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Capturing Concepts

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Addendum to #24

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). As conveyed in the preceding posts to this thread, the attainment of “object concept” is an intensely researched development. Piaget’s findings on this were famous at the time Rand was writing her treatise.

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. (And if there were not, it would be back to the drawing board on my Barnes-inspired general argument about identification in #24!)

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Thomas, I think it is too coarse-grained to run with only the idea extraction for the various levels of perceptual processing. Even the little features in preattentive perceptual awareness (within not more that one-fifth of a second) have not been simply extracted. At the retina, there is extraction (and some helpful enhancement by lateral-inhibition wiring), but on to the thalamus and cerebral cortex, with all their reentrant connectivity, “extraction” leaves out too much of what is going on. Abstraction has its own elaborateness, too, beyond extraction.

Then extraction has its simplicity, well shy of abstraction. So it is stretching the differences too far to call the activities of the retina abstractions. These two terms have their useful distinction.

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If the light hitting the cones and rods in the retina effects changes in them then this may be viewed as abstraction of some sort. This follows since light of different wavelength gives rise to different responses so it seems some "information", or better yet structure, is being abstracted by the cells. At each stage in the nervous system further abstraction occurs and ultimately you have "an object" - which is some sort of neural image. This applies equally well with the other senses however it seems light gives us much more useful information than other forms of energy since the visual cortex is by far the most evolved and complex.

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There is a further point I should add to the first three paragraphs of “My Reply” in #24. In the course of my argument that the infant to three weeks does not have percepts, therefore does not have knowledge, I mentioned that the memorial capabilities of the infant were too slight for the infant to have attained percepts. I want to add that Rand conceived of percepts as being required for the perception of entities.

The lower of the conscious species possess only the faculty of sensation . . . . A sensation is produced by the automatic reaction of a sense organ to a stimulus from the outside world; it lasts for the duration of the immediate moment, as long as the stimulus lasts and no longer.

The higher organisms possess a much more potent form of consciousness: they possess the faculty of retaining sensations, which is the faculty of perception. . . . which gives it the ability to be aware, not of single stimuli, but of entities, of things. An animal [of this level] is guided, not merely by immediate sensations, but by percepts. (OE 18–19)

I have shown in another note that Aristotle had also mentioned a requirement of memory for higher animal cognition.

For Rand’s account of the perception of entities, what is the minimal kind of memory required? I think the answer is pretty obvious in a coarse-grained way. The minimal memory commensurate with percepts is working memory, as distinct from iconic memory. (On the former, see also Postle A, B.)

The research of Renee Baillargeon, cited above, showing that the notion of enduring objects has been acquired by four months shows also that working memory is beginning by that time. I need to learn more about the development of working memory (e.g. Johnson 2005, Chapters 6 and 8) to find the exact degree of it attained at the four-month achievement of a notion of enduring objects.

I have held with Rand that all concretes, including entities, have measurable relations to other concretes. Furthermore, cognitive systems are measurement systems. That second conjecture is a claim to be set within the popular (and less constrained) approach of analyzing cognitive systems as computational systems. I have proposed that to qualify as an entity, a concrete existent must stand in (not only ordinal, but) interval or ratio measurement relations to other concretes. It will be interesting to see whether the perceptual and memorial powers engaged for the four-month notion of enduring objects are computational systems that are (not only ordinal, but) interval or ratio measurement systems.

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Gadfly, I received your thoughtful email by a pop-up notice. I have been unable to find a place where one’s emails are stored, so I’ll just have to go by memory from that one quick read.

Yes, I’m familiar with the finding of recognition by newborns of melodies to which they were exposed in the womb. Also, the newborn shows preference for a prose passage voiced prenatally by the mother (whether or not postnatal voice is mother); his auditory resolution of pitches and volumes is pretty good; he shows a preference for mother’s voice over stranger’s; and he distinguishes certain speech syllables. (Some of these capabilities were included in #24.) It is unknown whether he experiences sound sources as outside the head; head-turning towards sounds is also evoked by earphones. He has auditory recognition memory lasting for hours, and with conditioning, lasting for days.

You asked whether newborn recognition of prenatal melodies would have any implications for my picture of infants to three weeks as having no percepts. It is my understanding that the recognition capabilities of the infant in this period are this-versus-that discriminations categorized into his previous this-versus-that discriminations. That is a kind of identification, an identification of that and which, but not yet also an identification of what. (Cf. note 34.) I take the latter to be a requirement for sensory experience in which percepts are attained.

Ambiguous figures eliciting two different gestalt resolutions, such as with the 12-line see-through cube, show nicely the identifications of what by the perceptual system. The coming together of the elements to one of the two modes of competed cube in perspective is not accomplished by identification of what in thought. An example of the latter would be the quick reasoning one might make in concert with on-going percepts to discern whether the animal under the magnolia is a deer or a dog.

~~~~~~~~~~~~~~~~

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.

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Further note following on #24:

“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)

“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).

I argued in #22 to Daniel’s proposition that Rand erred in this second remark by allowing there could be a notion of existents as entities without a notion of their identities. I stand by that conclusion and my argument for it insofar as it pertains to as much identity as Rand described in that second remark. However, her first remark includes a further layer of identity of entities: that some cannot turn into others. There is no up-front problem with allowing the possibility that that more full-bodied (implicit*) notion of identities of entities does not obtain with the earliest notion of entities.

*[Note 35 of “Universals and Measurement”] – The sense of implicit here is extracted from the relevant cognitive-development research literature (viz., Gelman and Meck 1983, 344). The child is said to have implicit knowledge of the counting principles if she engages in behavior that is systematically governed by those principles, even though she cannot state them. (See Note 40 for the principles.) Gelman and Meck liken this implicitness of the counting principles at this stage of cognitive development to the way in which we are able to conform to certain rules of syntax when speaking correctly without being able to state those rules. That much seems right, but there is a further distinction I want to make. The child’s implicit counting principles are being learned (and taught) as an integral part of learning to properly count aggregations explicitly, expressly. In contrast, we can (or anyway, my preliterate Choctaw ancestors centuries past could) live out our lives, speaking fine in our mother tongue, following right rules of syntax, yet without being able to state those rules; indeed, without even knowing any of the terminology of syntax. Our learning of tacit rules of syntax is not for the sake of becoming able to follow them explicitly, only tacitly.

In the present developmental discussion, I shall reserve the term implicit to indicate that an operative rule is not only tacit, but has become operative as an integral part of becoming explicitly operative. The tacit logical principles, whose acquisition according to Macnamara is traced in the text, are not implicit in my present sense.

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The child will be about eight years old before she has a secure concept of materiality. Until then the child will tend to interpret “made out of” as “constructed from”

I don't know if this relates but I remember coming home one day from kindergarden and my mom asked, "Did you 'make' any friends today?" I was so confused. I had never been taught how to make friends. I imagined they must be formed out of plato. I still have trouble making friends, haha!

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The child will be about eight years old before she has a secure concept of materiality. Until then the child will tend to interpret “made out of” as “constructed from”

I don't know if this relates but I remember coming home one day from kindergarden and my mom asked, "Did you 'make' any friends today?" I was so confused. I had never been taught how to make friends. I imagined they must be formed out of plato. I still have trouble making friends, haha!

But as you have learned, your friends are not formed out of plato but out of aristocrates.

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The same is true of everything you perceive or experience; as an adult, you cannot perceive or experience in a vacuum, you do it in a certain automatized context—and the efficiency of your mental operations depends on the kind of context your subconscious has automatized.” (“The Comprachicos” 1971)

Once you posses a certain automatized context is that context capable of being rearranged or is it effectively permanent? Have studies been made on this? What if we had the ability to improve or reform our automated context as to enhance our learning capability. Is that plausible? Is it common?

Here you speak of the same topic:

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.

Ah ha! I pulled that last sentence out for a reason. You say reason proceeds upon whatever precepts are possible to us. Could the sensations be compared to the various inputs of a complex algorithm. With more accurate and neccessary inputs comes improved processing. The processor or program being the percept and the output being logic or reality...

I hope these questions don't cause you nausea(by being annoying). I'm very interested in child-development. I believe everyone should have a big interests their children's intellectual development. I grew up with a mother who encouraged naivety and a father who demanded reason without premise a.k.a (telepathy). It seems that so many parents want to keep their children dumb. I believe parents have a substantial influence on our sensations and percepts.

Edited by Aristocrates

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The child will be about eight years old before she has a secure concept of materiality. Until then the child will tend to interpret “made out of” as “constructed from”

I don't know if this relates but I remember coming home one day from kindergarden and my mom asked, "Did you 'make' any friends today?" I was so confused. I had never been taught how to make friends. I imagined they must be formed out of plato. I still have trouble making friends, haha!

But as you have learned, your friends are not formed out of plato but out of aristocrates.

That sounds like one of my jokes. Cheese topped with cheddar :D I'm happy to learn that you understand how i derived my screen name. I worry people might think I mispelled aristocrats.

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Pre- kindergarten my son "made" two imaginary friends, called Cracker Boy and Tunie, Tooney or Tuna maybe. In pre-cellphone days he had to stop at payphones everywhere we went and talk to them. Outings to the corner store took hours. I know we are in joking vein here, but this is true. I find the cheese connection zanily interesting, since I live in the land of Kraft Dinner.

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PS to the previous

You are too young to know the barenaked ladies "If I had a million dollars" but its best line is

I wouldn't have to eat Kraft Dinner

- but I would anyway

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Hey now! I may be young but I happened to be very cognisant when Barenaked Ladies' 'If I had a million dollars' made its debut

I'd buy a fur coat but not a real fur coat cause that's cruel. I use to sing the only two lines i knew over and over. Now I know a new one! Are they canadian?

its been one week since you looked at me, yesterday you've forgiven me but it'll still be 2 days till we say we're sorry. All the kids new these songs.

Edited by Aristocrates

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Hey now! I may be young but I happened to be very cognisant when Barenaked Ladies' 'If I had a million dollars' made its debut

I'd buy a fur coat but not a real fur coat cause that's cruel. I use to sing the only two lines i knew over and over. Now I know a new one! Are they canadian?

its been one week since you looked at me, yesterday you've forgiven me but it'll still be 2 days till we say we're sorry. All the kids new these songs.

Wow, I didn't know they were popular in the US. Yes they are Canadian, Steven Page's parents live in my neighbourhood and I have seen him out with them a few times. He is out on his own now since his troubles but I believe the Ladies still tour.

Keep not being cruel!

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The same is true of everything you perceive or experience; as an adult, you cannot perceive or experience in a vacuum, you do it in a certain automatized context—and the efficiency of your mental operations depends on the kind of context your subconscious has automatized.” (“The Comprachicos” 1971)

Once you posses a certain automatized context is that context capable of being rearranged or is it effectively permanent? Have studies been made on this? What if we had the ability to improve or reform our automated context as to enhance our learning capability. Is that plausible? Is it common?

Bobby,

In “The Comprachicos” Rand used the skill of walking (a, b) as an example of subconscious integration and automatization of perceptual and motor elements. She then said, more generally, “A mind’s cognitive development involves a continual process of automatization.”

She turns for example to one’s perception of a table as a toddler in comparison to one’s (normal) perception of a table as an adult. I see the horizontal surface on which objects, including my arms, presently reside. Rand thought that that much seeing is not subject to revision in normal (hallucination-free) adult perception of the table. Ever after a certain level of infant development, that much is given in perception. In further learning, we come to see the table as a place to put objects off the floor. This level of seeing, Rand takes us to experience as not separate from our perception of the geometrical table form itself. We experience that form and the further elementary function-affordance of the table “as a single, indivisible percept—yet all you see is a four-legged object; the rest is an automatized integration of a vast amount of conceptual knowledge which, at one time, you had to learn bit by bit. The same is true of everything you perceive or experience; as an adult, you cannot perceive or experience in a vacuum, you do it in a certain automatized context . . . .”

Like its perceived form, the everyday functional affordances of the table are taken as effectively permanent, in Rand’s picture. However, if we change the context of the object known as a table by taking it along with us on a weightless voyage in outer space, it will retain its perceived geometrical form, but (hopefully) will lose its perceived affordance as a place upon which to stow objects. I do not know if any research on adaptation to weightless environment has included data on this sort of reformation in habit.

The cognitive habits of special concern to Rand in this essay are of a higher conceptual order. That is, they are more removed from continual immediate correction by elementary perceptual and motor experience. Specifically, they are basic framework habits about thinking. What is one’s expectation of being able to comprehend the world and one's self? How alluring is comprehension? How far does one integrate one’s experiences? How well does one keep straight one’s feelings and perceptions?

In Section II of her essay, Rand says she thinks it is an open question how far these basic habits of higher cognition can be reformed. Still, she thought surely one could correct many faults in one’s mental functioning.

Concerning your idea that parents have a substantial influence on our sensations and percepts, I would say strictly speaking No. Under wide variability of family and culture, one will come to correctly recognize tables and some of what they afford in childhood. Perhaps you are using percept more widely than usual (Note). Signification of the word percept has been more restricted than perception in the following way: Perception is used in a narrow way meaning sensory perception, but it is also used to mean conscious grasp at any level of consciousness. Percept is not given that more general signification in addition to its narrow one.

Note

. . .

“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. . . . Percepts, not sensations, are the given, the self-evident” (1966, 5; cf. Aristotle). 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. I should add to Rand’s account that the minimal memory commensurate with percepts is working memory, as distinct from iconic memory. Rand characterizes the integration in the formation of percept differently than Peirce, as we shall see. Rand shares with later Peirce the idea that human knowledge begins with percepts, not with subsidiary sensory elements composing them. Rand may have adopted the term percept from Peirce for her concept nearby his; the third paragraph of the Foreward of her Introduction to Objectivist Epistemology (1966) is a quotation from American Pragmatism: Peirce, James, & Dewey (Moore 1961).

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""Concerning your idea that parents have a substantial influence on our sensations and percepts, I would say strictly speaking No. Under wide variability of family and culture, one will come to correctly recognize tables and some of what they afford in childhood. Perhaps you are using percept more widely than usual (Note). Signification of the word percept has been more restricted than perception in the following way: Perception is used in a narrow way meaning sensory perception, but it is also used to mean conscious grasp at any level of consciousness. Percept is not given that more general signification in addition to its narrow one.""

I see. I've assigned too much responsibility(of providing sensations and percepts) to the parents.

You caught me, I obviously wasn't sure of the definition of percept. I like this definition of percept: the object of perception. I'm learning, I'm learning. Knowing this definition sure does make parts of my posts look quite ridiculous.

I think it would be interesting to chart this process of perception, cognition, however you call it.

Quote from S.B. : "Like its perceived form, the everyday functional affordances of the table are taken as effectively permanent, in Rand’s picture. However, if we change the context of the object known as a table by taking it along with us on a weightless voyage in outer space, it will retain its perceived geometrical form, but (hopefully) will lose its perceived affordance as a place upon which to stow objects. I do not know if any research on adaptation to weightless environment has included data on this sort of reformation in habit."

Clever... Taking gravity away is one way in which our automated context is forced to be rearranged. I suppose that our automated context operates entirely below the level of conscious(I see you mentioned this at the top of your post), thus a conscious effort to change would be pointless... And then the idea of reforming it, as i suggested (o bother), would likely require removing previous perceptions. I doubt that's generally desirable or possible.

""The cognitive habits of special concern to Rand in this essay are of a higher conceptual order. That is, they are more removed from continual immediate correction by elementary perceptual and motor experience. Specifically, they are basic framework habits about thinking. What is one’s expectation of being able to comprehend the world and one's self? How alluring is comprehension? How far does one integrate one’s experiences? How well does one keep straight one’s feelings and perceptions?

In Section II of her essay, Rand says she thinks it is an open question how far these basic habits of higher cognition can be reformed. Still, she thought surely one could correct many faults in one’s mental functioning""

Those are all extremely provocative questions.

Edited by Aristocrates

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Research subsequent to "Capturing Concepts" (1990) on categorical perception and its relations to concepts and language:

Categorical Perception Effects Induced by Category Learning

Kenneth R. Livingston, Janet K. Andrews, and Stevan Harnad (1998)

Journal of Experimental Psychology: Learning, Memory, and Cognition 24(3):732–53

Relatively Speaking: An Account of the Relationship between Language and Thought in the Color Domain

Debi Roberson and J. Richard Hanley (2010)

In Words and the Mind: How Words Capture Human Experience

B. C. Malt and P. Wolff, editors

Color Term Knowledge Does Not Affect Categorical Perception of Color in Toddlers

Anna Franklin, Ally Clifford, Emma Williamson, and Ian Davies (2005)

Journal of Experimental Child Psychology 90(2):114–4

How Culture Might Constrain Color Categories

Debi Roberson and Catherine O’Hanlon (2005)

Behavioral and Brain Sciences 28(4):505–6

Effects of Stroop Interference on Categorical Perception in Simultaneous Color Discrimination

Takashi Suegami and Chikashi Michimata (2010)

Perceptual and Motor Skills 110(3):857­–78

ABSTRACT

Two experiments were conducted to assess the effects of verbal interference on categorical perception. The task involved simultaneous color discrimination with no memory demands, and a concurrent Stroop task was employed as verbal interference. Exp. 1 demonstrated that categorical perception was eliminated by an incongruent color word (i.e., Stroop interference), but not by a congruent color word or nonword fixation crosses. Exp. 2 demonstrated that categorical perception decreased when the intensity of Stroop Interference was increased, and it increased when correct verbal coding was enhanced. These results provide further evidence that interfering with verbal coding disrupts categorical perception, suggesting that verbal coding has a crucial role in categorical "perception." It is also suggested that categorical perception could occur at the encoding or decision level but not at the storage or memory level. The possible mechanisms for categorical perception are also discussed.

~~~~~~~~~~~~~~~~~~

Other supplements subsequent to "Capturing Concepts" and to "Universals and Measurement" are here.

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Capturing Quantity

Part 1

I completed “Capturing Concepts” and published it in 1990.* In the 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 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). (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.)

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.

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. Cambridge, MA: Harvard University Press.

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 twenty-two years ago:

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;* further, on working memory: a, b, c, d ). 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.)

References

Aguiar, A., and R. Baillargeon 1999. 2.5 Month-Old Infants’ Reasoning about When Objects Should and Should Not Be Occluded. Cognitive Psychology 39(2):116–57.

——. 2002. Developments in Young Infants’ Reasoning about Occluded Objects. Cognitive Psychology 45(2):267–336.

Baillargeon, R. 1986. Representing the Existence and the Location of Hidden Objects: Object Permanence in 6- and 8-Month-Old Infants. Cognition 23:21–41.

——. 1987. Object Permanence in 3.5- and 4.5-Month-Old Infants. Developmental Psychology 23:655–64.

Baillargeon, R., and S. Carey. 2012. Core Cognition and Beyond: The Acquisition of Physical and Numerical Knowledge. In Early Childhood Development and Later Outcome. S. Pauen, editor. Cambridge.

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

——. 1991a. Induction on Identity I. O 1(2):33–46.

——. 1991b. Induction on Identity II–XII. O 1(3):1–56.

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

Bennett, J. 1966. Kant’s Analytic. Cambridge.

Bremner, J. G. 1994. Infancy. 2nd edition. Blackwell.

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

Deacon, T. 1997. The Symbolic Species. Norton.

Diamond, A. 1989. Differences Between Adult and Infant Cognition: Is the Crucial Variable Presence or Absence of Language? In Thought without Language. L. Weiskrantz, editor. Oxford.

Feigenson, L., S. Carey, and M. Hauser 2002. The Representations Underlying Infants’ Choice of More: Object Files versus Analog Magnitudes. Psychological Science 13(2):150–56.

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

Friedenberg, J. 1993. Intricate Consciousness. O 1(5):29–67.

Jetton, M. 1991. Imagination and Cognition. O 1(3):57–92.

——. 1995. Time, Prescience, and Biology. O 2(2):59–104.

——. 1998. Pursuing Similarity. O 2(6):41–130.

Johnson, M. H. 2005 [1997]. Developmental Cognitive Science. Blackwell.

Kahneman, D., Treisman, A., and B. J. Gibbs. 1992. The Reviewing of Object Files: Object-Specific Integration of Information. Cognitive Psychology 24:175–219.

Macnamara, J. 1986. A Border Dispute – The Place of Logic in Psychology. MIT.

Mandler, J. M. 2010. The Spatial Foundations of the Conceptual System. Language and Cognition 2(1):21–44.

Nelson, K. 1996. Language in Cognitive Development. Cambridge.

Piaget, J. 1954. The Construction of Reality in the Child. Basic Books.

Rand, A. 1966–67. Introduction to Objectivist Epistemology. 2nd edition. Meridian.

——. 1970. The Comprachicos. In The New Left: The Anti-Industrial Revolution. Signet.

Touchstone, K. 1996. Mathematics and Intuition. O 2(4):93–182.

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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 1, 2, 8, 15, 17, 18, 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 than with that understanding twenty-three years ago, when I first opened Rand’s work in that light.

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.

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

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

Deacon, T. W. 1997. The Symbolic Species: The Co-Evolution of Language and the Brain. Norton.

Dehaene, S., and E. Brannon, editors, 2011. Space, Time and Number in the Brain. Academic Press.

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.

Hartshorn, K. et al. 1998. Ontogeny of Learning and Memory in the First Year-and-a-Half of Life. Developmental Psychobiology 32:69–89.

Jetton, M. 2011. The Sim-Dif Model and Comparison. JARS 11(2):215–32.

Johnson, M. H. 2005 [1997]. Developmental Cognitive Science. Blackwell.

Nosofsky, R. 1992. Similarity Scaling and Cognitive Process Models. Annual Review of Psychology 43:25–53.

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