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From Integrity to Calculus

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From Integrity to Calculus

Part 1

In July 1966, Ayn Rand issued the first installment of her treatise Introduction to Objectivist Epistemology. She opened the first section “Cognition and Measurement” with an idea that had not appeared in her previous writings: “Consciousness, as a state of awareness, is . . . an active process that consists of two essentials: differentiation and integration.” Integration was an old key in her work. Differentiation was new.

In The Fountainhead Rand had trumpeted the virtues of independence, reliance on reason (one’s own), honesty, creative achievement, love of one’s work, and courage (HR II 559–60, XVIII 739–40). She contended that a concept of justice will make human life and happiness impossible if the concept ignores the uniqueness of individuals and the unity and self-sufficiency required by the preceding virtues (HR II 559–60, XVIII 740). She held up integrity as the overarching virtue pronouncing this unity and self-sufficiency (PK XIII 166, HR VIII 625–28, XVIII 742).

Integrity is characterized as “the clean, consistent, reasonable, self-faithful, the all-of-one-style, like a work of art” (GW IX 532). Howard Roark is integrity in the flesh. And though each of Roark’s buildings is unique—as each human being is unique (GW V 495)—they all display the concept and virtue of integrity.

A client of Roark’s, Kent Lansing, says to Roark “‘Integrity is the ability to stand by an idea. That presupposes the ability to think’” (ET X 333; also GW IX 53). Roark remarks, speaking of a housing complex he wants to design: “I want to make it real, living, functioning, built. But every living thing is integrated. Do you know what that means? Whole, pure, complete, unbroken. Do you know what constitutes an integrative principle? A thought. The one thought, the single thought that created the thing and every part of it.” (HR VIII 628)

When we turn from Fountainhead to Atlas Shrugged, we find Rand’s ethical thought fully developed. Seven moral virtues are articulated, for all individuals: rationality, independence, integrity, honesty, justice, productiveness, and pride. By the time of Atlas (1957), loyalty to truth in all things by reason, which is termed rationality, is the premier virtue. As with integrity in Fountainhead (1943), integration is key to the overarching virtue rationality in Rand 1957.

Rand writes in Atlas:

Reason is the faculty that perceives, identifies, and integrates the material provided by the senses . . . .

All thinking is a process of identification and integration. Man perceives a blob of color; by integrating the evidence of his sight and his touch, he learns to identify it as a solid object. . . .

No concept man forms is valid unless he integrates it without contradiction into the sum total of his knowledge. (1016)

[Thinking] is the process of defining identity and discovering causal connections. (1038)

[Man’s senses] do not provide him with automatic knowledge in separate snatches independent of context, but only with the material of knowledge, which his mind must learn to integrate. . . . The evidence [the senses] give him is an absolute, but his mind must learn to understand it, his mind must discover the nature, the causes, the full context of his sensory material. (1041)

Subsumed under rationality in Atlas is our old friend integrity, trueness to the fact that one “is an indivisible entity, an integrated unit of two attributes: of matter and consciousness,” integrity as keeping unity of mind and body, action and thought, life and convictions (1019).

In Rand’s “For the New Intellectual” (1960), integration is express and is underscored everywhere in the properly functioning lives and minds of individuals. There is the integration of what she conceives as The New Intellectual, “an integrated man, that is: a thinker who is a man of action” (51). Rand sets forth her formula of perception and conception as shot through with integrative process in this essay and repeats the formula the following year in “The Objectivist Ethics.” She writes “Sensations are integrated into perceptions automatically, by the brain of a man or an animal. But to integrate perceptions into conceptions by a process of abstraction is a feat that man alone has the power to perform” (14; see also the remarks on Hume). She conceives of human consciousness as preserving some continuity and as demanding “a certain degree of integration, whether a man seeks it or not” (18). Two disintegrators of human consciousness are guilt and fear (44). Philosophy should formulate “an integrated view of man, of existence, of the universe” (22). “Man needs an integrated view of life, a philosophy, whether he is aware of his need or not” (18).

In “The Objectivist Ethics” (1961), Rand writes of animals capable of perception as having awareness of integral entities, not only receipt of separate sensations, and of such an animal being “directed by an integrated awareness of the perceptual reality confronting it” (19). Man is such an animal and beyond. By organizing his perceptual material into concepts, he is able “to identify and integrate an unlimited amount of knowledge” (20; see also Branden 1963, 9). And what is a concept? It is “a mental integration of two or more perceptual concretes, which are isolated by a process of abstraction and united by a specific definition” (20).

The lives of animals are repetitive cycles without conscious connection between them. “An animal’s consciousness cannot integrate its entire lifetime” (Rand 1961, 24). Humans can and, for their survival, they must.

Integration is key not only in cognition, but in emotion. “Emotions are the automatic results of man’s value judgments integrated by his subconscious” (Rand 1961, 27). “If a man desires and pursues contradictions—if he wants to have his cake and eat it, too—he disintegrates his consciousness” (28). Failure of rationality, failure of conscious, volitional integration, yields emotional dissonance. Success in rationality produces happiness (29).

In the premier issue of The Objectivist Newsletter, there is an excerpt from a composition of Nathaniel Branden’s, a composition for the book Who Is Ayn Rand? He writes:

Objectivism teaches man that his mind and his emotions do not have to be antagonists, that his conscious convictions and his desires do not have to clash; it teaches man how they are to be integrated, how to bring them into non-contradictory harmony; it teaches man how he can determine the content of his desires and emotions. (It defines the principles involved; to develop their full implementation is the task of the science of psychology.) (1962, 3)

In reason one identifies and integrates the material provided by the senses. In one’s subconscious, there are “super-rapid integrations of sensory and ideational material” (Branden 1966, 68).

In 1965 and in the months of ’66 preceding the unveiling ITOE, Rand and Branden (1965, 53) wrote some on child cognitive and emotional development, and Rand wrote of more integrations distinctive of human life.

The process of a child’s development consists of acquiring knowledge, which requires the development of his capacity to grasp and deal with an ever-widening range of abstractions. This involves the growth of two interrelated but different chains of abstractions, two hierarchical structures of concepts, which should be integrated, but seldom are: the cognitive and the normative. The first deals with knowledge of the facts of reality—the second, with the evaluation of these facts. The first forms the epistemological foundation of science—the second, of morality and art. (Rand 1965a, 10)

The following month, Rand penned “The Psycho-Epistemology of Art.” In this essay, there is a five-paragraph stretch summarizing the nature and function of concepts. There we read of integration, of identification and integration, and of the ability of concepts to extend the range of consciousness. And we read: “A concept is like a mathematical series of specifically defined units, going off in both directions, open at both ends and including all units of that particular kind” (1965b, 15). We do not read of differentiation.

In February and March of 1966, Rand issued in order “Philosophy and Sense of Life” and “Art and Sense of Life.” By a sense of life, she meant a person’s “generalized feeling about existence, an implicit metaphysics with the compelling motivational power of a constant, basic emotion—an emotion which is part of all his other emotions and underlies all his experiences” (1966a 17). This generalized feeling she took to be the result of a subconscious integration summing the history of one’s psychological activities, one’s reactions and conclusions. She extends her earlier notion that human consciousness preserves willy-nilly some continuity and demands a certain degree of integration (1961, 18). “The enormously powerful integrating mechanism of man’s consciousness is there at birth; his only choice is to drive it or be driven by it” (1966a, 18).

Recall Rand’s mention in Atlas of early integration of sight and touch to perception of solid objects as such. I noted in “Induction on Identity” (1991) research on that integration:

Infants four months old can determine whether two objects (rings) are connected by a solid (but occluded) link through bimanual grasping and displacement of the two objects. They can discriminate visually (without touching) which of two assemblies, linked rigidly or elastically, they previously explored haptically. Object perception, even in the earliest phases of our development, seems to be mediated by relatively central mechanisms; haptic and visual discriminations are coordinated even before the onset of active manual-visual explorations (Streri and Spelke 1988; Spelke 1989). (Boydstun 1991, 39)

In its selective recreations of reality, according to Rand, art isolates and integrates aspects of reality to yield a new concrete that can serve certain functions for the human psyche. The artist’s sense of life is the controller and integrator of the artwork he or she creates. Rand saw Cubism,* in its attempts to display objects from multiple perspectives at once, as aiming for disintegration in consciousness (Rand 1966b, 38). So once more, as with her discussions of concepts and their formation to this point, Rand writes of isolation and integration, but not of differentiation.

Then comes ITOE where Rand open its first section “Cognition and Measurement’ with a proclamation that consciousness is “an active process that consists of two essentials: differentiation and integration” (1966–67, 5). She says “discriminated awareness begins at the level of percepts” (ibid.) I would say this is not quite right in that a newborn can discriminate his mother’s voice from other voices, and he discriminates directions of sources of sounds (see Appendix and its links). But Rand is on to something. Recall that she had tied experience at the level of percepts to awareness of entities (1961, 19). It is true that in every percept, we discriminate its object from other things or from background. Then too, we discriminate components within its object.

Discrimination in perceptual consciousness had already been part of her conception of the perception of entities. In Atlas she had spoken of the infant learning to distinguish solid objects and learning that Mother is one thing, a curtain is another, and neither can turn into the other (1040–41). Of later, conceptual development, Branden had written: “As the child grows, his intellectual field widens: he learns language, he begins to grasp abstractions, he generalizes, he makes increasingly subtle discriminations, he looks for principles, he acquires the ability to project a distant and more distant future” (1965, 53; emphasis added).

Rand’s emphasis on integration in childhood cognitive and affective development was in tune with the thinking of Piaget (e.g. 1954). For Piaget the main English terms would be assimilation and accommodation, denoting technical concepts of his which, with coordination between them, are the essence of development . Here is an example from lectures in 1953:

In psychological assimilation viewed from the cognitive perspective [as distinct from the affective perspective], objects are perceived relative to existing perceptual schemes. Assimilation may, therefore, be perceptual, sensorimotor, or conceptual. . . . Conceptual assimilations are those where a new object is conceived or understood because it is incorporated into forms or structures of internal action or thought. In other words, they are assimilations into the systems of mental operations that the subject has constructed.

The cognitive aspect of psychological accommodation is seen if the object resists assimilation into any existing scheme. In that case, the schemes are not adapted to the new object and must be modified. (Piaget 1953, 4–5)

Not only integration, but differentiation too, and by those names, loom large in Piaget’s studies in generalization, which he composed in the years after Rand’s ITOE (Campbell 2009, 157–59; note also*). Herbert Spencer, in his Principles of Psychology, had written: “Under its most general aspect therefore, all mental action whatever is definable as the continuous differentiation and integration of states of consciousness” (1855, 333; see Smith 1981, 119). Perhaps Rand had been familiar with Spencer’s theme of differentiation and integration in his thinking about biology and epistemology.

Discrimination or differentiation had always been part of Rand’s conception of consciousness as identification. That was logically entailed. But it is with her closer analysis of concepts—their structure, function, and genesis—in ITOE that differentiation becomes prominent.

(To be continued.)

Appendix – Baby

This is a more complete picture of the competencies of the human infant from day of birth to three weeks. Prior to birth, one had the grasp reflex, responsiveness to sounds, and the gustofacial reflex. One’s eyelids were opening, the retinal layers of the visual system were complete, and one was responsive to light intensity.

By day of birth, one had the additional reflexes of pupil dilation, kneejerk, and startle. On that day, one had visual preference for 3-dimensional objects (one perceived something of the 3D of objects), visual discriminations of different static line orientations, visual correction for 3D size constancy under variation of distance and correction for shape constancy under variation of object orientation. One was unable to detect boundaries and unable to fill in invisible parts of objects. One’s visual acuity was poor (probably due to immaturity of both the retina and the visual cortex), and one’s contrast sensitivity was poor.

One’s significant body motions were in alternation with visual attending. One was capable of rough, saccadic tracking, which was not only not smooth, but not anticipatory. One fixed on interesting objects, and perhaps one had some slight control in this; perhaps it was not entirely passive capture. One may have had an early visual preference for faces in tracking. One could imitate two facial movements and one head turn; one could perform these imitations when forced to delay until the model movement was absent.

One’s auditory resolution of pitches and volumes was already pretty good. One had a preference for Mother’s voice over the voice of a stranger, and one could distinguish human language from other auditory input. One was engaged in early head-turning, in the horizontal plane, towards sound sources. As of the time I compiled—fifteen years ago—the developmental time line from which the items here are taken, it was unknown whether the sound source is experienced as outside the head; head-turning had been evoked also by earphones.

Let’s wrap up the first day. One cried when other infants cried. One had auditory recognition memory; retention was for days under conditioning, for 24 hours under habituation. One was sensitive to pain, to touch (coetaneous and active), and to changes in bodily position.

By the end of the second day, one could discriminate Mother’s face from a stranger’s face. One had a preference for infant-directed speech (motherese) over adult-directed speech.

By five days, one engaged in early reaching towards an object in the visual field, reaching that included a preparation for grasping. This reaching and visual detection may be an undifferentiated attention system.

By twelve days, one could imitate three facial (oral) movements and one set of sequential finger movements. By fourteen days, one had a preference for Father’s voice over that of other males. By three weeks, one expected the reappearance of visual objects that were gradually occluded by a moving screen, provided the occlusion time was short.

(See further, a, b, sections from “Universals and Measurement” (2004); see also the posts following them, starting here.)

References

Branden, N. 1962. Excerpt from Who Is Ayn Rand? The Objectivist Newsletter (ON) 1(1):3.

——. 1963. Mental Health versus Mysticism and Self-Sacrifice. ON 2(3):9–11.

——. 1965. What Is Psychological Maturity? ON 4(11):53.

——. 1966. Emotions and Values. The Objectivist 6(May):65–73.

Boydstun, S. 1991. Induction on Identity. Objectivity 1(2):33–46.

Campbell, R. L. 2009. Constructive Processes: Abstraction, Generalization, and Dialectics. In The Cambridge Companion to Piaget. U. Müller, M. Carpendale, and L. Smith, editors. Cambridge.

Piaget, J. 1953. Intelligence and Affectivity – Their Relationship During Child Development. T. Brown and C. Kaegi, translators. 1981. Annual Reviews.

—. 1954. The Construction of Reality in the Child. Routledge.

Rand, A. 1943. The Fountainhead. Bobbs-Merrill.

——. 1957. Atlas Shrugged. Random House.

——. 1960. For the New Intellectual. Title essay. Signet.

——. 1961. The Objectivist Ethics. In The Virtue of Selfishness. Signet.

——. 1965a. Art and Moral Treason. ON 4(3):9–10, 12–14.

——. 1965b. The Psycho-Epistemology of Art. ON 4(4):15–16, 18.

——. 1966a. Philosophy and Sense of Life. The Objectivist (O) 5(Feb):17–22.

——. 1966b. Art and Sense of Life. O 5(Mar):33–40.

——. 1966–67. Introduction to Objectivist Epistemology. Reprinted from O 5(July–Dec), 6 (Jan–Feb). Meridian.

Smith, G. H. 1981. Herbert Spencer’s Theory of Causation. The Journal of Libertarian Studies 5(2):113–52.

Spencer, H. 1855. The Principles of Psychology. Longman, Brown, Green, and Longmans.

Spelke, E. S. 1989. The Origins of Physical Knowledge. In Thought without Language. L. Weiskrantz, editor. Oxford.

Steri, A., and E. S. Spelke 1988. Haptic Perception of Objects in Infancy. Cognitive Psychology 20:1–23.

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From Integrity to Calculus

Part 2

Identification entails differentiation. To point to a leaf and say leaf is not to point to a stone and say stone. Leaf is leaf and not stone. An item that is leaf is a leaf and not stone, and what a leaf is is not what a stone is (Rand 1957, 1016). Differentiation enters in that way in Rand’s conception of concepts as identifications. Differentiation enters her conception of concepts in two other ways as well. “A concept is a mental integration of two or more perceptual concretes, which are isolated by a process of abstraction and united by a specific definition” (Rand 1961, 20). Integration and uniting entail prior differentiation, and definition is going to require differentiation within a genus.

At the perceptual level, one recognizes and distinguishes entities. At the conceptual level, one grasps relationships of similarity and difference between entities, between their identities, and treats the items as units in groups of similar items, such as tables or chairs (ITOE 6). Beyond concepts of perceptual concretes, Rand defines concepts more generally:

A concept is a mental integration of two or more units which are isolated according to a specific characteristic(s) and united by a specific definition.

The units involved may be any aspect of reality: entities, attributes, actions, qualities, relationships, etc.; they may be perceptual concretes or other, earlier-formed concepts. The act of isolation involved is a process of abstraction: i.e., a selective mental focus that takes out or separates a certain aspect of reality from all others . . . . The uniting involved is not a mere sum, but an integration, i.e., a blending of the units into a single, new mental entity which is used thereafter as a single unit of thought. (ITOE 10)

The concept chair is the single particle of thought that stands for any chair. Any chair will do. Any particular chair fills the slot chair. Particular chairs are what I call substitution units in the concept chair. This is one of Rand’s two uses of unit in her theory of concepts (Boydstun 2004, 273).

We can study logic in an elementary text. We can learn what is logic so far as the text takes the subject. We would not be able to understand what is being taught unless we already knew, less explicitly, some of that logic. But we do not have to understand how we came by our prior, less explicit knowledge of logic in order to understand logic as taught in the text. Rand’s treatise on concepts says much about what they are (or at least what they can and ought to be), but it also says much about how we come to have concepts. To a considerable extent, she can be mistaken on points of the genesis of concepts, yet correct, insightful, and fruitful in theory of what they are and how they can fail or be improved.

It is the substitution-unit character of concepts that Rand displayed when she wrote, shortly before ITOE, “a concept is like a mathematical series of specifically defined units, going off in both directions, open at both ends and including all units of that particular kind” (1965, 15). The open-endedness in this picture captures the idea that there are more items than we have encountered that are qualified to be units within this concept. In this preview to her theory of concepts, it might seem puzzling from this quote alone that Rand pictures a sequence open at both ends, but the context shows she was thinking of the circumstance that in addition to units encountered, there are equally legitimate units of the concept not encountered, and these could reside in either the past or the future (see also ITOE 17–18). That is not part of the basic invariant structure of a Randian concept, for the set of integers (. . . –3, –2, –1, 0, 1, 2, 3, . . .) is in a one-to-one correspondence with the natural numbers (1, 2, 3, . . .), which is the more elementary sequence. Form an interlaced sequence from the integers, losing no members from their original sequence: (0, 1, –1, 2, –2, 3, –3, . . .). That sequence can be put into one-to-one correspondence with the naturals.

To say that a collection of items, ones encountered and ones not encountered, fall under a concept is not to consider them as having one order rather than another within the concept. It is somewhat like counting objects in a collection. The items can be counted in any order to find the number.

Rand said a concept was “like a mathematical series.” This is reasonable, thinking of the units of the concept as substitution units, although we need to get persnickety about the word series. In ordinary parlance, we often say “mathematical series” to indicate what is called a sequence in mathematics. That is what Rand means in this quotation, a mathematical sequence. In mathematics, we reserve the term series for a sequence in which some operation is being performed on the elements as we go along the sequence. An example would be [1 + (1/2) + (1/3) + (1/4) + . . . ], which sums to 2. And the measure value 2, indeed the magnitude of quantities affording real-number measure values of 2, are sums of that series and many others.

It is with Rand’s second sense of unit, which is measure-value unit, that concepts begin to look a bit more like a mathematical series in the strict sense. Take the concept length. The lengths of the legs of the chair I sit on and the lengths of the legs of the table I write on have particular different values along the ordered dimension that is length. Such values of length can be operated on: physically, as when one adds the thickness of a matchbook to one leg of a table at the diner to keep the table from rocking or mentally, as in addition, taking logarithms, and so forth. Having such susceptibility to mathematical operations is a prerequisite for placement in mathematical series. Rand is staying with sequence, any sequence of instances, for concepts, even after measure values have been acknowledged of the substitution units, which could be arrayed in correct measure order along dimensions of the substitution units falling under the concept. For in the concept, the particular measure values are omitted.

Not all measure values are susceptible to arithmetic operations such as addition. Ordinal values, such as the scratch-hardness of solids, are not meaningfully additive. Nevertheless, along the dimension scratch-hardness, substitution units under the concept minerals can be truly linearly ordered in mathematical sequence.

If one has studied calculus, one has perhaps studied series. The latter are a nice way to approach integrals in calculus.* Because Rand speaks in ITOE of some likeness to mathematics in the differentiations and integrations that go into the constitution of concepts, because operations called differentiation and integration are basic operations of calculus, and because Rand mentions integration in calculus when speaking about measurement (14), one may slide from sequence to series as that which is more like the constitution of Randian concepts. Do not make that slide. Stay with sequence, indifferently any one with the proper members.

There is impetus as well to slide from sequence/series to integral. Because the units of a concept are contracted into a single item, the concept, when their particular measure values along the concept’s dimensions are suspended, one can slip into thinking the contraction to be that which occurs in the evaluation of a definite integral in calculus. An example of the latter would be the evaluation of integrals over some figure to yield the single value that is the area of the figure or to yield the single value that is the volume of the figure. Do not make this slide either. No mathematical integration occurs along the measure-value dimensions of a concept when its measure values are omitted and the instances falling under the concept lose (in thought) their array in sequences in proper orders along those dimensions.

Another likeness Rand observes between concepts in general and a concept learned in mathematics is the likeness between concepts and variable in algebra. Merlin Jetton has noted that this likeness was observed also by Berkeley, though the bishop did not attain a measurements-omitted conception of concepts (Jetton 1998, 66–67; see also). Rand writes:

The basic principle of concept-formation (which states that the omitted measurements must exist in some quantity, but may exist in any quantity) is the equivalent of the basic principle of algebra, which states that algebraic symbols must be given some numerical value, but may be given any value. In this sense and respect, perceptual awareness is the arithmetic, but conceptual awareness is the algebra of cognition.

The relationship of concepts to their constituent particulars is the same as the relationship of algebraic symbols to numbers. In the equation 2a = a + a, any number may be substituted for the symbol “a” without affecting the truth of the equation. . . . In the same manner, by the same psycho-epistemological method, a concept is used as an algebraic symbol that stands for any of the arithmetical sequence of units it subsumes. (ITOE 18)

Rand was using the equation 2a = a + a only to illustrate the sense in which a concept is like a variable in algebra. She was not saying that two instances of a concept somehow combine to make a magnification of the concept. She was not insinuating that propositions in which a concept is explicated can be modeled on algebraic statements of equality or inequality such as a = b + c. She has not said that the is or has of a proposition is reducible to the equals in mathematical equations. (It remains, of course, that equations are propositions.) Lastly, she has not insinuated that the measure values of a unit falling under a concept are necessarily additive.

Rand knew some algebra. I expect she knew also something of the basic operations of calculus: differentiation and integration. In Atlas Shrugged Dagny’s father had been inspecting

a complex system of pulley’s which Francisco, aged twelve, had erected to make an elevator to the top of a rock. . . . Francisco’s notes of calculation were still scattered about on the ground; her father picked them up, looked at them, then asked, “Francisco, how many years of algebra have you had?” “Two years.” “Who taught you to do this?” “Oh, that’s just something I figured out.” She did not know that what her father held on the crumpled sheets of paper was the crude version of a differential equation. (93)

A differential equation is something beyond an algebraic equation in that the terms in the differential equation are not only functions of variables, but derivatives of such functions, where “differentiations” or “taking derivatives” of functions is one of the two fundamental operations on functions distinctive of calculus. Having some elementary knowledge of what is a differential equation, such as the author of Atlas evidently had, would likely include knowledge that implicit in the differential equation are various specific relations among the variables whose differentials are as specified in the differential equation. One would likely know that solving a differential equation—sometimes by integration, the other operation of calculus—that correctly describes a physical system yields those various specific relations among the variables and that those relations tell how the system will behave.

A humanities person in mid-twentieth century could learn these sorts of broad strokes by opening an encyclopedia, if a mathematics text were too forbidding. Perhaps Rand knew nothing of the use of integration in solving a differential equation. Still, she could know that differentiation and integration are the two basic operations of calculus, and even that the latter operation is at hand in convergent continuous infinite series and the value(s) to which they converge.

I alluded earlier to the remark of Rand’s recognizing that integration in calculus can be used to measure areas (ITOE 14). After discussing the attribute of shape as one of the characteristics by which one clusters chairs with chairs and tables with tables, she wrote:

Similarity is grasped perceptually; in observing it, man is not and does not have to be aware of the fact that it involves a matter of measurement. It is the task of philosophy and of science to identify that fact.

As to the actual process of measuring shapes, a vast part of higher mathematics, from geometry on up, is devoted to the task of discovering methods by which various shapes can be measured . . . . (Integral calculus, used to measure the area of circles, is just one example.) (ITOE 14)

“Measuring” by integral calculus is to be understood as a calculation using a measured value (the radius of a circle, in Rand’s example) to deliver another measure value (the area of the circle) that one did not measure directly. Rand drifts away from shape by her example, for area is not shape. I have repaired that in “Universals and Measurement.” We characterize shapes by distributed sets of measure-values of curvatures or torsions, creatures in the regions of mathematics called analytic and differential geometry, which indeed consort with calculus (see further Boydstun 2004, 275, 293, 296n8, 299n27).

In 1991 Leonard Peikoff remarked:

Ayn Rand regarded her theory of concepts as proved, but not as completed. There are, she thought, important similarities between concepts and mathematics still to be identified; and there is much to be learned about man’s mind by a proper study of man’s brain and nervous system. In her last years, Miss Rand was interested in following up on these ideas—in relating the field of conceptualization to two others: higher mathematics and neurology. Her ultimate goal was to integrate in one theory the branch of philosophy that studies man’s cognitive faculty with the science that reveals its essential method and the science that studies its physical organs. (109)

Rand was headed in the right direction on both counts. Her conception of cognitive systems as measurement systems needs to be situated within the modern program of understanding CNS as a computational system. As for the other avenue of research, it is voiced too softly to say there are “important similarities between concepts and mathematics still to be identified.” Similarities should be replaced with relationships in that prospect, so as to include similarities, but more intimate relationships as well. The relation of particular instance to concept is the same as the relation of particular numerical value to algebraic variable; they are the same, not only similar. Then too, measurement-omission theory is to be joined to real measurement and solid modern measurement theory, including geometry (synthetic, analytic, and differential) and calculus (see Boydstun 2004, a, b).

To conclude the story of Rand and her long-time friend, integration, we shall pick up, in ITOE and beyond, her further employments of that friend in theory of concepts and related powers.

(To be continued.)

References

Boydstun, S. 2004. Universals and Measurement. The Journal of Ayn Rand Studies 5(2):271–305.

Jetton, M. 1998. Pursuing Similarity. Objectivity 2(6):41–130.

Peikoff, L. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Rand, A. 1957. Atlas Shrugged. Random House.

——. 1965. The Psycho-Epistemology of Art. The Objectivist Newsletter 4(4):15–16, 18.

——. 1966–67. Introduction to Objectivist Epistemology. Reprinted from The Objectivist 5(July–Dec), 6 (Jan–Feb). Meridian.

Edited by Boydstun

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From Integrity to Calculus

Part 3

Rand wrote:

When a child observes that two objects (which he will later learn to designate as “tables”) resemble each other, but are different from four other objects (“chairs”), his mind is focusing on a particular attribute of the objects (their shape), then isolating them according to their differences, and integrating them as units into separate groups according to their similarities. (ITOE 6; further, 21–26)

All concepts are formed by first differentiating two or more existents from other existents. All conceptual differentiations are made in terms of commensurable characteristics (i.e., characteristics possessing a common unit of measurement). (13)

Notice that in the second quotation there is implicit the mild claim that concepts have implicitly in their constitutions differentiations from things excluded from the concept. That mild claim can be true even if some particulars of Rand’s conception of how the concepts came to be formed were incorrect. The “two or more” requirement could be dropped, for example, as when one forms the concept existence in the most general sense. “There is nothing antecedent to existence, nothing apart from it—and no alternative to it” (Peikoff 1967, 313; see also ITOE Appendix, 245–47, 149–50).

Rand overstated the case when she wrote “all conceptual differentiations are made in terms of commensurable characteristics.” Existence and non-existence have no commensurable characteristics, as the latter has no characteristics. Moreover, incommensurable characteristics such a long and green do have characteristic in common as a genus. Rand readily allows, in follow-up verbal discussion, that at higher levels of genus, from higher abstraction, common characteristics can be found for incommensurable characteristics at a lower level (ITOE Appendix, 145–47). I have worked out the implications of these hierarchical circumstances for measurement theory of similarity and measurement-omission theory of concepts in the sections* Superordinates and Similarity Classes and Analytic Constraint of my “Universals and Measurement” (2004; see also Kelley and Krueger 1984, 55–57; Jetton 2011, 215–28).

It remains that long and short are literally commensurable; they are of a common measurable dimension. Likewise it stands with blue and green. If all concretes stand in measurable relations with other concretes, then it remains that all concretes can be placed under concepts having the measurement-omission structure. (Usually the concepts will have multiple dimensions. For the concrete that is the whole universe, the concretes commensurable with it will have to be its own parts.) And it remains that long and green are incommensurable, a profound difference between them.

David Kelley and Janet Kruger remarked “The natural place to look for a process of abstraction would be in modes of selective attention to determinate features and relations in the perceptual field, especially to concrete similarities among objects brought out by their common differences from other objects in the background” (1984, 64; further, Kelley 1984, 339). As we have seen, in Rand’s view, “You cannot form a concept by integration alone or by differentiation alone. You need both, always. You need to observe similarities in a certain group of objects and differences from some other group of objects . . .” (ITOE Appendix, 138).

In “A Theory of Abstraction” David Kelley defends Rand’s model of concept-formation, describing “a sequence of cognitive states and processes occurring at the preconceptual level” (1984, 331). He draws out Rand’s formation process in fine lines, but at a remove from any consideration of child development. His is a logical contour of the process of forming concepts from perceptions, consistent with cognitive psychology, but guided by various general constraints on empirical, objective accounts of concept generation and by past failed attempts at such an account. Allan Gotthelf notes the excellent developments by Kelley from Rand’s basics on similarity and concepts in Kelley’s 1984 papers in his own “Ayn Rand on Concepts” (2007).*

While Rand was putting integration to work in theory of concepts in the installments of ITOE, Branden was extending the work of integration in psychology.

Recall that the Objectivists to this point and beyond regarded emotions as from automatic subconscious integrations of value judgments and that one’s conscious rational integrations of existence, including one’s own existence and possibilities, pave the road to happiness; whereas, evasion of integration, success in disintegration, leads to unhappiness. Recall also that Rand had lately added that there is a generalized emotion she called a sense of life, which is the result of subconscious integrations of one’s entire experience and reactions. (Rand had used the phrase sense of life once in Fountainhead, twice in Atlas, and evidently used it routinely in conversation [Branden 1999, 38, 56, 101, 105, 168, 171], before beginning to write about the meaning of the phrase in 1965.) Lastly, recall that she had conceived of human consciousness as naturally preserving a certain amount of continuity and as pushing forward to some amount of integration best it can without deliberate, controlled integration by the person.

Branden proposed understanding pathological anxiety as subversion of “the clarity and integrity of [one’s] own mental processes, by evasion, repression, and rationalization” (1966, 170–71). He portrayed the subconscious as registering that there are things one needs to think about that one has refused to think about. He portrayed the subconscious as registering and summing implications about a policy of disconnect between one’s convictions and actions. These disintegrating policies and their subconscious records lead to self-distrust, the ramifications of which are drawn out in the essay. This is an example of the Rand-Branden idea that the science of psychology deals with the implications of the broad need identified in Rand’s philosophy, the need for integration, for integrity, for rationality (Branden 1962, 3).

Rand and her circle had been using the term psycho-epistemology to refer to an individual’s method of awareness: Is the time scope of his outlook brief or long? Is his concern only with what is physically present? Does he recoil into his emotions in the face of his physical life and need for action? How far does he integrate his perceptions into conceptions? Is his thinking a means of perceiving reality or justifying escape from reality? (Rand 1960, 14, 19, 21). Chris Sciabarra reports that Barbara Branden was the one who originated the concept (and, I presume, the word) psycho-epistemology (1995, 194). In her lecture series Principles of Efficient Thinking, Ms. Branden had defined psycho-epistemology as “the study of the mental operations that characterize a man’s method of dealing with reality” (1962, 178). Nathaniel Branden had further specified the compass of psycho-epistemology in an essay with that title (1964). Rand had written on the psycho-epistemological function of art, integration of consciousness by concretization of abstraction and value (1965b).

In 1967 Mr. Branden portrayed mental illness as at root a cognitive, psycho-epistemological disorder. Such disorders are disintegrations of consciousness, contravening a fundamental need. There is continuity in the need for integration from organic life to the life of consciousness.

Biologically, life is a state and process of integration: the physical integrity of an organism, and the integration of its actions in the direction of life-serving goals, is the pre-condition and essence of biological well-being—of an organism’s success at the task of survival. Any forces that work against integration, work against life; disintegration is motion towards death.

Integration is basic to the cognitive process—and to mental health. Disintegration and conflict are the hallmark of mental illness. (1967a, 219)

That picture of mental health and illness resonates with Rand’s contentions about egoism and the psyche in Galt’s speech. Branden’s picture of psychological maturity resonates with Rand’s stress on a policy of conceptualizing in “The Objectivist Ethics.” He outlines ways in which that policy is basic in psychological maturity and in fullest self-esteem (Branden 1967b, 230–31).

The following year, Branden attempted to give further psychological explication of Rand’s concept sense of life: its development in childhood, its forms, and the affinity it buoys between persons. How much disparity there is between a person’s avowed philosophy and his sense of life depends on “such factors as how rational he is, how conceptually reflective about his own life, how well-integrated psychologically” (1968, 401).

It seems to me that Rand and Branden portrayed one’s sense of life as too well integrated in itself, and their sets of philosophical issues going into the make-up of (all the facets of) one’s sense of life may well be incomplete. That is another story, for a separate essay.

I mentioned in Part 1 that Rand’s emphasis on integration in childhood cognitive and affective development was in tune with the thinking of Jean Piaget. I quoted from some 1953 lectures of his, wherein he defined two of his key concepts in development, which are assimilation and accommodation. Those are processes coming under the concept integration in Rand’s system. The similarity between Rand and Piaget is even plainer when we turn to John Flavell’s The Developmental Psychology of Jean Piaget (1963). Unlike the lectures from which I was quoting, Flavell’s text was easily available to Rand and her circle. (By the late ’60’s, my college friends and I were very wrapped up in ITOE and Piaget’s genetic epistemology and their relationships; works about Piaget’s work were readily available. See further, Hsueh 2009.)

In this text, one reads of Piaget’s concept of the schema in cognitive development, which is especially important for thought prior to language (see also Jetton 1998, 104–7). Rand makes brief contact with a schema of the image type when she speaks of the child’s iconic representation in drawing man as with oval torso, round head, and stick limbs (ITOE 13; Boydstun 1990, 16–18).

In Piaget’s conception,

A schema is a cognitive structure which has reference to a class of similar action sequences . . . . / In discussing sensory-motor development, Piaget speaks of the schema of sucking, the schema of prehension, the schema of sight, and so on. . . . / To say that a grasping sequence forms a schema is to imply more than the simple fact that the infant shows organized grasping behavior. It implies that assimilatory functioning has generated a specific cognitive structure, an organized disposition to grasp objects in repeated occasions. It implies that there has been a change in over-all cognitive organization such that a new behavioral totality has become part of the child’s intellectual repertoire. (Flavell 1963, 52–53)

With Piaget’s concept of schema in hand, one reads a little later “A Piagetian schema . . . is always the product of the differentiation, generalization, and integration of earlier schemas . . .” (Flavell 1963, 73). Differentiation and integration, Rand will announce in 1966, are active processes essential to all consciousness as a state of awareness (ITOE 6).

That not only integration but differentiation should become highlighted when Rand came to presenting her theory of concepts and definition was natural. Why did she go on to announce at the outset of setting forth her theory of concepts that differentiation and integration are essential in all consciousness? To be sure, it is understandable that she should be setting her view of conceptual consciousness in wider patterns of consciousness. Then too, yes, differentiation and integration are implicit in the idea of consciousness as identification. And yes, years past Rand might have read Spencer saying differentiation and integration are essential to all consciousness.

In Principles of Efficient Thinking, Barbara Branden had spoken of full mental clarity as “a state in which one perceives, judges, connects, and integrates the full conceptual meaning of every aspect of that with which one is dealing” (1962, 161). She had included some discussion of definitions and concepts (165–70), and that material, including the examples, was later reused by Rand in ITOE. This 1962 discussion of definitions and concepts by Ms. Branden is replete with references to processes of differentiation and integration. There is no mention of the essential role of that dynamic duo in all consciousness.

It is possible that Rand had the generalization in her hand for some time, but never thought to show that card until ITOE. In the alternative, it is possible she had only recently come squarely to the generalization.

I mentioned that in Flavell 1963 Rand and her circle could readily come across the Piagetian dynamics of differentiation and integration at the preconceptual level of cognition, at the level of schemas. That would be an inch towards the differentiation-integration generalization.

Here would be another, though I'll trace a loop whose line integral is nine inches by way of gaining this other inch. In “The Objectivist Ethics,” Rand had written:

Now in what manner does a human being discover the concept of ‘value’? By what means does he first become aware of the issue of ‘good or evil’ in its simplest form? By means of the physical sensations of pleasure or pain. Just as sensations are the first step of the development of a human consciousness in the realm of cognition, so they are its first step in the realm of evaluation. (1961, 17)

Leonard Peikoff reports “Miss Rand used to be a strong advocate of what she called ‘the pleasure-purpose principle’. She meant the idea that on any level, whether we’re talking about thought or action, you cannot function without a purpose that brings you pleasure, something you want to achieve, that you enjoy achieving” (1983, 325). Suppose Rand had that idea far back and, like the dual cognitive-evaluative development she expressed in the preceding paragraph (1961), had this idea in advance of 1963.

Flavell 1963 included a summary of Piaget’s Paris lectures I quoted in Part 1. Flavell reported Piaget’s view that

affect and cognition can be necessarily involved in all human adaptation. The affective-motivational aspect provides the énergetique of behavior while the cognitive aspect provides the structure (affect cannot of itself create structures, although it does influence the selection of the reality content upon which the structures operate). (81)

Flavell writes that in Piaget’s view “cognitions with primary affective, interpersonal content function like those of a more purely intellectual sort” (1963, 80). He quotes from a 1951 paper of Piaget:

This generalized application of initial affective schemas raises no particular problems with regard to the mechanism of assimilation which is necessarily involved. It is the same as that of sensory-motor or intuitive assimilation. . . . It is the same assimilation because personal schemas, like all others, are both intellectual and affective. (quoted in Flavell 1963, 81)

With ITOE just around the corner, Rand writes that a child’s development involves growth of two interrelated chains of abstractions, one cognitive, the other normative (1965a, 10). “While cognitive abstractions identify the facts, normative abstractions evaluate the facts, thus prescribing a choice of values and a course of action” (1965b, 15). Rand soon adds a third distinct chain of abstractions interrelated with the two previous chains. Like Piaget, she sees the cognitive chain as basic.

Cognitive abstractions are formed by the criterion of: What is essential? (epistemologically essential to distinguish one class of existents from all others). Normative abstractions are formed by the criterion of: What is good? Esthetic abstractions are formed by the criterion of: What is important? (1966, 34–35)

If Rand had exposure to Piaget, through Flavell 1963 or through other conduits, she could have adapted what was concordant therein to her own vocabulary and picture of child cognitive and affective development. Piaget’s discernment of the dual strands, cognitive and affective, in child development and in all human adaptation would strike a chord with Rand’s “pleasure-purpose principle.” That chord could resound in the dual chains, cognitive and normative, in Rand 1965a and 1965b, reformed into the triple chain in Rand 1966. Rand first took the normative chain to be the epistemological foundation of art (1965a, 10), but then gave art its own abstraction chain interrelated to the two other chains (1966, 34). Art does not evoke exactly an emotion, rather a sense or feel (ibid.).

It is the artist’s sense of life that controls and integrates his work, directing the innumerable choices he has to make, from the choice of subject to the subtlest details of style. It is the viewer’s or reader’s sense of life that responds to a work of art by a complex, yet automatic reaction of acceptance and approval, or rejection and condemnation. (Rand 1966, 33)

“Art is man’s metaphysical mirror” (Rand 1966, 37), and it permits him “to contemplate his abstractions outside his own mind” (36). The stylized subject of a work of art tokens aspects of existence known abstractly in one’s cognitive chain and in one’s esthetic chain of evaluations as to metaphysical importance. The style tokens psycho-epistemology (37). In the creation of a work of art, on Rand’s understanding, there is selection, which entails differentiation, followed by integration in a new concrete for respite-perception in concrete: human life in its abstract and protracted effort of thought and valuation. The process of art’s creation or appreciation is a type of construction requiring differentiations and integrations, cousin to those at work in the journey of consciousness from perceptions to abstract concepts (1965b, 15–16). That is our second inch toward the generalization that all consciousness as an active process involves both differentiation and integration in an essential way.

Piaget’s discernment of dual processes of differentiation and integration in preconceptual, schematic cognition would strike a chord with Rand’s stress on those dual processes in definitions of concepts (which had been voiced by Ms. Branden in 1962). That chord advanced the other, more straightforward inch towards Rand’s differentiation-integration generalization. The Piaget of Flavell 1963 was in happy time for the dawn that is ITOE.

(To be continued.)

References

Boydstun, S. 1990. Capturing Concepts. Objectivity 1(1):13–41.

——.2004. Universals and Measurement. The Journal of Ayn Rand Studies 5(2):271–305.

Branden, B. 1962. Principles of Efficient Thinking. In The Vision of Ayn Rand. 2009. Cobden.

Branden, N. 1962. Excerpt from Who Is Ayn Rand? The Objectivist Newsletter (ON) 1(1):3.

——. 1964. Psycho-Epistemology. ON 3(10):41, 43–44; 3(11):46–47.

——. 1966. The Nature of Anxiety. The Objectivist (O) 5(Nov):167–72.

——. 1967a. The Concept of Mental Health. O 6(Feb):216–22.

——. 1967b. Self-Esteem I. O 6(Mar):225–31.

——. 1968. Self-Esteem and Romantic Love III. O 7:(401–5).

——. 1999. My Years with Ayn Rand. Jossey-Bass.

Flavell, J. H. 1963. The Developmental Psychology of Jean Piaget. D. Van Nostrand.

Gotthelf, A. 2007. Ayn Rand on Concepts – Another Approach to Abstraction, Essences, and Kinds. Conference Proceedings Nature and Its Classification. Birmingham, UK.

Hsueh, Y. 2009. Piaget in the United States 1925–1971. In The Cambridge Companion to Piaget. U. Müller, M. Carpendale, and L. Smith, editors. Cambridge.

Jetton, M. 1998. Pursuing Similarity. Objectivity 2(6):41–130.

——. 2011. The Sim-Diff Model and Comparison. JARS 11(2):215–32.

Kelley, D. 1984. A Theory of Abstraction. Cognition and Brain Theory 7(3&4):329–57.

Kelley, D., and J. Krueger 1984. The Psychology of Abstraction. Journal for the Theory of Social Behavior 14(1):43–67.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy IV. The Objectivist 6(Aug):311–16.

——. 1983. Understanding Objectivism. M. Berliner, editor. 2012. NAL.

Rand, A. 1960. For the New Intellectual. Title essay. Signet.

——. 1961. The Objectivist Ethics. In The Virtue of Selfishness. 1964. Signet.

——. 1965a. Art and Moral Treason. The Objectivist Newsletter (ON) 4(3):9–10, 12–14.

——. 1965b. The Psycho-Epistemology of Art. ON 4(4):15–16, 18.

——. 1966. Art and Sense of Life. The Objectivist 5(Mar):33–40.

——. 1966–67. Introduction to Objectivist Epistemology. Reprinted from The Objectivist 5(July–Dec), 6(Jan–Feb). Meridian.

Sciabarra, C. M. 1995. Ayn Rand: The Russian Radical. Penn State.

Edited by Boydstun

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From Integrity to Calculus

Part 4

“Reason is the faculty that perceives, identifies and integrates the material provided by [man’s] senses” (Rand 1957, 1016). Reason is man’s only tool of knowledge. Reason is the tool by which the individual achieves his specific choice of happiness (1018). “Reason is the source, the precondition of his productive work” (Rand 1961, 25). “Productive work is the central purpose of a rational man’s life, the central value that integrates and determines the hierarchy of all his other values” (ibid.).

All the fruit trees lend themselves to work of this kind. The gathering in of the almonds interests even the smallest children, who do a really useful bit of work, so diligent are they in seeking out the hidden almonds and gathering them into baskets. Hunting out the strawberries, lurking under the leaves is a work no less pleasing than of seeking for sweet violets. (Montessori 1906, 94)

It is by work that children organize their personalities. (Montessori quoted in Lillard 1996, 42)

In addition to ordering their environment and ordering themselves in their outward personalities, they have also ordered the inner world of their minds. / The didactic material, in fact, does not offer to the child the “content” of the mind, but the order for that “content.” It causes him to distinguish identities from differences, extreme differences from fine gradations, and to classify, under conceptions of quality and of quantity, the most varying sensations appertaining to surfaces, colors, dimensions, forms and sounds. The mind has formed itself by a special exercise of attention, observing, comparing, and classifying. . . . / Language now comes to fix by means of exact words the ideas which the mind has acquired. (Montessori 1914, 82–83)

All work is noble. The only ignoble thing is to live without working. It is essential to understand the value of work in all its forms, be they manual or intellectual. Practical experience will cause the adolescent to understand that the two forms are complementary and they are equally essential in a civilized existence. (Montessori 1948, 109)

The affinity between Maria Montessori’s educational methods and Rand’s view of reality, human nature, and theory of knowledge is several. Rand praised those methods for the way Montessori’s didactic material sets an appropriate place for the child to explore and find the natures of things by his or her own interest and effort, guided by the instructor, corrected by the reality of the material (Hessen 1970, 871). My third quotation above from Montessori is also quoted by Rand, who writes that in this passage “it is psycho-epistemological training that Dr. Montessori had in mind (though this is not her term)” (Rand 1970, 886). Recall that psycho-epistemology pertains to “cognitive processes from the aspect of the interaction between man’s conscious mind and the automatic functions of his subconscious” (883).

Cognitive integration is another grand concordance between Rand’s view of knowledge and Montessori’s culture for its growth from infancy to child to adolescent and adult. Montessori education provides meaningful contexts for learning. What is being learned is given connection to real-life endeavors, and it is given connection to the child’s existing knowledge (Lillard 2005, 224–56). A Montessori educator* who is an Objectivist observes: “One of the cornerstones of the Montessori method is the presentation of knowledge as an integrated whole, emphasizing conceptual relationships between different branches of learning and the placement of knowledge in its historical context.”*

The metaphor of the human mind as a blank slate on which experience must write in order for us to acquire knowledge is a favorite of Aristotelians and Lockeans. Rand concurred in that general picture, but she went on to add that to acquire knowledge one must acquire skills, not only content reached with those skills (Rand 1970, 883–86). The mind is no empty vessel into which knowledge is poured (Lillard 2005, 9–14, 225). Context matters for real learning in an actively integrating, living intelligence.

“The function of psychological integrations is to make certain connections automatic, so that they work as a unit and do not require a conscious process of thought every time they are evoked. (All learning consists of automatizing one’s knowledge in order to leave one’s mind free to pursue further knowledge.)” (Rand 1966, 34). One first acquires knowledge “by fully conscious, focused attention and observation,” one then establishes “mental connections which make the knowledge automatic (instantly available as a context)” (ITOE 65).

The process of automatization was discussed by William James in his Principles of Psychology in connection with the nature and functions of habits. He quotes from one author who speaks of human modes of movement and sense perception become automatic, but which first had to be acquired—they are not congenital—through nervous-system development in the course of self-education (James 1890, 111). He quotes another author who describes the attention upon the movement of his fingers required by the beginning piano player, which movements will later flow easily upon the slightest stimulus (112–13). James concluded that habits simplify “movements required to achieve a given result, makes them more accurate and diminishes fatigue” (112).

James quotes a third author:

If an act became no easier after being done several times, if the careful direction of consciousness were necessary to its accomplishment on each occasion, it is evident that the whole activity of a lifetime might be confined to one or two deeds—that no progress could take place in development. A man might be occupied all day in dressing and undressing himself . . . . Think of the pains necessary to teach a child to stand, of the many efforts which it must make, and of the ease with which it at last stands, unconscious of any effort. (Henry Maudsley quoted in James 1890, 113–114)

James concluded that habit “diminishes the conscious attention with which our acts are performed” (1890, 114).

The great difference between (i) James’ understanding of habit formation and habit utility and (ii) Rand’s understanding of automatization and its role in cognition is the salience of the concept integration for Rand and its absence for James. He was aware of Spencer’s pervasive use of the concept integration for understanding cognition, but he did not pick up this element for his own account of our powers (James 1890, 151–52).

Integration is today pervasive and salient in cognitive science. Consider the work Attention and Performance XVI, whose chapters dig into how a wide range of constraining influences

are integrated in perception, attention, language comprehension, and motor control. [The chapters] consider the mechanism of information integration in the brain; examine the status of the modularity hypothesis in light of efforts to understand how information integration can be successfully achieved; and discuss information integration from the viewpoints of psychophysics, physiology, and computational theory. (Inui and McClelland 1996, back cover)

Integration is today put to work as well in philosophies not Rand’s: “What I am getting at here is the question of how to integrate art into the practical, goal-driven pursuits that we take to be particularly important” (Stroud 2011, 2).

Recall that in Rand’s view acquisition of knowledge includes acquisition of the specific skills by which the content of knowledge is obtained. By age two, the human being is running around and speaking in two-word sentences. In his first two years, the little one has learned an amazing lot.

To focus his eyes (which is not innate, but an acquired skill),[*] to perceive the things around him by integrating his sensations into percepts (which is not innate, but an acquired skill), to coordinate his muscles for the task of crawling, then standing upright, then walking—and, ultimately, to grasp the process of concept-formation and learn to speak—these are some of an infant’s tasks and achievements . . . . (Rand 1970, 883)

The creation of art also requires the acquisition of specific skills. The art that is literature—literary fiction—deals in language and therefore concepts, but the skill required to write fiction is additional to the skills I had to acquire to be able to write this essay for you. The accomplished writer of drama, libretto, short story, or novel creates a story and its characters that illustrates, or concretizes, a theme, conveyed with a definite style (Rand 1968, 481). In the literary art, the concretization resides in imagination. Its concretes are told of and induced into the imagination of the audience.

The hoist swung like a pendulum above the city. It sped against the side of the building. It passed the line where the masonry ended behind her. There was nothing behind her now but steel ligaments and space. She felt the height pressing against her eardrums. The sun was in her eyes. The air beat against her raised chin. —Rand

Style is not told, but entered (cf. Peikoff 1991, 422–23; Rand 1958, 89–93, 123–25). A poem can have all the concretes, the imagined and the entered, as have the other genre of literature, but it need not. Its concretization of a theme may be attained without story or characterization (Rand 1968, 481). Poetry has style, of course, and it can utilize more focally than can the other literary forms the concrete sound and rhythm of language* (see also Branden c. 1968, 170, 466).

Not a whisper, not a thought,

Not a kiss nor look be lost. —Auden

The theme and style of a good poem must be worked together to unified result. In a good novel, the theme, plot, characterization, and style must likewise be worked together to “an individual sum” (Rand 1968, 481). The theme of the novel is about human existence. The theme serves as integrator of the novel, guiding the writer’s selections in plot, which is to express the theme in actions. Plot integrates the events of the story, making them entirely purposeful towards a resolving climax (but see Cox 2000, 323–24*). The logic of the plot is the logic of final causation in human action. The cardinal principle of a good novel is that “the theme and the plot must be integrated—as thoroughly integrated as mind and body or thought and action in a rational view of man. . . . / . . . The integration of an important theme with a complex plot-structure is the most difficult achievement possible to a writer, and the rarest” (Rand 1968, 485).

A character in a work of fiction should “be an abstraction, yet look like a concrete” (Rand 1968, 486). The good writer unfolds, mainly in action and dialogue, the motivations of a character, showing the character’s nature and actions intelligible (see also Minsaas 2005*). Characters enact the plot, and the events of the plot contribute to characterizations; and theme is required for a plot. “This is the kind of integration required by the nature of a novel” (Rand 1968, 500).

Literary style is a novelist’s distinctive or characteristic mode of execution. It includes what is selected to include in a passage as well as choice of words and sentences. In Rand’s view, a novel must tell a story, and style in a novel is only a means to that end. An accomplished style effecting moods is part of the craft of the novelist. But style and mood alone without a re-creation of reality by plot and characterization does not occasion for the reader existence and purpose as drivers of human consciousness.

Rand overextended those points by applying them beyond the art-form that is the novel. She says that literary mood studies, “little pieces conveying nothing but a certain mood, . . . are not an art-form” (1968, 503). Then, what about poetry without characterization or story? Rand took such poetry as literary art. On the face of it, she is in contradiction.

Rand conceived as art-forms only elaborate, highly developed forms. Consider dance. To attain what Rand takes as an art-form, dance must be a fully developed system, such as ballet, or at least it must have “the key elements on which a fully developed system could be built,” such being the case with tap dancing (Rand 1971, 1034; cf. Crowther 2007). The system test could be passed by some poetry that evidently would not also pass the test of Rand’s definition of art.

Rand was aiming for what has been called a “‘wrapper definition’ that attempts to cover the entire extension of a concept,” rather than only “an evaluative characterization of what the best forms of art aspire to be like” (Stroud 2011, 5). Rand took up the challenge of showing literary and nonliterary art-forms to be distinctive and explicable under a definition, her definition of art, which is, we recall: “a selective re-creation of reality according to an artist’s metaphysical value-judgments.” (On this definition, see Torres and Kamhi 2000; Hunt 2001, 261–62;* Enright 2001, 353–56*). In the course of her examinations of various art-forms, we learn more about what she means by re-creation of reality in the way of art. For poetry without story or characterization—say, Rossetti’s Silent Noon—Rand does not take up the challenge of articulating how such poetry differs from so-called mood studies, thence, with that difference, how such poetry is art.

Silent Noon has a scene and an event.* (The idea microcosm comes quickly to mind; see Bissell 1997;* 2004.*) In this poem, existence and human act are told of. They are re-creations of reality and the basic draw of the consciousness aroused in the readers. Imagined perceptions and induced feelings are aroused by what is said in the poem and how it is said, all well integrated. I don’t have an example of what Rand was calling literary mood studies, so I don’t know how it might differ from this sort of poem. Do such mood studies concretize a theme, but without re-creation of reality, without any showing of existence and purpose driving consciousness?

This much is clear by Rand and satisfactory by me: an artistic selective re-creation is a re-integration, and for all art, not only literary, there will be a theme. For arts not literary, the theme will not be so fully expressed in words as in the medium, but it is there and is the large integrator.

The form of a building that is a work of art has a theme that is integral with the building’s purpose and its site. (See the pertinent passages from The Fountainhead I quoted in “Soul, Structure, Struggle,”* also Enright 2001, 347–53.) “Literature starts with concepts [expressed by the words] and integrates them to percepts—painting, sculpture and architecture start with percepts and integrate them to concepts” (Rand 1971, 1010). How the integrations in artistic drawing, painting, sculpture, and music differ from the integrations of percepts into concepts into propositions into theories, that is, into the integrations of knowledge, will be the subject of the next, final installment.

(To be continued.)

References

Bissell, R. E. 1997. The Essence of Art. Objectivity 2(5):33–65.

——. 2004. Art as Microcosm: The Real Meaning of the Objectivist Concept of Art. The Journal of Ayn Rand Studies (JARS) 5(2):307–63.

Branden, N. c. 1968. The Basic Principles of Objectivism. In The Vision of Ayn Rand. 2009. Cobden.

Cox, S. 2000. The Art of Fiction. JARS 1(2):313–31.

Crowther, P. 2007. Defining Art, Creating the Canon. Oxford.

Enright, J. 2001. Art – What a Concept? JARS 2(2):341–59.

Hessen, B. 1970. The Montessori Method III. The Objectivist 9(Jul):870–74.

Hunt, L. 2001. What Art Does. JARS 2(2):253–63.

Inui, T., and J. L. McClelland 1991. Attention and Performance XVI – Information Integration in Perception and Communication. MIT.

James, W. 1890 [1950]. The Principles of Psychology. Vol. 1. Dover.

Lillard, A. S. 2005. Montessori: The Science Behind the Genius. Oxford.

Lillard, P. P. 1996. Montessori Today. Shocken.

Minsaas, K. 2005. The Stylization of Mind in Ayn Rand’s Fiction. In The Literary Art of Ayn Rand. W. Thomas, editor. The Objectivist Center.

Montessori, M. 1906 [1948, 2004]. The Discovery of the Child. Aakar.

——. 1914. Dr. Montessori’s Own Handbook. Frederick A. Stokes.

——. 1948 [1976]. From Childhood to Adolescence. Schocken.

Rand, A. 1943. The Fountainhead. Bobbs-Merrill.

——. 1958 [2000]. Lectures on Fiction Writing. In The Art of Fiction. T. Boeckmann, editor. Penguin.

——. 1966. Art and Sense of Life. The Objectivist (O) 5(Mar):33–40.

——. 1968. Basic Principles of Literature. O 7(Jul):481–88; 7(Aug):497–504.

——. 1970. The Comprachicos I. O 9(Aug):881–91.

——. 1971. Art and Cognition. O 10(Apr):1009–17; 10(May):1025–47.

Stroud, S. R. 2011. John Dewey and the Artful Life – Pragmatism, Aesthetics, and Morality. Penn State.

Torres, L., and M. M. Kamhi 2000. What Art Is: The Esthetic Theory of Ayn Rand. Open Court.

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From Integrity to Calculus

Part 5

Before touring on to the promised conclusion of Rand’s work for integration, that is, to the rest of her esthetics, let us pull off the road for a scenic overlook.

Immanuel Kant was an integrating son-of-a-gun. Synthesis and unity are leading ideas in his Critical philosophy.

Human cognition has two stems, viz., sensibility and understanding, which perhaps spring from a common root, though one unknown to us. Through sensibility objects are given to us; through understanding they are thought. (KrV A15 B29)

By means of sensibility objects are given to us, and it alone supplies us with intuitions. Through understanding, on the other hand, objects are thought, and from it arise concepts. But all thought must, by means of certain characteristics, refer ultimately to intuitions, whether it does so straightforwardly or circuitously. (A19 B33)

The kind of presentation that can be given only through a single object is intuition. (A32 B47)

Intuition is that by which a cognition refers to objects directly, and at which all thought aims as a means. (A19 B33)

Our intuition, by our very nature, can never be other than sensible intuition; i.e., it contains only the way in which we are affected by objects. Understanding, on the other hand, is our ability to think the object of sensible intuition. . . . Thoughts without content are empty; intuitions without concepts are blind. Hence it is just as necessary that we make our concepts sensible (i.e., that we add the object to them in intuition) as it is necessary that we make our intuitions understandable (i.e., that we bring them under concepts). (A51 B76)

In Kant’s view, our experience of space does not consist of separate disconnected bits nor of less than three dimensions. We experience spatial form directly and as a unified whole. The presentation that is space is an intuition. That presentation is one whose constituent parts are not prior their whole, not parts whose accumulation makes their whole, and not instances under a concept of that whole. Rather, the parts of intuitive presentations, such as the parts of space, are by limitations and divisions of a singular, unified whole. All objects encountered or even possible in sensory experience have their places in that unitary space. Our abstract geometric reasoning, Euclidean geometry, is not disconnected from the space of our sensory experience (KrV A22–30 B37–45; B162; A140–42 B180–82; A162–66 B202–7; A223–24 B271–72; A712–24 B740–52).

Our experience of time, in Kant’s view, is also of a continuous unified whole. All objects, whether in sensory or inner experience, have their places in that one time. Physical things endure and have their motions in determinate ways obligating our perception of them in just those temporal and spatial ways (KrV A30–41 B46–58; A103–10; B136–40; B150–56; B162–63; A140–45 B181–85; A189–211 B232–56).

Kant’s faculty of understanding is a part of what has traditionally been called reason. The power of understanding is the power of concepts. Our rational faculties beyond the understanding are two, which Kant called the faculties of judgment and reason. The powers of reason, in this narrower sense, are of inference and cognitive management (KrV A130–31 B169–70; A686–87 B714–15; A723–38 B751–76). The three higher faculties work together, and each is a grand cognitive unifier (A67–234 B93–294; A669–704 B697–732).

Kant joined his philosophy of experience and understanding to fundamental physics (1786). He further elaborated our cognitive powers to enfold our esthetic capabilities (1790). In the power he called reason, he located the keys to morality. Between reason and morality, there is no divide (KrV A800–819 B828–47; 1785, 4:389–90, 403–4, 408, 411–13, 426–40, 446–48, 453–63; 1788, 5:15–16, 31, 42–57, 89–110, 119–21, 131–32, 134–48; 1797, 6:213–21, 375–78, 396–97). Yet, the reality of moral law, free will, and God largely transcend reality accessible by our intuition and understanding.

Kant inherited entrenched problematic divides in philosophy. Older among them would be the divide between the material world of the senses and the immaterial realm of thought, soul, and God; the divide between inclination and moral obligation; and the divide between reason and faith. More recent among them would be the divide between the deterministic world of science and the inner world of freedom; the divide between the value-absent world of reason and the value-full world of action and feeling; and the divide between things and their effects on us.

Where Kant attempted to smooth together those divisions, he succeeded little. Kant deepened and hardened the divide between inclination and moral obligation. However many ties he made between sensing and thinking, he deepened and hardened the divide between them. Moreover, he deepened and hardened the divide between things and their effects on us. His embrace and expansion of that divide entailed that all the unity and structure he would give to experience, understanding, and morality must come from the side of the subject. Space, time, objects, identity, causality, and moral reasons—all of them, systematically and fantastically, and seductively to many bright thinkers, must come from the constitution of an articulate subject striving for and touched by things as they are in themselves, things as they cannot be in our grasp, things with their own articulation unknowable to us.

Reinhold, Fichte, Schelling, and Hegel would innovate their own further integrations to bridge or dissolve problematic divides as they stood in Kant’s philosophy, but their solutions further increased the crafting of reality by subjectivity and, of course, continued to make room for the supernatural. Leonard Peikoff was partly right, though in considerable exaggeration, to call Kant’s philosophy anti-integration (2012, 34–35). That was part of Kant’s endeavor, a result overachieved, alongside his achievements of integration.

Rand’s world and ours is only one world. There is life, condition of consciousness and value. Human consciousness and valuation are open to human choice, within the one, natural world. In all the one world, existence is identity. Consciousness is identification, the grasp of what is and exclusion of what is not. Consciousness is an active process of differentiation and integration. We grasp the world in its given particulars, settings, dimensions, interactions, and magnitude structures. We detect and measure in perception, joined to the magnitude structures there in the world. Our concepts, at their best, rearticulate the world’s own articulation, including its magnitude structures. We are highly integrated in our cognitive powers and highly integrated with the only world, the one available for perception, comprehension, enjoyment, and action.

References

Kant, I. 1781, 1787. Critique of Pure Reason (KrV). W. S. Pluhar, translator. 1996. Hackett.

——. 1785. Groundwork of the Metaphysics of Morals. M. J. Gregor, translator. In Immanuel Kant – Practical Philosophy. 1996. Cambridge.

——. 1786. Metaphysical Foundations of Natural Science. M. Friedman, translator. In Immanuel Kant – Theoretical Philosophy after 1781. H. Allison and P. Heath, editors. 2002. Cambridge.

——. 1788. Critique of Practical Reason. M. J. Gregor, translator. In Practical Philosophy. 1996. Cambridge.

——. 1790. Critique of Judgment. W. S. Pluhar, translator. 1987. Hackett.

——. 1797. The Metaphysics of Morals. M. J. Gregor, translator. In Practical Philosophy. 1996. Cambridge.

Peikoff, L. 2012. The DIM Hypothesis. NAL.

Edited by Boydstun

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Part 6

. . .

In those oral remarks, Rand spoke of the positively ugly. I do not take that as an affirmation of the views of some modern estheticians that ugliness is some sort of fundamentally positive antithesis to beauty, another positive reality. (On history, analysis, and resolution of the issue, see PB 250–59.) It would seem most natural in Rand’s philosophy to see ugliness vis-à-vis beauty as parallel evil vis-à-vis goodness (AS 1024). That is, ugliness would be lack of beauty and not equally a positive reality, but a real lack and, moreover, a positive opposition to beauty.

. . .

I should add a note to that paragraph.

In her 1963, Rand characterized misery, disease, disaster, and evil as negatives in human existence and “not proper subjects of contemplation for contemplation’s sake. In art, and in literature, these negatives are worth re-creating only in relation to some positive, as a foil, as a contrast, as a means of stressing the positive—but not as an end in themselves” (38).

 

Within Géricault’s The Raft of the Medusa, one sees people who have already died, people in despair, and people with hope, waving to get the attention of a very distant ship. This painting fits squarely within what Rand described as having a subject containing negatives of human existence, yet also a positive in contrast, and worthy of contemplation for contemplation’s sake.

When it comes to the great negatives in life, I have some reservations concerning Rand’s idea that negatives are unworthy as whole subjects of a work of art. Sometimes there is widespread common background of the beholders, who know the subject is from a larger story with its road to a positive; such would be a painting showing only that the dead Jesus is being taken down from the cross. War scenes as subjects of artworks, containing no positive aspects in the subject, may have viewers who know some history from which the scene is taken and some evaluation of that history, possibly positive. On the other hand, a war scene—say, a massacre—as subject of a painting, might be effective in inducing the horribleness of such an event to a viewer and nothing more than that horror. I would not want to contemplate that painting so much that I put it on the living room wall opposite me just now, in place of the triptych of Monet’s water lilies spanning that wall. However, the well-executed massacre painting might be worth my contemplation in a memorial museum of the event or in an art museum, where one passes from one feeling of life to another.

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I have mentioned in the essay, conceptions among Rand’s contemporaries and predecessors closely akin to Rand’s idea that all active awareness consists of joint processes of differentiation and integration. I’d like to note one more kin for possible further study and comparison in the future. Here the notion appears in Aquinas, who writes that “truth resides in the intellect composing and dividing” (ST I 16.2).

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There is another prior kin of Rand’s idea that all active awareness consists of joint processes of differentiation and integration. This philosopher, like Rand,* was associating these processes with differentiation and integration in calculus. However, he kept quantity and quality in their old-fashioned separate departments, whereas Rand wove them together to a notable extent in her theory of concepts. Also, though he yoked the activity of pure mathematics to another conscious activity, the latter is not all consciousness, but a type of reflection in metaphysics.

 

“The object of metaphysics is to perform qualitative differentiations and integrations” (53). The philosopher is Henri Bergson, writing in “An Introduction to Metaphysics,” which issued in 1903 and appeared in English in 1912. He was familiar with writings of Spencer, though he does not mention any connection of this thought to the corresponding one in Spencer.

 

From Integrity to Calculus
. . .
Rand’s emphasis on integration in childhood cognitive and affective development was in tune with the thinking of Piaget (e.g. 1954). For Piaget the main English terms would be assimilation and accommodation, denoting technical concepts of his which, with coordination between them, are the essence of development . Here is an example from lectures in 1953:

Not only integration, but differentiation too, and by those names, loom large in Piaget’s studies in generalization, which he composed in the years after Rand’s ITOE (Campbell 2009, 157–59). Herbert Spencer, in his Principles of Psychology, had written: “Under its most general aspect therefore, all mental action whatever is definable as the continuous differentiation and integration of states of consciousness” (1855, 333; see Smith 1981, 119). Perhaps Rand had been familiar with Spencer’s theme of differentiation and integration in his thinking about biology and epistemology.
. . .

 

From Integrity to Calculus
. . .
I mentioned in Part 1 that Rand’s emphasis on integration in childhood cognitive and affective development was in tune with the thinking of Jean Piaget. I quoted from some 1953 lectures of his, wherein he defined two of his key concepts in development, which are assimilation and accommodation. Those are processes coming under the concept integration in Rand’s system. The similarity between Rand and Piaget is even plainer when we turn to John Flavell’s The Developmental Psychology of Jean Piaget (1963). Unlike the lectures from which I was quoting, Flavell’s text was easily available to Rand and her circle. (By the late ’60’s, my college friends and I were very wrapped up in ITOE and Piaget’s genetic epistemology and their relationships; works about Piaget’s work were readily available. See further, Hsueh 2009.)

In this text, one reads of Piaget’s concept of the schema in cognitive development, which is especially important for thought prior to language (see also Jetton 1998, 104–7). Rand makes brief contact with a schema of the image type when she speaks of the child’s iconic representation in drawing man as with oval torso, round head, and stick limbs (ITOE 13; Boydstun 1990, 16–18).
. . .
With Piaget’s concept of schema in hand, one reads a little later “A Piagetian schema . . . is always the product of the differentiation, generalization, and integration of earlier schemas . . .” (Flavell 1963, 73). Differentiation and integration, Rand will announce in 1966, are active processes essential to all consciousness as a state of awareness (ITOE 6).

That not only integration but differentiation should become highlighted when Rand came to presenting her theory of concepts and definition was natural. Why did she go on to announce at the outset of setting forth her theory of concepts that differentiation and integration are essential in all consciousness? To be sure, it is understandable that she should be setting her view of conceptual consciousness in wider patterns of consciousness. Then too, yes, differentiation and integration are implicit in the idea of consciousness as identification. And yes, years past Rand might have read Spencer saying differentiation and integration are essential to all consciousness.
. . .

Edited by Boydstun

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Plas,

 

Yes, Bergson thought we have an intellectual power he called intuition, and it is in contrast and in some tension with (but also in some support of) what we call the intellect, our power of concepts, judgments, and inference.

 

In Creative Evolution, he writes:

Consciousness, in man, is pre-eminently intellect. It might have been, it ought, so it seems, to have been also intuition. Intuition and intellect represent two opposite directions of the work of consciousness: intuition goes in the very direction of life, intellect goes in the inverse direction, and thus finds itself naturally in accordance with the movement of matter. A complete and perfect humanity would be that in which these two forms of conscious activity should attain their full development. . . . In the humanity of which we are a part, intuition is, in fact, almost completely sacrificed to intellect. It seems that to conquer matter, and to reconquer its own self, consciousness has had to exhaust the best part of its power. This conquest, in the particular conditions in which it has been accomplished, has required that consciousness should adapt itself to the habits of matter and concentrate all its attention on them, in fact determine itself more especially as intellect. Intuition is there, however, but vague and above all discontinuous. It is a lamp almost extinguished, which only glimmers now and then, for a few moments at most. But it glimmers wherever a vital interest is at stake. On our personality, on our liberty, on the place we occupy in the whole of nature, on our origin and perhaps also on our destiny, it throws a light feeble and vacillating, but which none the less pierces the darkness of the night in which the intellect leaves us. (291–92, Random House 1944)

 

 

That is a bit odd.

 

In addition to Bergson’s resonance with Rand on differentiation and integration, at least in a certain consciousness we have in metaphysics, I see also the following rough resonance:

All the operations of our intellect tend to geometry, as to the goal where they find their perfect fulfillment. But, as geometry is necessarily prior to them . . . it is evident that it is a latent geometry, immanent in our idea of space, which is the main spring of our intellect and the cause of its working. We shall be convinced of this if we consider the two essential functions of intellect, the faculty of deduction and that of induction.

 

. . . Prior to the science of geometry, there is a natural geometry whose clearness and evidence surpass the clearness and evidence of other deductions. Now, these deductions bear on qualities, and not on magnitudes purely. They are, then likely to have been formed on the model of the first, and to borrow their force from the fact that, behind quality, we see magnitude vaguely showing through. We may notice, as a fact, that questions of situation and of magnitude are the first that present themselves to our activity, those which intelligence externalized in action resolves even before reflective intelligence has appeared. . . .

. . .

Deduction, then, does not work unless there be spatial intuition behind it. But we may say the same of induction. . . . It is a far cry from a mechanical expectation and reaction of the body, to induction properly so called, which is an intellectual operation. Induction rests on the belief that there are causes and effects and that the same effects follow the same causes. . . . For the system of today actually to be superimposed on that of yesterday, the latter must have waited for the former, time must have halted, and everything become simultaneous: that happens in geometry, but in geometry alone. Induction therefore implies also that qualities can be superposed on each other like magnitudes. If, in imagination, I place the stove and fire of today on that of yesterday, I find indeed that the form has remained the same; it suffices, for that, that the surfaces and edges coincide; but what is the coincidence of two qualities, and how can they be superposed one on another in order to ensure that they are identical? Yet I extend to the second order of reality all that applies to the first. The physicist legitimates this operation later on by reducing, as far as possible, differences of quality to differences of magnitude; but, prior to all science, I incline to liken qualities to quantities, as if I perceived behind the qualities, as through a transparency, a geometrical mechanism. The more complete this transparency, the more it seems to me that in the same conditions there must be a repetition of the same fact. Our inductions are certain, to our eyes, in the exact degree in which we make the qualitative differences melt into the homogeneity of the space which subtends them, so that geometry is the ideal limit of our inductions as well as of our deductions. The movement at the end of which is spatiality lays down along its course the faculty of induction as well as that of deduction, in fact, intellectuality entire. (230–37)

 

 

Integration of this perspective with the circumstance of our broader geometries such as ordered geometry or affine geometry and with Rand’s cast of similarity classes of qualities in terms of measure-value suspensions could pay dividends.

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From Integrity to Calculus

Part 2

. . .

If one has studied calculus, one has perhaps studied series. The latter are a nice way to approach integrals in calculus.* Because Rand speaks in ITOE of some likeness to mathematics in the differentiations and integrations that go into the constitution of concepts, because operations called differentiation and integration are basic operations of calculus, and because Rand mentions integration in calculus when speaking about measurement (14), one may slide from sequence to series as that which is more like the constitution of Randian concepts. Do not make that slide. Stay with sequence, indifferently any one with the proper members.

 

There is impetus as well to slide from sequence/series to integral. Because the units of a concept are contracted into a single item, the concept, when their particular measure values along the concept’s dimensions are suspended, one can slip into thinking the contraction to be that which occurs in the evaluation of a definite integral in calculus. An example of the latter would be the evaluation of integrals over some figure to yield the single value that is the area of the figure or to yield the single value that is the volume of the figure. Do not make this slide either. No mathematical integration occurs along the measure-value dimensions of a concept when its measure values are omitted and the instances falling under the concept lose (in thought) their array in sequences in proper orders along those dimensions.

. . .

 

To the values of what the publisher notes to be inside the following book, I'd add the sheer joy of it: Inside Interesting Integrals by Paul J. Nahin (Springer 2014)

Edited by Boydstun

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