969 [1954]. Gottfried Wilhelm Leibniz – Philosophical Papers and Letters. 2nd ed. Kluwer. Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian. Spinoza, B. c. 1662. Emendation of the Intellect. In The Collected Works of Spinoza. E. Curley, translator. Princeton. Stekeler-Weithofer, P. 1992. On the Concept of Proof in Elementary Geometry. In Proof and Knowledge in Mathematics. M. Detlefsen, editor. Routledge. Sutherland, D. 2010. Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early Kant. In Discourse on a New Method. M. Domski and M. Dickson, editors. Open Court.
  19. Part 2 – Locke and Leibniz John Locke thought that extension, the terminations of it, and figure are primary qualities of nature and are among our perfectly simple ideas. They are things “really in the world as they are, whether there were any sensible being to perceive them or no” (EU 2.31.2). Though they exist in nature, Locke thought of the exact figures of geometry as not simple primary qualities of nature. They are not objects simply given objectively to the mind. They are our voluntary assemblies, “without reference to any real archetypes, or standing patterns existing anywhere” (EU 2.31.3). Such assemblies, Locke calls ideas of modes. A mode is a complex not self-subsistent, but depending on self-subsistent things, which is to say, depending on substances. A triangle is a complex, as is any mode, but among modes, it is relatively simple. And it is an idea clear, distinct, and certain (EU 2.12.4, 2.31.3, 3.3.18, 3.9.19, 4.4.6, 4.7.9). Locke calls an idea adequate if it does not lack anything. It is complete, perfect. The figures of geometry are adequate ideas. It is never a triangle per se that is in nature. “The general idea of a triangle . . . must be neither oblique nor rectangle [right-angled], neither equilateral, equicrural [isosceles], nor scalon; but all and none of these at once” (EU 4.7.9). The general idea of triangle is a tool in communication and in the enlargement of knowledge, according to Locke. There are some strains of Aristotle in that. Locke, however, does not base certainty ultimately on certainty of the existence of the world and what it contains, as would Aristotle or Rand. He finds the ultimate ground of certainty in some of the agreements of our ideas with each other. “A man cannot conceive himself capable of a greater certainty than to know that any idea in his mind is such as he perceives it to be; and that two ideas, wherein he perceives a difference, are different and not precisely the same” (EU 4.2.1; see also 3.8.1; 4.7.4, 19). Such immediate knowledge is called intuitive by Locke. Is the sum of the angles of a triangle the same or variable from one triangle to another? Seeing the sameness of that sum and the sameness of that sum to the angle of a half-circle for all triangles in the Euclidean plane is not immediately evident. It is not a single perceptive act of intuition, rather it requires demonstration (EU 4.2.2). Each step of a demonstration, in Locke’s view, requires an intuitive knowing (EU 4.2.6). The mind can perceive immediately the agreement or disagreement of each step in the demonstration (EU 4.2.5). Locke thought the reason mathematics has demonstrative certainty is that in mathematics the mind can perceive the immediate agreement and difference between its ideas, which are the ideas of “extension, figure, number, and their modes”(EU 4.2.9). Locke goes on to remark that simple ideas other than those given for mathematics—say the non-mathematical simple ideas of color or brightness—cannot be compared in quantity, but only in degree, which is to say not so thoroughly as with mathematical simple ideas (EU 4.2.11–13). Locke is using the concept quantity to indicate what we would call today a magnitude affording ratio scaling (which is more narrow than what Rand, Gotthelf, or I take for the class determinate magnitude, or quantity.) Locke does allow that non-mathematical ideas such as color or brightness could be entered into effective systematic reasoning, though those ideas are precise only to the level of the least differences we can perceive, unlike ideas of line or figure in geometry. There are degrees of difference in our intuitive perceptions of a simple secondary quality, such as brightness, and such degrees of difference suffice to found inferential knowledge beyond subjects such as geometry or mechanics. Where the difference in discerned difference in a secondary quality “is so great as to produce in the mind clearly distinct ideas, whose differences can be perfectly retained, there these ideas or colors, as we see in different kinds, as blue and red, are as capable of demonstration as ideas of number and extension” (EU 4.2.13). Where Locke has written “as capable of demonstration,” I think he means “as capable of use in demonstration.” Locke is sensitive to the work of constructing figures and auxiliary lines in Euclid’s proofs. That much is good. Our way of learning geometric possibilities for geometries possibly physical (definitely physical in Euclid) is not our everyday way or scientific way of learning non-geometric physical possibilities. That much of Locke’s account is also right. However, in opposition to Locke’s account, it should be stressed that the difference between geometric and non-geometric objects is not that the former are quantitative, whereas the latter are not. Coolness to the touch is registration of a rate of heat flow, and that is a quantitative object, in Locke’s sense, and a non-geometric object of knowledge. We do not proceed in thermodynamics as we proceed in geometry, notwithstanding the circumstance that for both the objects are quantitative, in the elementary sense of Locke and his era. That would be acceded by Locke, but I add that it is not getting anywhere to say the precision of geometry is on account of its objects being amenable to quantification. Rather, precise determinateness is one of the requirements of the quantifiable. More deeply, his theory is defective in basing the certain truth of Euclid’s geometry, derived in part from constructions, ultimately on certainty in the agreement between our ideas rather than on certainty of some possibilities for acts in the world along with possibilities of the world affording those acts. Locke’s attempt at accounting for the extent to which demonstrative certainty has been attained in mathematics has the merit of not foreclosing an essential justificative role of the lettered diagram that is appealed to in most of Euclid’s demonstrations. It allows, at least implicitly, for the possibility that we think with the lettered diagram—though the thought is of the exact form there within the lettered diagram—and that this kind of thinking forms part of the justification of the certain truths of Euclid’s geometry. Gottfried Wilhelm Leibniz responded to Locke’s Essay concerning Human Understanding (EU – 1690) with New Essays on Human Understanding (NEU – 1704). Leibniz opposes much of the Lockean view of what is going on in Euclidean demonstration. Admittedly, the Euclidean diagrams are “helping judgment to gain demonstrative knowledge” (NEU 385; further, 352–53). Sensory experience can aid in the geometric proof, but only as a crutch for our progression of thought. It is not essential to thought and is not part of the justification for the geometric truth. Sensory experience is confused perception, not distinct perception, and it is not essential to thought. Distinct ideas may accompany sensory ideas, but it is only the former that serve for demonstrations (NEU 137, 487). Leibniz thinks the ideas of extension and figure come from “the common sense, that is, from the mind itself; for they are ideas of the pure understanding (though ones which relate to the external world and which the senses make us perceive), and so they admit of definitions and of demonstrations” (NEU 128; also 1682–84, 286). Ideas are not images, and imagination is not thought (NEU 261–62; c. 1691). “One can have the angles of a triangle in one’s imagination without thereby having clear ideas of them. Imagination cannot provide us with an image common to acute-angled and obtuse-angled triangles, yet the idea of triangle is common to them” (NEU 375; further, 451–53 and De Risi 2007, 35–39). Concerning necessary, geometric truths: In geometry, as in syllogistic inference, thought is apart from sense (NEU 370–72). Geometry’s elementary ideas are innately, implicitly within us, and we expose them and their relations in the discipline (NEU 50, 77, 392). “Neither a circle, nor an ellipse, nor any other line we can define exists except in the intellect, nor do lines exist before they are drawn, nor parts before they are separated off” (1689, 34). Nevertheless, “number and line are not chimerical things . . . for they are relations that contain eternal truths, by which the phenomena of nature are ruled” (1695, 146–47; further, Garber 2009, 158–62). “As for the proposition The square is not a circle, . . . in thinking it one applies the principle of contradiction to materials which the understanding itself provides” (NEU 83). Yes and no. That building blocks stack and balls roll is learned by the infant prior to language. Incompatible shapes are available to see and handle. Leibniz will allow that. He allows also that a child having language can know what is a square and its diagonal without yet knowing a square’s diagonal is incommensurable with its side (NEU 102). To grasp the perfectly exact figures and relations that enter geometry—such as the incommensurability of a square’s diagonal with its side or the equality of the sum of angles in a triangle to two right angles—requires a high level of conceptual understanding. Leibniz errs, however, in thinking perfectly exact figures and relations are only from abstract thought. They are partly taken from the world, they may obtain perfectly in physical space, and without mind. Unlike Locke and like the other rationalists, Leibniz takes Euclid’s figures and their specific natures to be objective givens in the mind (1675, 1–2). In the view of Leibniz, extension and its geometric modes, though they are related to physical objects, come forth in the mind and are therefore definite objects and relations suited for entering demonstrations. No material objects have shapes so precise and determinate as geometric objects (1687, 86–87; 1689, 34; 1704 or 1705, 183). The boundaries of figures we draw on paper in a geometric proof are not exactly the boundaries and figure in mind for the proof (further, NEU 360; Norman 2006, Ch. 6; Azzouni 2004*]), and while the former, as with all matter, are compositions, the latter are not (1695, 146–47). Leibniz observes that even where we have distinct physical knowledge, as in a definition of gold, we yet have knowledge incomplete, for we know not much (in Leibniz’ day) about the processes yielding the traits in the definition of gold. Not knowing much, that is, in comparison to knowledge in geometry, wherein ideas are so distinct that all their components are distinct (NEU 266–67, 308–9, 346–48). Such completeness is perfect knowledge, which Locke and many others called adequate knowledge. In geometry “we can prove that closed plane sections of cones and cylinders are the same, namely ellipses; and we cannot help knowing this if we give our minds to it, because our notions pertaining to it are perfect ones” (NEU 267). The inner natures of geometrical figures can be reached by the human mind; not nearly so swiftly might the inner natures of “the incomparably more composite species in corporeal nature” be reached (NEU 348). Leibniz thought possible and hoped for an algebraic rendering of geometry. He thought possible and hoped both could be brought under an art of formal deduction, one subsuming syllogistic logic (NEU 478–79; 1666-67; 1678; 1679; c. 1691; c. 1692; De Risi 2007, 40–41, 63–80, 85–89, 95–98, 569–75; Capozzi and Roneaglia 2009; Sutherland 2010, 155–63). Some of his attempts to achieve this program proposed a philosophical definition of similarity, from which he hoped both similarity in Euclidean geometry and similarity in its non-mathematical occasions might logically stem. Rand, on the other hand, proposed a mathematical, mensural account of similarity in general, one applicable to similarity in geometry and, she hoped, to similarity everywhere else (ITOE 13–14; ITOE Appendix 139–40; cf. Heath 1956, 132–33). (To be continued.) References Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton. Ariew, R., and D. Garber, translators, 1989. G. W. Leibniz – Philosophical Essays. Hackett. Belot, G. 2011. Geometric Possibility. Oxford. Capozzi, M., and G. Roncaglia 2009. Logic and Philosophy of Logic from Humanism to Kant. In The Development of Modern Logic. L. Haaparanta, editor. Oxford. De Risi, V. 2007. Geometry and Monadology – Leibniz’s Analysis Situs and Philosophy of Space. Birkhäuser. Garber, D. 2009. Leibniz: Body, Substance, Monad. Oxford. Euclid c. 300 B.C. The Thirteen Books of The Elements. T. L. Heath, translator. 2nd ed. 1956 [1908, 1925]. Dover. Leibniz, G. W. 1666. Dissertation on the Art of Combinations. In Loemker 1969 (L). ——. Letter to Foucher. In Ariew and Garber 1989 (AG). ——. 1678. Letter to Tschirnhaus. (L) ——. 1679. Studies in a Geometry of Situation with a Letter to Huygens. (L) ——. 1682–84. On the Elements of Natural Science. (L) ——. 1687. Letter to Arnauld, 30 April. (AG) ——. 1689. Primary Truths. (AG) ——. c. 1691. Ars Representatia. V. De Risi, translator. Leibniz Review 15:134–39. ——. c. 1692. Uniformis Locus. V. De Risi, translator. Leibniz Review 15:140–51. ——. 1695. Note on Foucher’s Objection. (AG) ——. 1702. Reply to Bayle. (L) ——. 1704. New Essays on Human Understanding. P. Remnant and J. Bennett, translators. Cambridge. ——. 1704 or 1705. Letter to de Volder. (AG) ——. 1714. Principles of Nature and Grace, Based on Reason. (AG) Locke, J. 1690. Essay Concerning Human Understanding. 1959. Dover. Loemker, L. E. 1969 [1954]. Gottfried Wilhelm Leibniz – Philosophical Papers and Letters. 2nd ed. Kluwer. Norman, J. 2006. After Euclid – Visual Reasoning & the Epistemology of Diagrams. CSLI. Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian. Sutherland, D. 2010. Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early Kant. In Discourse on a New Method. M. Domski and M. Dickson, editors. Open Court.
  20. I wrote this series a few years ago. This philosophical look covers Aristotle, Locke, Leibniz, Wolff, and precritical Kant. I hope to continue it in the next few years, bringing it up to present-day mathematics. This series is a strand to be used in pulling together the 3-ply cord of mathematics, logic, and metaphysics, as well as in formulating a concept of objective analyticity. It will figure into completion of my thread Peikoff’s Dissertation as well as into completion of my book. ~~~~~~~~~~~~~~~~ Truth of Geometry – Necessity in Geometry Part 1 – Aristotle “Without an image thinking is impossible. For there is in such activity an affection identical with one in geometrical demonstrations. For in the latter case, though we do not make any use of the fact that the quantity in the triangle is determinate, we nevertheless draw it determinate in quantity” (On Memory 450a1–4). According to Aristotle, the subject matter of geometry consists of things contained in or bounding natural bodies, things such as surfaces, volumes, lines, and points. A light-ray or a line scored in a stone are studied by the geometer only as lines. It is likewise for surfaces, volumes, and points. In thought the geometer separates these things from bodies. She does not treat them as the limits of a natural body, and no falsity results from this. Neither does she consider these attributes as the attributes of bodies. “Geometry investigates natural lines but not qua natural” (Physics 194a10). Geometric objects are not prior to sensibles in being, though sensible bodies presuppose geometric objects. In claiming geometric objects are in or bound bodies and that they can be separated in thought from bodies, Aristotle is not saying they exist in bodies or apart from bodies in the way we say body parts exist (Metaphysics 1077a15–b17). “If we suppose things separated from their attributes and make any inquiry concerning them as such, we shall not for this reason be in error, any more than when one draws a line on the ground and calls it a foot long when it is not; for the error is not included in the propositions” (Metaph. 1078a16–21). Suppose we are trying to see if some furniture at the store can be fitted into a nice arrangement for your living room. We take some graph paper and say “let the squares this size be each a square foot.” We draw the outline of your living room floor to scale on the paper. To the same scale, we cut another sheet of graph paper into the horizontal cross-sectional areas of the furniture pieces. We then test their arrangement on the paper with the floor outline, knowing our results apply to the room, even though what we took as a square foot on paper was not actually a square foot. The truths of geometry are of the real much as Aristotle’s foot-representing line is of a line one foot long. In Euclid’s geometry, we reason about perfectly exact planes, points, lines, and figures. As the objects of Euclidean geometry, planes are perfectly flat, straight lines are perfectly straight, circles are perfect circles, and so forth. We draw on paper icons of those perfect elements. The icons can deviate somewhat from the perfect items we conceive and about which we reason using those icons. This does not mean that the perfect objects of geometry, thence their relations, are not instantiated in the material world. In Aristotle’s view, they are (Lear 1988, 240–43; Lennox 1986, 33–38). Descartes and Newton concurred with Aristotle in that view. That something can only be accessed by abstraction does not entail that the something does not exist concretely, beyond thought. It is no great mystery in Aristotle’s view that geometry can supply the reasons the rainbow has some of the characteristics it has (Meteorology 271b26–29, 375b17–76b21, 376b28–77a11; R; Lennox 1986, 44–49). In the diagrams and reasoning of Greek geometry, unlike the thinking with the furniture floor-plan, we are not making an (indirect) empirical test. The natures of geometric objects as geometric are uncovered by proofs from assumed starting propositions, including definitions, and from permitted elementary acts for constructing diagrams. It is of the nature of a triangle in Euclidean geometry that its interior angles sum to two right angles. One does not need to accept that on authority; one can follow Euclid’s proof and thereby understand that this proposition on the nature of the triangle is true and why (Elements I.32; 34–35)*. Assuming only what a triangle most simply is, one shows that it necessarily has angles summing to two right angles (Phys. 200a16–17). Euclid lived after Aristotle, but geometric proof by the time of Aristotle was as we have it in Euclid (Netz 1999, 275). Aristotle discovered logic, in particular the theory of the syllogism. Geometers were proving geometric propositions before then. Aristotle thought the kind of syllogisms he counted as demonstrations, or proofs, were what geometric proofs came down to. That was incorrect (Prior Analytics 40b18–41a20, 48a29–39; see Lear 1980, 10–14, 39, 48, 51–53, 65; also Friedman 1992, 57–66; 2000, 187, 202). But geometric proofs are like his syllogistic demonstrations in carrying necessary truth of starting points by strict rules of development into necessarily true conclusions explained by the premises. The geometers and Aristotle have also been in accord in thinking that a geometric proof is not only necessary for establishing that the sum of interior angles of a triangle is two right angles (180°), but that the proof is sufficient to establish that truth. The construction of lines and figures is usually part—an essential justificatory part—of a proof in Euclid and in later Greek geometers, such as Apollonius and Archimedes (Norman 2006, 20, 79–86; Netz 1999, 26–43, 187–88, 264–65). In the technology of geometric diagrams, there are two elements: (i) straight-edge and variable compass, both without scale marks, and (ii) letters naming points of line intersection in the diagram. David Hilbert famously recast Euclidean geometry into a logical order that took propositions implicit in Euclid’s diagrams and explicitly stated them. In calling Hilbert’s topic geometry in this accomplishment Euclidean geometry, I mean for example that Hilbert was in this work doing plane geometry on the same plane that Euclid was studying, as distinct from, say, the hyperbolic plane of another geometry that had been developed in the nineteenth century. Hilbert’s axiomatization of Euclidean geometry fortifies the truth of Euclidean geometry beyond Euclid, that is, the truth of that geometry concerning its objects, which are objects of our understanding and which have instances in the world (see also Norman 2006, 19, 73–86; further, Mumma 2011*) I should mention that the non-Euclidean geometry that is hyperbolic geometry has instances in the world, such as the plane geometry of a half-sphere, which is of course a surface and geometry embedded in Euclidean space. An important potential instance of hyperbolic geometry is the large-scale structure of four-dimensional spacetime, in which local three-dimensional space is Euclidean; whether this potential is actual will be decided by physics, which is to say by empirical test (Martin 1975, chap. 26; Friedman 2000, 206–8; Bolte and Steiner 2012). Aristotle had it that “the objects of mathematics exist, and with the character ascribed to them by mathematicians” (Metaph. 1077b33). Instances of geometry are in the sensible, but in geometry one does not treat its occurrence in the sensible as sensible, and geometry does not on account of its sensible occasion become a science of the sensible (1078a2–4). “There are attributes of things which belong to things merely as lengths or as planes” (1078a7–8). Furthermore, “in proportion as we are dealing with things which are prior in formula [even though not prior in being] and simpler, our knowledge will have more accuracy, i.e. simplicity” (1078a9–10). Leonard Peikoff took the proviso “in the present context of knowledge” as not applicable to truth of mathematical axioms, because of the very delimited subject matter in mathematics (cf. Peikoff [Prof. E] in ITOE Appendix 203). The following oral exchange took place between Allan Gotthelf and Ayn Rand in her epistemology seminar (c. 1970). Gotthelf and Rand are here in accord with Aristotle’s thought that our knowledge in geometry has such great accuracy due to its simplicity and cognitive self-sufficiency when we have abstracted its objects from their embodiments. In the next two installments, I shall consider what Locke and Leibniz, then Kant say about sources of the exactitude of geometric truth, as well as its certainty, inferential validity, generality, and immutability. Here I want to pause over what Gotthelf and Rand remarked informally concerning immutability of mathematical concepts. Where Rand answered “That’s right” to Gotthelf, I answer “That’s roughly right.” The basic elements for arithmetic and for geometry are set in the beginning of those disciplines and are occasioned every day all around us. New observations sometimes stimulate introduction of new or revised concepts in mathematics. But for the most part, changes in mathematical concepts come by way of creative resolutions of tensions within mathematics itself. (We should notice too that what were the elementary concepts in Euclidean geometry as Euclid wrote it are not the complete set of elements we identify for that geometry today; when doing it rigorously, at an advanced level, we now know there are further, less obvious elements on which that geometry is logically based.) Concepts can rationally change in the referents they subsume when it becomes evident that the concept should be (i) narrowed, recognizing that not all referents previously subsumed are of the same type at the level of the concept in the (possibly shifting) conceptual hierarchy or (ii) broadened, recognizing that referents previously known but excluded or referents previously not conceived should be included under the concept. Furthermore, in the development of science and mathematics, new concepts are introduced. Sometimes that is because although an old concept was picking out a kind, it was conceived within a theory later seen to be false. An example from chemistry would be replacement of the concept dephlogisticated air with the concept oxygen. An example from mathematics would be replacement of the concept number-whose-square-is-minus-one, where the concept of number really allowed no such number, with the concept imaginary number, where the concept of number had been broadened such that complex numbers were included (Kitcher 1984, 175–77). A new mathematical concept may be introduced as a distinction of subspecies under a current concept. An example would be introduction of the concept uniformly continuous function under the concept function. An example of broadening a mathematical concept would be the history of the concept function (Kline 1972, 338–40, 403–6, 505–7, 677–78). An example of narrowing would be the history of the concept integrability (Kline 1972, 959–61). In sum mathematical concepts and definitions do change as the discipline advances. What concepts are most basic in an area of mathematics can also change, though we are able to locate the old basics in the new framework. Euclid’s geometry is in the class we today call synthetic geometry. If one took geometry in high school, it was probably Euclid’s geometry, and one knows some synthetic geometry. It is geometry as synthetic relations that can be geometry as concrete relations in the world independently of mind. When we see the word geometry without qualification, it is fairly safe to suppose the reference is to synthetic geometry, rather than analytic geometry. When we see accounts of how geometry is rooted in sensory experience, it is the empirical origin of some of the concepts in synthetic geometry that is being proposed. Analytic geometry uses algebraic methods and equations to study geometric problems (Boyer 1956; Kline 1972, 302–24, 544–66; Netz 2004). Experiential origins of concepts in the arithmetic, the algebraic, and the mathematically analytic are issues for the epistemologist, but they are distinct from the issue of experiential origins of concepts in (synthetic) geometry. With the discovery of non-Euclidean geometries elliptical and hyperbolic, which like Euclidean geometry are geometries with planes and spaces of constant curvature (Euclidean has constant curvature of zero), we can awaken to possible empirical sources in some of the concepts underlying all three of these synthetic geometries and thereby wake to possible empirical sources for Euclidean geometrical concepts additional to volume, surface, line, and point. Helmholtz proclaimed this new, specific, and very plausible possibility (Friedman 2000, 200–202; DiSalle 2006). There can be unidentified empirical sources for unidentified elements implicit in our mathematical thought. Physical concepts are much more than their mathematical characters. Establishing new truth in physical science requires observation and experimental tests. In the history of mathematics, there have been some episodes in which finding a physical exemplification of a mathematical innovation has drawn the mathematics community into taking the innovation more seriously. But physical exemplification is unnecessary for, and empirical testing is irrelevant to establishing new mathematical truth. The deliberate simplicity and delimitation of mathematical concepts, which Aristotle noted, are surely some part of the story of why that is so. (To be continued.) References Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton. Bolte, J., and F. Steiner, editors, 2012. Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology. Cambridge. Boyer, C. B. 1956. History of Analytic Geometry. 2004. Dover. DiSalle, R. 2006. Kant, Helmholtz, and the Meaning of Empiricism. In The Kantian Legacy in Nineteenth Century Science. MIT. Euclid c. 300 B.C. The Thirteen Books of The Elements. T. L. Heath, translator. 2nd ed. 1956 [1908, 1925]. Dover. Friedman, M. 1992. Kant and the Exact Sciences. Harvard. ——. 2000. Geometry, Construction, and Intuition in Kant and His Successors. In Between Logic and Intuition. G. Sher and R. Tieszen, editors. Cambridge. Kitcher, P. 1984. The Nature of Mathematical Knowledge. Oxford. Kline, M. 1972. Mathematical Thought – From Ancient to Modern Times. Oxford. Lear, J. 1980. Aristotle and Logical Theory. Cambridge. ——. 1988. Aristotle and the Desire to Understand. Cambridge. Lennox, J. G. 1986. Aristotle, Galileo, and “Mixed Sciences.” In Reinterpreting Galileo. W. A. Wallace, editor. Catholic University of America. Martin, G. E. 1975. The Foundations of Geometry and the Non-Euclidean Plane. Springer. Mumma, J. 2011. The Role of Geometric Content in Euclid’s Diagrammatic Reasoning. Les Ètudes Philosophiques 97:243–58. Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. Cambridge. ——. 2004. The Transformation of Mathematics in the Early Mediterranean World – From Problems to Equations. Cambridge. Norman, J. 2006. After Euclid – Visual Reasoning & the Epistemology of Diagrams. CSLI. Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian.
  21. Clive, that quote is apparently from Rand in some personal notes or correspondence, as indicated on page 43 of the book linked here. The next sentence in that book on Rand is surely false if one confines Christianity to biblical text alone: the Bible, including the New Testament, does not teach putting others above self, only to love them as one loves oneself. The doctrine of the moral virtue of regularly sacrificing oneself for the benefit of certain others is apparently a pretty modern stance for Christianity. Sacrifice of self in conforming to the rules of God seems the more constant doctrine down from ancient Christianity. Those libertarians are right, again concerning the Christianity squarely contained in the New Testament. It is a fact that some Christians have used force aiming to do good from ancient times to the present, but that is really not supportable by the New Testament. It says pay your taxes and mentions that God puts the authorities in place for, of course, good purpose; but it does not say to get involved in the state, indeed it talks as if the state is something of a outsider as far as the righteous community is concerned. If you ever desire to have a copy of the New Testament on hand for reference, I recommend the translation of J. B. Phillips. It's a breeze to read and get meaning of statements unobscured by the English of the King James era.
  22. Interesting, on Top Ten. I incline to favor Harris, Klobachar, and Rice among these. Warren is smart and speaks to substance, but I disagree with that substance mostly. / In the 2008 primary, my first choice was Biden because he was the most fiscally conservative. But he got nowhere in the primary. Just hope he doesn't croak before the 2020 general election is held. / Mr. Trump has proven to be pretty much a Democrat on fiscal matters, just putting balancing the budget off into some vague distant future somewhere beyond his term(s). In April of 2017, with both chambers Republican, he should have not signed the budget that Congress sent to his desk, rather, he should have sent it back to Ryan saying cut it all across the board, keeping all proportions the same, cutting so far as to match expected revenue. That would have been true significance in a good way. Pragmatists are myopic when it comes to what is actually practical. The first Trump administration has reminded me of what happened with so many Republican governors after the 2010 gains of that office: few actually got anywhere on fiscal matters; what they got much more easily was culture-wars wins of this or that. What Mr. Trump got of enduring significance during this first term was appointment of anti-Roe Supreme Court Justices. The rest was circus (such as trying to figure out how to move his hands with "evangelicals" in the office), continue Obama in foreign affairs while shouting NEW, and having yet another conventional-wisdom, greater-than-ever stimulus package designed by his supposedly free-market Treasury man (reminds me of Snow's about-face under G W Bush).
  23. Atlas Shrugged was published on 10 October 1957. A brief interview with Rand by Lewis Nichols was published in the New York Times three days later. On the writing of Atlas Shrugged she remarked: “‘It goes back a long way. I was disappointed in the reaction to The Fountainhead. A good many of the reviewers missed the point. A friend called me to sympathize, and said I should write a non-fiction book about the idea back of The Fountainhead. ‘While I was talking, I thought, “I simply don’t want to do this. What if I went on strike?” My husband [Frank O’Connor] and I talked about that all night, and the idea was born then. ‘. . . From the first night idea of the thinking people being on strike, it was natural to move to the mind on strike. With this as a theme, I decided to touch on industry, and to use a railroad as the connecting link for the story. . . . ‘In front of the desk I had a plain railroad map of the country, and marked in the Taggart lines on that. There also was a furnace’s foreman’s manual, which I studied for steel making, and I had one very pleasant ride in the engine cab of a train.’ ‘. . . The greatest guarantee of a better world is a rational morality . . . the collectivist cause is really dead. The capitalist case never has been clearly presented. . . . The doctrine of Original Sin is a monstrous absurdity, a contradiction in terms. Morals start only when there is a choice. . . . ‘The fault of the American system goes back to the Constitution. It is so vague on general welfare that the looters get in.’” “Looking into the future a bit, into the new world beyond page 1168 {the last page}, Miss Rand would see the Taggart lines being rebuilt, first between New York and Philadelphia, then, in ten years, across the continent. . . . “And Miss Rand herself? She will be sitting still for a long time, now, resting and playing records. Not her invention, the Halley’s Fifth Concerto, which runs like the Third Man Theme through Atlas Shrugged, but Rachmaninoff.”
  24. 14 March 2020 PNC Ground Shifts to the Side of the Subject – Kant IV-b Bernard Bolzano’s masterwork Theory of Science issued in 1837. In this work, we find him objecting to Kant’s definition of logic as conveyed by Jäsche: “The science of the necessary laws of understanding and reason in general, or of the mere form of thinking, is logic.” Bolzano named seven writers of logic texts since Kant who had followed Kant in that definition of logic as a science. Taking thinking in its usual wide sense, Bolzano objected to that definition. It is not a plausible characterization of logic to say logic is merely the law-governed use of reason and understanding. One could then be regarded as engaging logic when thinking fallaciously or with an aim to evasion or when thinking in a whimsical entertainment. Quite better, in Bolzano’s assessment, were several writers who had required that such laws be restricted to those serving the chosen purpose of recognizing truth (the purpose of identification, in Rand’s vocabulary and explication) to warrant the title logic. We have seen Kant’s view that “general logic . . . abstracts from all content of cognition, i.e., from all reference of cognition to its object” (KrV A55 B79). Bolzano’s own conception of logic was as the set of rules for dividing truths into perspicacious domains of specific sciences, for their cultivation, and for their perspicacious presentation. Logic so understood is itself a science as well. We use some of these rules before turning to isolate what they are. “Once these rules are known, every science, including the theory of science itself can be further elaborated and presented in writing. This amounts to no more than arranging certain known truths in an order and connection that they themselves prescribe” (§2). Bolzano thought the ultimate goal of logic the discovery of truth (§7). In that ambition, one is reminded of the expansion of logic envisioned in Francis Bacon’s New Organon (although Bolzano, contra Bacon, did not regard Aristotle’s syllogistic as useless). It seemed to Bolzano that “one of Kant’s literary sins was that he attempted to deprive us of a wholesome faith in the perfectibility of logic through an assertion very welcome to human indolence, namely, that logic is a science which has been complete and closed since the time of Aristotle” (§9). Bolzano 1837 looked forward to future developments of logic that would be a boon to all the sciences. It turned out that, after Bolzano, there were advances in deductive logic. However, these did nothing to advance or clarify knowledge in empirical science. They did illuminate mathematics and its connections to logic, and they illuminated and extended the Aristotelian (and Stoic) logic of old. Bolzano criticized the Germans such as Kant, Jakob (1791), Hoffbauer (1794), and Maimon (1794) for their slippage from the topic-neutrality of a syllogism form presented as “all A are B, all B are C, therefore all A are C” to taking the objects A, B, and C for indeterminate as to all their characteristics. That is, they erred in taking A, B, and C as empty of absolutely all content. “If we think of an object as altogether indeterminate, then we cannot claim anything about it” (§7). The signs A, B, and C need the determination that B can be rightly predicated of A, and so forth. Logical form in Bolzano’s view was not fundamentally about thoughts, but relations of truths. Bolzano correctly objected as well to contemporaries trying to make logic into some kind of empirical science of the mind. Today when we say science, we usually mean empirical science. The term science had been used more broadly until recent times, such that science encompassed also the organized disciplines of mathematics, ethics, and logic, especially when considered in their allegedly pure, necessary, nonempirical, and applications-suspended mode. I said in “Kant IV-a” that Kant proposed to get out of a bind—the bind of holding logical rules to be absolutely necessary rules of human mind given a priori by human mind, yet rules capable of being transgressed in operations of human mind—by a distinction of pure general logic from its application, a distinction between pure and applied logic, and by a dissection of the latter in terms of posited cognitive powers. I want to press on the soundness of Kant’s distinction of pure and applied logic (and whether problems for Kant in this area also bear against Hanna). I also want to press on Kant’s conception that necessity in empirical science is a function of application of mathematics and of basic (Kantian) metaphysics in the empirical science. Aristotle had noted the import of necessity by import of geometry into his account of the gross form of the rainbow.* Although, that sort of geometry application was a tidbit compared to the use of geometry by Descartes in theory of the rainbow, let alone the use of geometry by Newton in remaking the world. It was amid these modern roles for geometry that Kant did his thinking, of course. Kant knew of Aristotle’s general doctrines on science. And via Leibniz, Kant was still hankering after them and to some extent resisting the scheme for making science brought on by Newton. Aristotle had appealed to a mental faculty in describing how a logical principle, specifically PNC, is ascertained. That, as we have seen, was what in our time has been known as a power of intuitive induction or abstractive induction. That posit of faculty was quite opaque, and its (fallible) attachment to formal character in the world by the human mind’s proposed assimilation of said form—mind itself becoming external formalities—and Aristotle’s form-matter aspect of metaphysics were pretty roundly judged false in the modern era of philosophy (outside the preservation of Scholasticism by Catholic scholars). I’ll close this installment by getting before us Kant’s basic treatment of the pure/applied distinction in logic. “A logic that is general but also pure deals with nothing but a priori principles. Such a logic is a canon of understanding and of reason, but only as regards what is formal in our use of then—i.e., we disregard what the content may be (whether is is empirical or transcendental). A general logic is called applied, on the other hand, if it is concerned with the rules of the understanding as used under the subjective empirical conditions taught us by psychology. Hence such a logic empirical principles, although it is general insofar as it deals with our use of the understanding without distinguishing the understanding’s objects. . . . In general logic, therefore, the part that is to constitute the pure doctrine of reason must be separated entirely from the part that is to constitute applied (though still general) logic. Only the first of these parts is, properly speaking, a science . . . . In such pure general logic, therefore, the logicians must always have in mind two rules: As general logic, it abstracts from all content of the cognition of understanding and from the difference among the objects of that cognition, and deals with nothing but the mere form of thought. As pure logic, it has no empirical principles. Hence it does not (as people have sometimes come to be persuaded) take anything from psychology; and therefore psychology has no influence whatever on the canon of the understanding. Pure general logic is demonstrated doctrine, and everything in it must be certain completely a priori. “What I call applied logic is a presentation of the understanding and of the rules governing its necessary use in concreto, viz., its use under the contingent conditions attaching to the subject {the mind–SB}, conditions that can impede or promote this use and that are, one and all, given only empirically. . . . Pure general logic relates to applied general logic as pure morality relates to the doctrine proper of virtue. Pure morality contains merely the moral laws of a free will as such; the doctrine of virtue examines these laws as impeded by the feelings, inclinations, and passions to which human beings are more or less subject. The doctrine of virtue can never serve as true and demonstrated science; for, just like applied logic, it requires empirical and psychological principles.” (KrV A53–55 B77–79 – Werner Pluhar translation). To be continued. (This post and all those preceding it in this thread were restored from my word-processing file for the thread today because I accidentally deleted the entire thread this morning. I apologize to all who had posted in the thread, which posts were not entered into my word-processing file. At the time I deleted the thread, by the way, the number of hits on it had been 7038.)
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