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Critical Faculty

Rand wrote that the higher animals “possess the faculty of retaining sensations, which is the faculty of perception. A ‘perception’ is a group of sensations automatically retained and integrated by the brain of a living organism, which gives it the ability to be aware, not of single stimuli, but of entities, of things. An animal is guided, not by immediate sensations, but by percepts. Its actions are not single, discrete responses to single, separate stimuli, but are directed by an integrated awareness of the perceptual reality confronting it. It is able to grasp the perceptual concretes immediately present, and it is able to form automatic perceptual associations” (OE, 1961, 19).

Nathaniel Branden maintained that higher animals can engage in perceptual abstractions (Mortimer Adler’s term), which is recognition of a number of sensible particulars as of the same kind. The animal does not have the further, human conceptual power, which entails “identifying explicitly of what the kind consists” (Psy of S-E 1969, 30).

Leonard Peikoff remarked that perceptual similarities are seen by animals, but humans go beyond that. We isolate similar concretes and form concepts standing for an unlimited number of concretes (Phil of Obj, Lecture 4, 1976).

David Kelley wrote that in judging of an object, which one is perceiving, that it is a such-and-such, one “classifies the object on the basis of similarities that have been explicitly isolated and named” (ES, 1986, 219).

Rand wrote that “an animal has no critical faculty; he has no control over the function of his brain and no power to question its content. To an animal, whatever strikes his awareness is an absolute that corresponds to reality—or rather, it is a distinction he is incapable of making: reality to him is whatever he senses or feels” (FNI, 1961, 17). She remarked orally that “an animal does not have the capacity to isolate critically the fact that there is something and he is conscious of it” (ITOE, c.1970, 246). An animal lacks the human ability “to apply introspection to the processes of one’s own consciousness and check them” (256).

In the preceding statements, notice these phrases especially: identifying explicitly /// isolate similar concretes /// explicitly isolated /// isolate critically the fact that there is something.

In their Scientific American article “The Biological Basis of Learning and Individuality” (Sep 1992), Kandel and Hawkins distinguish between two kinds of learning found in animals with nervous systems. Higher animals, including humans, learn in both ways. One kind of learning is called implicit, the other is called explicit.

Habituation, sensitization, and classical conditioning are forms of what is called implicit learning. Such learning “is slow and accumulates through repetition over many trials. It often [viz., classical conditioning] involves association of sequential stimuli and permits storage of information about predictive relations between events” (80). In contrast, “explicit learning is fast and may take place after only one training trial. It often involves association of simultaneous stimuli and permits storage of information about a single event that happens in a particular time and place; it therefore affords a sense of familiarity about previous events” (80). Explicit learning requires consciousness. Explicit learning evidently occurs only in vertebrates; it requires structures in the temporal lobes.

Within the broad category known as explicit learning, there are degrees of explicitness. In Explaining Behavior, Fred Dretske characterizes those degrees: Consider a rat that has learned through operant, or instrumental, conditioning that to obtain food it should press a bar. The rat could reasonably be said to be guided in its behavior by a belief that is relatively implicit. An implicit belief (explicit, but relatively implicit) is applied in fairly narrow circumstances. A fully explicit belief “can enter into combinations with other beliefs to generate a wide range of different actions” (1988, 118).

Prior to the acquisition of language, our beliefs lie in the more implicit zones of explicit learning and belief. Even at the relatively implicit levels of belief guiding a rat’s operantly conditioned behavior, it is possible to err and correct, although it be in a very local way. Nevertheless, I decline to call such beliefs knowledge. I incline to reserve knowledge for beliefs attaining truth where those beliefs are interconnected within and improvable by a critical consciousness holding them. This lies in the more fully explicit zone of explicit belief, a zone not far from linguistically held beliefs of an intact human mind.



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—a dozen 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.

Through the third week, I don’t think the infant has yet had a percept. She is not yet able to engage in explorations sufficiently controlled, and with enough memory, to have what we would call a percept. Sensory experience, discriminations, detections, responses—but no percepts. If no percepts, then no knowledge in the strict sense.

If the infant in the first couple of days, and even in the first three weeks, has yet to have a percept, do his sensory registrations and preferences, his reaches, and his expectations amount to identifications? Well, yes, she does seem bent on singling things out, especially Mother. It’s just that the strivings and discriminations do not yet coalesce into a percept.

Rand had expressed already in 1957, how she thought about the beginning of the infant’s mind in his first days after delivery. She interpreted the baby as having not yet grasped that A is A. This is the stage “ when a consciousness acquires its initial sensory perceptions and has not learned to distinguish solid objects.” To a baby at this stage, “the world appears as a blur of motion, without things that move—and the birth of his mind is the day when he grasps that the streak that keeps flickering past him is his mother and the whirl beyond her is the curtain, that the two are solid entities and neither can turn into the other . . .” (AS 1040–41)

There was some revision or refinement of Rand’s views on identifications of what between 1957 and 1966. She wrote in the earlier exposition: “The task of [man’s] senses is to give him the evidence of existence, but the task of identifying it belongs to his reason, his senses tell him only that something is, but what it is must be learned by his mind” (1016). To reason as an identifying faculty, Rand would later add the mind’s automatic formation of percepts from sensory inputs. “A percept is a group of sensations automatically retained and integrated by the brain. It is in the form of percepts that man grasps the evidence of his senses and apprehends reality. . . . Percept, not sensations, are the given, the self-evident” (1966, 5). Sensations are parts of the percept, but their integration to form the whole percept has already been accomplished by automatic brain processing. Consciously directed identification proceeds, reason proceeds, upon whatever percepts are possible to us.

By way of speaking truth, and in conformity with Rand’s later view, take the statement “his senses tell him only that something is, but what it is must be learned by his mind” thusly: the telling of the senses is automatic sensory processing up to but not including the automatic formation of percepts. (So for vision, that would include pickup and peripheral processing at the retinas, then preattentive processing [within one-fifth of a second] by LGN, V1–4, and others.) Let “learning by his mind” include some automatic learning, not only volitional, directed learning. However much Rand’s view shifted between ’57 and ’66, it is the latter view that settled as her final, considered picture.

In 1966 Rand added: “The (implicit) concept ‘existent’ undergoes three stages of development in man’s mind. The first stage is a child’s awareness of objects, of things—which represents the (implicit) concept ‘entity’. The second and closely allied sage is the awareness of specific, particular things which he can distinguish from the rest of his perceptual field—which represents the (implicit) concept ‘identity’”(ITOE 6). Only at a third stage are concepts attained, in Rand’s cameo portrait.

“The ability to regard entities as units is man’s distinctive method of cognition, which other living species are unable to follow.

“A unit is an existent regarded as a separate member of a group of two or more similar members. (Two stones are two units; so are two square feet of ground, if regarded as distinct parts of a continuous stretch of ground.) Note that the concept unit involves an act of consciousness (a selective focus, a certain way of regarding things), but that it is not an arbitrary creation of consciousness: it is a method of identification or classification according to the attributes which a consciousness observes in reality. . . . The criterion of classification is not invented, it is perceived in reality. Thus the concept “unit” is a bridge between metaphysics and epistemology: units do not exist qua units, what exists are things, but units are things viewed by a consciousness in certain existing relationships.” (ITOE 6–7)

Leonard Peikoff renders Rand’s idea of an existence-identity sequence in early development on page 12 of OPAR. He refers to this idea of Rand’s as something of a suggestion, and I think that is fair enough.

“The three axioms [axiomatic concepts] are not [implicitly] learned by the developing child simultaneously. ‘Existence’, Miss Rand suggests, is implicit from the start; it is given in the first sensation. To grasp ‘identity’ and (later) ‘consciousness’, however, even in implicit form, the child must attain across a period of months a certain perspective on his mental contents. He must perform, in stages, various processes of differentiation and integration that are not given in the simple act of opening his eyes.

“Before a child can distinguish this object from that one, and thus reach the implicit concept of ‘identity’, he must first come to perceive that objects exist. This requires that he move beyond the chaos of disparate, fleeting sensations with which his conscious life begins; it requires that he integrate his sensations into the percepts of things or objects. . . . At this point, the child has reached, in implicit form, the concept of ‘entity’.”

(The summer after this book was published, I attended a two-week conference at which Dr. Peikoff was presenting a series of lectures. I recall that at the dining hall, he had a little girl in tow. His parental experience of infant development was recent.)

I think Peikoff’s piecing together of Rand’s existence-identity sequence has a certain neatness. He has identity wait for entity, where entity is in the form object (all these notions being only implicitly held, once they are held at all). However, I do not think it is reasonable to have the identity that is at hand with this versus that wait on object (say, four months). There are other existents with their this-versus-that assimilated by the infant mind sooner than that.

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Capturing Quantity

Part 1

I completed Capturing Concepts and published it in 1990. In its list of References, there is much relevant research in developmental cognitive psychology published through 1989. There is only one work in that list published in 1990, and that is the expanded second edition of Introduction to Objectivist Epistemology. The expansion included new material, transcripts from recordings of Rand’s seminar on ITOE a couple of years after ITOE had been published in The Objectivst (1966–67).

I have often pointed out that Rand could be wrong in some ways concerning how concepts are formed, yet, in the logical presuppositions of such a proposed process, land on true character of all concretes and a true capability we possess for conceptualizing them around that character. In particular Rand can be wrong in supposing all concepts at all stages of our cognitive development are formed, at least implicitly, by a process of measurement omission, yet be correct in the presupposition that all concretes stand in measurement relations to other concretes and that we can and should conceive things with the appropriate measurement relations recognized and their particular values, within appropriate ranges, omitted for the contraction of concretes into concept.

Recall Rand’s picture of our earliest conceptual and mathematical capabilities. In coming to concepts and language we moved from regarding things only as entities to regarding them also as units.

"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.

"This is the key, the entrance to the conceptual level of man’s consciousness. The ability to regard entities as units is man’s distinctive method of cognition, which other living species are unable to follow.

"A unit is an existent regarded as a separate member of a group of two or more similar members. (Two stones are two units; so are two square feet of ground, if regarded as distinct parts of a continuous stretch of ground.) Note that the concept unit involves an act of consciousness (a selective focus, a certain way of regarding things), but that it is not an arbitrary creation of consciousness: it is a method of identification or classification according to the attributes which a consciousness observes in reality. . . . The criterion of classification is not invented, it is perceived in reality. Thus the concept 'unit' is a bridge between metaphysics and epistemology: units do not exist qua units, what exists are things, but units are things viewed by a consciousness in certain existing relationships.

"With the grasp of the (implicit) concept 'unit' man reaches the conceptual level of cognition, which consists of two interrelated fields: Conceptual and Mathematical. The process of concept-formation is, in large part, a mathematical process." (ITOE 6–7)

In “Capturing Concepts” I cited evidence from research that indicates our earliest concepts could not be of the Randian form for concepts. Subsequent research on infant cognition of quantity calls for a revision of my report and conclusion. Here is the pertinent part of what I had written in that 1990 paper:

"There is much evidence to suggest that preschoolers typically contrast objects in terms of holistic, dimensionally nonspecific relations of magnitude. Objects intense in size and color are grouped together and segregated from objects not intense in both size and color. It is moreness and lessness across both dimensions that matter for these young children. They are observed trying to seriate objects on two dimensions at once. Quite possibly, they have not only notions of global magnitude polarities, but notions of dimensionally undifferentiated directions of difference. The preschooler’s understanding of global magnitude, however, does not entail an understanding that greater than and less than are necessarily opposing directions of difference. That understanding comes later, with an understanding of the ordering of values along single dimensions (Smith 1989, 165–72; Halford 1984; Chapman and Lindenberger 1988; Minsky 1985, 99–107, 149, 241).

"The formation of concepts according to Rand’s formula of measurement omission requires a dimensional understanding of isolated attributes. The dimensions must be accessible but might be scalable only ordinally (ITOE 31–33, 14–15). A child knows size as a dimension when he regards big and little as attributes of a single kind; part identities have then become organized by dimensional kind. He knows size as an ordinal dimension when he regards bigger and littler as necessarily opposing directions of difference. Preschoolers have only a very incomplete and fragmented understanding of these quantitative relations. They do have the beginnings of such understanding (Smith 1989, 168–72; Gelman and Gallistel 1986, 160–67), and perhaps there is enough in some of the preschooler’s domains of knowledge to begin to form concepts along the lines of Rand’s formula.

"It is doubtful, though, that preschoolers could hold the full-blown Randian form of concepts; they cannot yet view numbers algebraically. (Could they nonetheless view different values along some dimensions as variables?) It is doubtful, furthermore, that the preschooler’s concepts engage universals full-force; they do not yet realize that counting numbers are endless (Gelman and Meck 1983, 357–58)." (Boydstun 1990, 33–34)

Rand had written that “man’s mathematical and conceptual abilities develop simultaneously” (ITOE 9). Research in recent years on infant cognition supports that contention. I am going to present those research results and look at what they say about the relation of first lexical concepts to the quantitative knowledge possessed by the infant at that time. Rand had gone on to say in support of her contention just quoted that “a child learns to count when he is learning his first words.”

That latter contention was incorrect. As I noted in “Universals and Measurement” (2004):

"The child has gone far beyond learning first words (roughly months 12 to 18) by the time she is learning to count. By 30 months, the basic linguistic system has become established and is fairly stable (Nelson 1996, 106). Not until around 36 months or beyond does the child have an implicit grasp of the elementary principles of counting: assign one-label-for-one-item, keep stable the order of number labels recited, assign final recited number as the number of items in the counted collection, realize that any sort of items can be counted, and realize that the order in which the items are counted is irrelevant (Gelman and Meck 1983; Butterworth 1999, 109–16)." (301n40)

This error of Rand’s was pointed out to her in her epistemology seminar (c. 1971). Professor H (who may be Michael Berliner) reminded Rand of her remark on counting and first words. She replied that counting is almost simultaneous with first words, but not quite.

H: “You say [counting] occurs shortly afterward. From what I have observed, it seems to occur quite a bit later, so that it seems to be a much higher-level process.

AR: It isn’t so much higher-level, but the fact is that you cannot begin to count objects until you have learned to distinguish them, and you cannot distinguish them firmly until you have learned some words—i.e., formed some concepts. Therefore, it is part of the same general development. But a child does have to acquire some conceptual vocabulary, meaning: learn to identity some concretes in reality, before he can begin to count.

H: I was taking it too literally.

AR: No, if I said “when he is learning his first words,” I meant in the same general period of development. (ITOE Appendix, 200)

The question of whether our earliest express concepts, our concepts at the one-word stage of language acquisition, are simultaneous with some quantitative knowledge need not hang on ability to count. It turns out we had some other quantitative knowledge at the dawn of our concepts. We can give Rand a little gold star, or anyway a silver star, for a little prescience in thinking the conceptual and mathematical work abreast each other from the outset.

It turns out also that I deserve a little star for something I said about units in “Universals and Measurement.”

"Concepts are mental integrations of 'two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted' (ITOE 13).

"The units spoken of in this definition are items appropriately construed as units by the conceiving mind. They are items construed as units in two senses, as substitution units and as measure values (ITOE Appendix, 184, 186–88). As substitution units, the items in the concept class are regarded as indifferently interchangeable, all of them standing as members of the class and as instances of the concept. Applied to concept units in their substitution sense, measurement omission means release of the particular identities of the class members so they may be treated indifferently for further conceptual cognitive purposes. This is the same indifference at work in the order-indifference principle of counting. The number of items in a collection may be ascertained by counting them in any order. Comprehension of counting and count number requires comprehension of that indifference.

"The release of particular identity for making items into concept-class substitution units is a constant and necessary part of Rand's measurement-omission recipe. But this part is not peculiar to Rand's scheme. What is novel in Rand's theory is the idea that in the release of particular identity, the release of which-particular-one, there is also a suspension of particular measure values along a common dimension." (Boydstun 2004, 273–74)

My new little lucky star is on account of research indicating that in infancy we already had two branches of quantitative knowledge that are precursors to and instruments for our later grasp of units in the dual sense of substitution units and measure-value units. Remember, this new material we are going to assimilate concerns only genesis of our human cognition, not analysis of its structure, actual or desirable. Still, the stars are sweet.

My window into recent research bearing so heavily on our genesis questions pertinent to Rand’s theory of concepts is Susan Carey’s The Origin of Concepts (2009).

(To be continued.)



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

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

Butterworth, B. 1999. What Counts: How Every Brain is Hardwired for Math. New York: Free Press.

Carey, S. 2009. The Origin of Concepts. Oxford.

Chapman, M., and U. Lindenberger 1988. Functions, Operations, and Decaloge in the Development of Transitivity. Developmental Psychology 24(4):542–51.

Gelman, R. and C.R. Gallistel 1986 [1978]. The Child’s Understanding of Number. Harvard.

Gelman, R., and E. Meck. 1983. Preschoolers’ Counting: Principles before Skill. Cognition 13:343–59.

Halford, G. 1984. Can Young Children Integrate Premises in Transitivity and Serial Order Tasks? Cognitive Psychology 16:65–93.

Minsky, M. 1985. The Society of Mind. New York: Simon & Schuster.

Nelson, K. 1996. Language in Cognitive Development. Cambridge: Cambridge University Press.

Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. H. Binswanger and L. Peikoff, editors. Meridian.

Smith, L. 1989. From Global Similarities to Kinds of Similarities. In Similarity and Analogical Reasoning. Vosniadou and Ortony, editors. 1989. Cambridge.

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Capturing Quantity

Part 2

Psychologists cited in my 1990 and 1991 include Renée Baillargeon, Randy Gallistel, Rochel Gelman, Alan Leslie, and Elizabeth Spelke. With those writers, Susan Carey records her agreement “that the cognition of humans, like that of all animals, begins with highly structured innate mechanisms designed to build representations with specific content. I call the mental structures that represent them ‘core cognition’” (2009, 67).

It is the word content in Carey’s description of these core cognitions that makes them decidedly innate cognitive systems (cf. Jetton 1995, 73; Touchstone 1996, 117–18). Those contents are not obtained by individual learning. This would draw a smile from Leibniz, but frowns from Locke, Helmholtz, Piaget, and Rand. It gets frowns from some of Carey’s colleagues as well (2009, 68–69). The dissenters argue that all cognitive content and the cognitive systems composing that content are learned ultimately from elementary perceptual and sensory-motor experience. For the pure empiricists, it is learning all the way down. Rand was of that view.

"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, 191; cf. Bennett 1966, 98–99; Touchstone 1996, 166)

Carey works assiduously, through laboratory research, to specify innate core cognitive systems. These cognitive systems she identifies as core are specific to different domains of experience. There are only a handful of these core systems (at least only a handful discerned so far), and two pertain to cognition of magnitude. Another pertains to object, another to intentional agency. All innate core cognitive systems are perceptual input analyzers, all aid acquisition of experientially learned skills and knowledge, and all remain in use throughout child and adult life. These innate core cognitive systems are to be understood as gotten by evolution and instituted in the individual simply by brain maturation, which may or may not be sufficient by time of birth (Carey 2009, 55–69; cf. Macnamara 1986, 72–74, 191–92).

In an extended sense, that could be said to be a purely empirical view, with the human species and its earlier lineages being said to have learned by population experience (cf. Jetton 1995, 67–68). But Carey does not select that wide arc as a type of learning, and I shall accede to her denomination of the core cognitions as innate, rather than empirical. It should be understood that these innate cognitive systems and their outputs upon their analyses of experiential inputs are devoid of the spiritualism endemic in the old rationalist theories of innate ideas.

I should mention that Carey does not confine the possibility of innate representations to the core cognitive systems. We may also have innate cognitive systems poised to deliver content (e.g. the representation cause) that spans various core cognitions (Carey 2009, 215–18, 240–46; see further, Baillargeon and Carey 2012).

Integrating the research reported and analyzed in Carey 2009, Origin of Concepts (OC), with Rand’s thinking on concepts and mathematics in infant and child development is not most importantly concerned over the issue of whether the core systems Carey calls innate are best said to be innate in Carey’s mild and purely natural sense. What matters most are sequence, functions, interrelations among skills, and interrelations among contents.

There is a key choice of terminology made by Carey to which I strongly object. She calls the innate core cognitions conceptual. Their contents are concepts, in her choice of terminology.

Core cognition has rich integrated conceptual content. By this I mean that the representations in core cognition cannot be reduced to perceptual or sensori-motor primitives, that the representations are accessible and drive voluntary action, and that representations from distinct core cognition systems interact in central inferential processes. (OC 67; see also 97)

Carey goes on to specify that “the format of representation of core cognition is iconic rather than involving sentence-like symbol structures” (OC 68; further, 104–5, 134–35, 195–96). My instant response would be: “Then those representations are schemas, not concepts” (Flavell 1963, 52–58; Jetton 1991, 70–72; 1998, 102–11). Very occasionally, Carey herself refers to a preverbval infant general representation by schema rather than by concept (OC 195–96, 241, 244).

It has been my view, consistent with Rand’s, that what C. S. Peirce and Terrence Deacon call purely iconic representations are not concepts. They are prerequisites for indexical and fully symbolic representations, but only the latter two are concepts by Rand’s account and mine. With more reflection, I need to revise my view, though it will yet be a view consistent with Rand’s and one under which I shall yet object to Carey’s choice concept, rather than schema, as name for the representations exercised in core cognitions.

The photo of me displayed at the top of the left margin is an iconic representation of me because it resembles me, and one can associate it with me on that account. It occasions also an indexical representation of the approximate elevation of the sun above the horizon, indicating that circumstance in one’s association through grasped causal relations (rather than simply through perceived similarity). That the photo is associated with photo is by convention, and this is symbolic representation. So say Peirce and Deacon. Digging further into Carey’s book, I find that when she wrote iconic she meant it in the more general sense of the pictorial, which would include not only the iconic in the strict Peirce-Deacon sense, but the indexical as well.

To now I have thought of Rand’s view of earliest concepts, those marked in the single-word stage of language acquisition, as being indexical representations, not symbolic representations in the strict sense. That is incorrect, and it is an internal tension in my 2004. On the one hand, I had written:

By 24 months, the child is using two-word utterances such as . . . “I know [how to do it]” (Bremner 1994, 252–53; Nelson 1996, 112, 124–25). Up to about this time, when grammar begins to develop, “words learned remain tied to their world models and do not form systems of their own” (Nelson 1996, 128). In terms of Deacon’s iconic, indexical, and symbolic levels of representation (1997, 70–83), I should say that concepts at the single-words stage are indexical representations, and these concepts will become symbolic representations with the onset of grammar. Rand’s conceptual level of representation cuts across Deacon’s indexical and symbolic levels. (Boydstun 2004, 300n37)

On the other hand, I had noted a way in which Rand’s thesis “a concept identifying perceptual concretes stands for some implicit propositions” is true even for the toddler at the single-word stage of language acquisition (ITOE 48, 21). He cannot state propositions, but by the time he can say ball (ba) for a ball, the word can mean a suite of image and action schemas into which he knows a ball fits (Boydstun 2004, 290). I was correct in thinking the representation of named classes of objects at the single-word stage to be indexical, but incorrect in thinking of them as not also symbolic in the strict sense of Peirce-Deacon. At the single-word stage the child has gotten symbolic representation, in which the word is a conventional representation for the class, not associated with the class simply by support of causal connection or simply by similarity. Having the reliable word for the concept caps the concept’s definiteness, even though the word is not yet set in grammatical expressions. So it remains that, contrary to Carey, I shall reserve concept for the representation in the first word of language competence and beyond. Before that, schema (see also Mandler 2010).

I reported in 1991:

"Jean Piaget (1954) concluded from his investigations that infants first acquire a notion of enduring objects at nine months of age. Until then, if a toy attractive to the infant were covered with a cloth, the infant would make no attempt to lift the cloth and grasp the toy, even though she were capable (after the fifth month) of performing each of these actions. Piaget concluded that, until the ninth month, the infant does not regard the toy as an enduring entity that continues to exist while not in view; only after the ninth month does the infant begin to infer the continued existence of objects.

. . .

"Piaget’s observations have been reconfirmed many times, but Piaget’s interpretation is now widely questioned. Infants younger than nine months may fail to lift the cover and grasp the toy not because they lack a notion of enduring objects, but because they are not yet able to coordinate such a means-ends sequence (Baillargeon 1986, 38–39). It is likely that this inability is due to immaturity of the frontal cortex (Diamond 1989). Recent experiments, not demanding coordination of means-ends-sequences, indicate that infants as young as six months (Baillargeon 1986), even four months (Baillargeon 1987), understand not only that objects continue to exist when not seen, but that moving objects continue along trajectories when not seen and that those trajectories could not be through other solid objects." (Boydstun 1991a, 38–39; see also 1991b, 6–7 and Friedenberg 1993, 32–33.)

The experimental methods of Baillargeon and other contemporary developmental cognitive psychologists were not available to Piaget. Since my reports in 1990 and 1991, those experimental methods have shown that infants have much of what Piagetians called “object concept” (e.g. Flavell 1963, 129–35) as early as two months. (On brain maturation factors, see also Johnson 2005, 82–85, 146–57, 172–75.) “By two months of age, infants represent objects as spatio-temporally continuous. Not only do they represent objects as continuing to exist behind barriers, they also take evidence of spatio-temporal discontinuity as evidence for numerical distinctness” (OC 40; see also Aguiar and Baillargeon 1999; 2002).

Carey analyzes the laboratory evidence that during the first two months, infants acquire the ability to represent and quantify objects from sensory and perceptual primitives. She finds that that evidence can be interpreted otherwise, and she gives some empirical reasons to conclude that “perceptual input analyzers that yield representations of objects are most likely innate” (OC 55; see 55–63).

In working memory, we have object representations used to track their individual identity as we perceive them in different locations. These representations are called object files (Kahneman, Treisman, and Gibbs 1992. Characteristics of our operations with these representations include: "(1) privileging spatio-temporal information over property/kind information in individuation and computations of numerical identity; (2) a set-size limit on the number of objects that may be simultaneously attended to and represented in working memory (on the order of three or four); and (3) the capacity to track individual objects through occlusion, with specific spatio-temporal information distinguishing cessation of existence from occlusion" (OC 71–72).

Carey adduces evidence that those our adult characteristics of object representation are also found in young infants (OC 72–87). The set-size limitation increases during infancy, reaching the adult level by 10 months.

Two core cognitions underlie infant performance in tasks reflecting sensitivity to magnitude. One is parallel individuation of small sets, which supports an infant and adult sensitivity to number. This cognition entertains a small number of object-file representations. Object files of some individuals are held in one bin in working memory while others are held in another. In one experimental task eliciting these operations, an infant watches as graham crackers are placed into each of two opaque containers, which have been shown empty to the baby prior to the deposits. In view of the baby, one cracker is placed in one container. One cracker and another are placed in the other container. Baby (10–12 months) is released to crawl to the containers. Baby heads for the container with two crackers. (Monkey’s also perform such tasks.) Next 2 versus 3. Baby succeeds again. 1 versus 3? Success. 1 versus 4? Failure. 2 or 3 versus 4? Failure. (Feigenson, Carey, and Hauser 2002).

The limits on the number of objects that can be simultaneously attended to and represented in working memory are manifest in those results. This characteristic indicates that in such tasks the infants’ sensitivity (implicitly) to number is supported by parallel individuation in small sets, rather than by a second core cognition concerning magnitude: analog magnitude representations of number. In such cognitions, ability to discriminate any two magnitudes is a monotonically increasing function of their ratios. A ratio of 4:1 is easier to discriminate than a ratio of 2:1 or 3:2. The analog magnitude system is “an evolutionary ancient representational system in which number is encoded by an analog magnitude proportional to the number of objects in the set. These representations support computations of numerical equivalence and numerical order” in preverbal infants (OC 123).

The individuals whose set sizes are discriminated by that core cognition can be objects or tones (OC 124). Cognition of magnitudes through parallel individuation of small sets has been elicited in preverbal infants using not only objects, but events and tones (OC 148–49).

It is parallel individuation in small sets I expect for first rung in the ladder to units in the substitution sense in Rand’s theory of concepts. It is analog magnitude representations of number I expect for first rung to units in the measure-value sense in Rand’s theory. We shall see.

(To be continued.)



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Capturing Quantity

Part 3

“With the grasp of the (implicit) concept “unit” man reaches the conceptual level of cognition which consists of two interrelated fields: Conceptual and the Mathematical. The process of concept-formation is, in large part, a mathematical process” (ITOE 7).

Rand registers the substitution sense of units by the locution some . . . any. That expression is logical quantification, which is a logical support of the mathematical concept number and which is applicable to mathematical objects and their principles. An elementary mathematical application would be at the start of a proof saying, “Construct any triangle ABC.” One understands that one needs to construct some triangle, but it could be any triangle and that each vertex is to be labeled one (some) of A or B or C, but any one of them, provided the letter is not already selected for another vertex. Or a proof might begin by saying, “Consider any rational number q.” One understands that one is reflecting on some rational number, that it can be any rational number, and that although the particular number is unspecified, it is to be regarded as a particular number throughout the proof.

Rand applies the logical quantification some . . . any three times to any concept in her mensural objectivist theory of concepts. Firstly, a concept means some one of its instances, any one (also true, degenerately, for proper-noun concepts). Secondly, the some . . . any quantification is used in regarding common distinctive dimensioned characteristics of items indicated by a defined concept in her theory. For example, Rand takes locomotion and consciousness to be the basic common characteristics distinguishing animals within the broader class of living things. (Sponges are animals, but she was apparently taking animal somewhat more narrowly.) An animal must possess some form of locomotion and consciousness, but it can be any form of these. Similarly, being a living thing, an animal must possess some form of internally generated actions, but these may be any such form, including the actions that are locomotion and consciousness (ITOE 24–25).

Thirdly, the logical quantification some . . . any is applied to measurement values of the common distinctive dimensioned characteristics. For example, the form of consciousness of a type of animal must have some scope, but may have any finite scope (ibid.). Then too, functions, including functions of consciousness, have measurable dimensions. So in saying the consciousness of a type of animal serves to preserve the life of the individual animal and its species, one is saying consciousness of the animal type has a measurable scope in perpetuating the life of that animal. To be an animal, it must have some scope of consciousness and some coordinate scope to which consciousness preserves its life, but any measure-value of such scopes will qualify for animal (ITOE 22; OE 18–21).

In Part 2, I introduced the core cognitions Carey discerns pertaining to quantity, or magnitude: (i) parallel individuation of small sets and (ii) analog magnitude representations of number. Carey observes that those core cognitions require something like linguistic and logical quantification in the prelinguistic infant. She calls this cognitive power set-based quantification. The term set here and in (i) means only definite collection. The prelinguistic infant must be able to select collections of individuals to quantify arithmetically in analog magnitude number representations (OC 136–37). That ability to select collections in the tasks exhibiting either of the two core quantitative cognitions need not have “the representational force of set-based quantification” (OC 255). In neither of those two core cognitions are prelinguistic symbols required for the logical quantifications marked in English by singular/plural constructions or by the terms each, every, any, some, more, a, and so forth (OC 254–57).

Along with many other authors, Carey thinks “human evolution bequeathed us with innate knowledge of language, which we can imagine takes the form of a language acquisition device (LAD). The LAD guides the learning of natural language syntax, morphology, and phonology” (OC 247).

From experimental results, Carey concludes that the process of learning one’s native language likely helps to avail and deploy the logical cognition that is set-based quantification. But that logical cognition is not a creature contrived by language.

The young infant distinguishes between individuated entities such as objects and nonindividuated entities such as sand, and compares representations of sets of objects with respect to both continuous and discrete quantification. Also, at least by the time the infant is about a year old, he or she has constructed her first object kind-sortals, distinguishing kind-sortals from properties in categorization, individuation, inductive inference, and language. . . . Natural languages most likely have the quantification machinery they do because of the quantificational resources of prelinguistic mental representations (both ontogenetically and phylogenetically), as these were drawn upon in creating the language-acquisition device. (OC 284–85)

Carey allows that the prelinguistic capability that is set-based quantification underlies natural language quantification, although prelinguistic set-based quantification does not invoke symbols marking it, such as will be available in a natural language such as English (OC 296). Our language has markers for distinguishing singular and plural, and it has approximate mathematical quantifiers such as more, some, and many. As we know from logic texts, the logical quantifiers a, some, and every (or all or any) work in the absence of the successor function. The latter takes thought beyond logic to mathematics, specifically to the infinite set of natural numbers, the counting numbers.

Part 1 included a reiteration (from Boydstun 2004, 301n40) that it is not until around age 3 that children demonstrate an implicit grasp of the principles of counting. Those five stated principles, which are from Gelman and Gallistel 1978, do not include an implicit grasp of the successor function nor a generative base for an inexhaustible supply of number names. By age 5, some children will have extended the number to which they can count to 20. They do not yet comprehend that the integers are endless, but they can count and have been doing so for a year or two (Gelman and Meck 1983, 357–58).

By age 3, the child has been talking and thinking in concepts for a couple of years. First year, first word is the rule of thumb. Rand’s measurement-omission theory of concept formation is readily adaptable to that circumstance. Grasp of units under a concept in their substitutional aspect need not include grasp that items potentially qualified for placement under the concept are endless. Neither does it require the 3-to-4 year old’s implicit mastery of the five principles of counting. The limited scope of prelinguistic set-based quantification implicated in parallel individuation of sets may suffice for the some . . . any logical quantification Rand proposed as implicit in the grasp of a first common-noun word such as ball. (That word and the concept it marks is an example of what Carey calls an object kind-sortal in the last block quote above [see OC 263–83].)

Rand’s idea that entry into conceptual thought is attained with implicit grasp of the concept unit includes the four claims designated below. I shall use the concept ball as example of an earliest concept because that was the first word of the child whose development I have been privileged to witness since his birth.

LS – Substitution unit is logically implicit in ball.

LM – Measure-value unit is logically implicit in ball.

ES – Substitution unit is implicitly employed in comprehending ball.

EM – Measure-value unit is implicitly employed in comprehending ball.

The first two are true, and between them it is LM, when greatly generalized, that is distinctive in Rand’s analysis of concepts, including universal concepts. However, the present essay is concerned only with ES and EM. I have related the result that ES looks true, based on research on infants, as reported in Carey 2009. ES is not a claim about concept formation uniquely Rand’s. When greatly generalized, EM is Rand’s distinctive claim concerning concept formation.

Carey’s core cognitions and representations in preverbal infants are tacit; they are not articulated in language. They are privately articulated iconically, wordlessly. Nevertheless, for the mind cognizing them—preverbal infant, nonverbal animal, or adult human—they are conscious articulations, they are explicit in that mind, even if wordless. Additionally, there is knowledge in that mind “embodied in the computations carried out over explicit representations” (OC 64). This is like our meaning in saying that a competent speaker of natural language is systematically governed by certain rules of syntax without being able to state those rules (Boydstun 2004, 300n35; cf. Campbell 2002). When Carey argues that a cognitive power of set-based quantification is required for the two core cognitions pertaining to magnitude, she is arguing not only a logico-mathematical entailment, but an implicitly operating cognition. Similarly, Carey’s elementary set-based quantification is implicitly in operation in core cognitions of quantity, or magnitude. Such implicit set-based quantification is required for ES.

Preverbal set-based quantification is operationally implicated in both ES and in parallel individuation of small sets for the infant at 10-to-12 months. At that age, parallel individuation of small sets is limited to having no more than three items. Why is ball applied to any of more than 3 of them in the infant’s experience? I think this is because applying ball to one in hand or one recalled does not create the demands on working memory present in the parallel-individuation experiments eliciting comprehension of numbers to 3. Application of ball requires only recognition memory and long-term memory. (On the developmental time line of explicit memory, see Hartshorn et al. 1998.)

I should head off a natural error. It is natural to think that parallel individuation of small sets is a primitive ability to count to 3. It is not. At its adult level of development, parallel individuation of small sets has advanced from 3 to 4. A child with the ability to count to say 20 has a numerical skill and comprehension markedly distinct from that in parallel individuation of small sets. (Note that the other core cognition pertaining to quantity—analog magnitude representation of number—is not limited to a small number of distinct ranked magnitudes; see also Piazza 2011.)

Moreover, an adult with Williams syndrome can have some command of language, but be unable to count (Johnson 2005, 144). That is not a counterexample to ES. The individual with that syndrome could possess the (delayed) power of parallel individuation of small sets and the set-based quantification that entails, which latter is required for ES to be true. The Williams individual has working memory, so I venture the possibility that these individuals have the power of parallel individuation of small sets and its implicit set-based quantification prior their first word.

I cannot venture as a possibility with any backing that the Williams child has the power of analog magnitude representation of number, for these children have deficits in spatio-visual skills. If such a child were not in possession of the perceptual ratio-scaling discrimination characteristic of analog magnitude representation of number, yet had ball, then that child’s concept of ball would seem unlikely to require EM. It could still be the case that ball for the normal one-year-old might require such perceptual ratio discriminations in addition to set-based quantification, application of both to balls of different sizes, and a grasp of the invariant shape across ball size. That is, consistent with research, it could still be the case that EM is at work in the first word of the normal child.

Notice that capabilities for parallel individuation of small sets and capabilities for analog magnitude representations of number, as well as the set-based quantification operative in them, are possessed by animals other than the human. It is a live option that Rand was right in saying that grasp of unit is key to conceptual capability. Concerning neurological distinctiveness supporting that distinct human grasp, I would look to the secondary sensory cortices (greater surface relative-area in humans compared to chimps) and to language areas (OC 268–85; Dehaene and Brannon 2011, chapters 17, 19; Johnson 2005, chapter 7; Deacon 1997).

Rand’s EM is probably on the mark for ball and other early concepts. It is off the mark for colors. When the toddler tells you a light on the Christmas tree to which you point is blue, that the next is red, and so forth, there is no EM at work. The fact that we can, with scientific advance, resolve colors into dimensions and scale them is not to the point of the operationally implicit (ITOE 13–14). There are other means available for delivering similarities given in perception for conceptual identification, means that do not operationally entail implicitly the scaling of quantities along dimensions (Boydstun 2004, 291–93, Analytic Constraint, and Nosofsky 1992 cited therein; Jetton 2011, 225–27). ES is enough in such cases.

Rand was off the mark, as I have noted before, in her complete generalization of LM and EM to all concepts. They cannot hold for concepts logically presupposed by measurement in its essentials (e.g. essentials in Suppes 2002, 25–26). For instance set-based quantification in schematic form is implicit in all early concepts for which LM and EM hold, and not the other way around. All the same, Rand’s theory of concept genesis has more (even more) consonance with cognitive psychology’s understanding of the infant and toddler today (2013) than with that understanding twenty-three years ago, when I first opened Rand’s work in that light.



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

Campbell, R. L. 2002. Goals, Values, and the Implicit: Explorations in Psychological Ontology. JARS 3(2):289–327.

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Gelman, R., and E. Meck. 1983. Preschoolers’ Counting: Principles before Skill. Cognition 13:343–59.

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Suppes, P. 2002. Representation and Invariance of Scientific Structures. CSLI.

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