Susan Carey - IA de Investigación especializada en el MIT

Susan Carey

Bootstrapping &
the origin of concepts

All animals learn. But only human

beings create scienti½c theories, mathe-
matemáticas, literature, moral systems, y
complex technology. And only humans
have the capacity to acquire such cultur-
ally constructed knowledge in the nor-
mal course of immersion in the adult
world.

There are many reasons for the differ-

ences between the minds of humans
and other animals. We have bigger
brains, and hence more powerful infor-
mation processors; sometimes differ-
ences in the power of a processor can
create what look like qualitative differ-
ences in kind. And of course human
beings also have language–the main
medium for the cultural transmission
of acquired knowledge. Comparative
studies of humans and other primates
suggest that we differ from them as well
in our substantive cognitive abilities–
Por ejemplo, our capacity for causal anal-

Susan Carey, professor of psychology at Harvard
Universidad, has played a leading role in transform-
ing our understanding of cognitive development.
A Fellow of the American Academy since 2001,
she is the author of numerous articles and essays
and the book “Conceptual Change in Childhood”
(1985).

ysis and our capacity to reason about
the mental states of others. Cada uno de
these factors doubtless contributes
to our prodigious ability to learn.

But in my view another factor is even
more important: our uniquely human
ability to ‘bootstrap.’ Many psycholo-
gists, historians, and philosophers of
science have appealed to the metaphor
of bootstrapping in order to explain
learning of a particularly dif½cult sort–
those cases in which the endpoint of the
process transcends in some qualitative
way the starting point. The choice of
metaphor may seem puzzling–it is self-
evidently impossible to pull oneself up
by one’s own bootstrap. Después de todo, el
process I describe below is not impos-
sible, but I keep the term because of
its historical credentials and because it
seeks to explain cases of learning that
many have argued are impossible.

Sometimes learning requires the cre-
ation of new representational resources
that are more powerful than those pres-
ent at the outset. Early in the cultural
history of mathematics, por ejemplo,
the concept of the number included only
positive integers: with subsequent de-
velopment the concept came to encom-
pass zero, rational numbers (fractions),
negative numbers, irrational numbers
like pi, etcétera.

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Susan
carey
en
aprendiendo

Bootstrapping is the process that un-

derlies the creation of such new con-
cepts, and thus it is part of the answer to
the question: What is the origin of con-
cepts?

Individual concepts are the units of
pensamiento. They are constituents of larger
mental structures–of beliefs that are
formed out of them and of systems of
representation such as intuitive theories.
Concepts are individuated on the basis
of two kinds of considerations: their ref-
erence to different entities in the world
and their role in distinct mental systems
of inferential relations.

How do human beings acquire con-
cepts? Logic dictates three parts to any
explanation of the origin of concepts.
Primero, we must specify the innate repre-
sentations that provide the building
blocks of the target concepts of interest.
Segundo, we must describe how the target
concepts differ from these innate repre-
sentations–that is, we must describe de-
velopmental change. And third, we must
characterize the learning mechanisms
that enable the construction of new con-
cepts out of the prior representations.

Claims about all three parts of the ex-

planation of the origin of concepts are
highly controversial. Many believe that
innate representations are either percep-
tual or sensory, while others (incluido
mí mismo ) hold that humans and other ani-
mals are endowed with some innate rep-
resentations with rich conceptual con-
tent. Some researchers also debate the
existence, even the possibility, of quali-
tative changes to the child’s initial repre-
sentaciones. One argument for the impos-
sibility of such radical changes in the
course of development is the putative
lack of learning mechanisms that could
explain them. This is the gap that my
appeal to bootstrapping is meant to ½ll.
To make clear both what the problem
es, and what role bootstrapping may play

in solving it, I will examine how children
acquire one speci½c set of concepts: el
positive integers–i.e., concepts such as
uno, two, tres, nueve, eighteen, etc..

Before they acquire language, infantes

form several different types of represen-
tation with numerical content, al menos
two of which they share with other ver-
tebrate animals.

Uno, described by Stanislas Dehaene
in his delightful book The Number Sense,
uses mental symbols that are neural
magnitudes linearly related to the num-
ber of individuals in a set. Because the
symbols get bigger as the represented
entity gets bigger, they are called analog
magnitudes. Cifra 1 gives an external
analog magnitude representation of
number, where the symbol is a line, y
length is the magnitude linearly related
to number. Mental computations using
these symbols include comparison, a
establish numerical difference or equal-
idad, and also addition and subtraction.
Mental analog magnitudes represent

many dimensions of experience–for
ejemplo, brightness, loudness, and tem-
poral duration. In each case as the physi-
cal magnitudes get bigger, it becomes
increasingly harder to discriminate be-
tween pairs of values that are separated
by the same absolute difference. You can
see in ½gure 1 that it is harder to tell that
the symbol for seven is different from
(and smaller than) that for eight than it
is to tell that the symbol for two is dif-
ferent from (and smaller than) that for
tres. Analog magnitude representa-
tions follow Weber’s law, according to
which the discriminability of two values
is a function of their ratio.

You can con½rm for yourself that you
have an analog magnitude system of rep-
resentation of number that conforms to
Weber’s law. Tap out as fast as you can
without counting (you can prevent your-

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Cifra 1
Analog magnitude models

Number represented by a quantity linearly related
to the cardinal value of the set

uno: ––

two: ––––

tres: ––––––

Siete: ––––––––––––––

eight: –––––––––––––––

self from counting by thinking ‘the’ with
each tap) the following numbers of taps:
4, 15, 7, y 28. If you carried this out
several times, you’d ½nd the mean num-
ber of taps to be 4, 15, 7, y 28, con el
range of variation very tight around 4
(usually 4, occasionally 3 o 5) and very
great around 28 (de 14 a 40 taps, para
ejemplo). Discriminability is a function
of the absolute numerical value, as dic-
tated by Weber’s law. Since you were not
counting, some other numerical repre-
sentation must have been guiding your
tapping performance–presumably ana-
log magnitudes, as your adherence to
Weber’s law, de nuevo, would seem to indi-
cate.

Space precludes my reviewing the ele-
gant evidence for analog magnitude rep-
resentations of number in animals and
human infants, but let me give just one
ejemplo. Fei Xu and Elizabeth Spelke
showed infants arrays of dots, one dot
array at a time, until the infants got
bored with looking at them. All other
variables that could have been con-
founded with number (total array size,
total volume of dots, density of dots,
etcétera) were controlled in these stud-
es, such that the only possible basis for
the infants’ discrimination was numeric.
Seven-month-old infants were habituat-
ed either to arrays of eight or sixteen

dots. After habituation they were pre-
sented with new displays containing ei-
ther the same number of dots to which
they had been habituated or the other
number. Xu and Spelke found that the
infants recovered interest to the new
number, and so concluded that they are
capable of representing number. Xu and
Spelke also found evidence for Weber’s
law: infants could discriminate eight
from sixteen and sixteen from thirty-
two, but not eight from twelve or six-
teen from twenty-four.1

Infants and animals can form analog
magnitude representations of fairly large
conjuntos, but these representations are only
approximate. Analog magnitude repre-
sentations of number fall short of the
representational power of integers; en
this system one cannot represent exactly
½fteen, or ½fteen as opposed to fourteen.
Sin embargo, analog magnitude repre-
sentations clearly have numerical con-
tent: they refer to numerical values,
and number-relevant computations
are de½ned over them.

A second system of representations

with numerical content works very
differently. Infants and nonhuman pri-
mates have the capacity to form sym-
bols for individuals and to create men-
tal models of ongoing events in which
each individual is represented by a single
symbol. Cifra 2 shows how, in this sys-
tema, sets of one, two, or three boxes
might be represented. The ½gure repre-
sents three different possibilities for
the format and content of the symbols.

1 For an overview of the evidence for analog
magnitude representations of number in both
nonhuman animals and human adults, ver
Stanislas Dehaene, The Number Sense (Oxford:
prensa de la Universidad de Oxford, 1997). For evidence
in human infants, see Fei Xu and Elizabeth S.
Spelke, “Large Number Discrimination in 6-
Month-Old Infants,” Cognition 74 (2000):
B1–B11.

Bootstrapping
& the origin
of concepts

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Susan
carey
en
aprendiendo

There is one symbol for each box, entonces
number is implicitly represented; el
symbols in the model stand in one-one
correspondence with the objects in the
world.

To give you a feel for the evidence that
infants indeed employ such models, dis-
tinct from the analog magnitude repre-
sentations sketched above, consider the
following experiment from my laborato-
ry. Ten- to fourteen-month-old infants
are shown a box into which they can
reach to retrieve objects, but into which
they cannot see. If you show infants
three objects being placed, one at a time
or all at once, into this box, y luego
allow them to reach in to retrieve them
one at a time, they show by their pattern
of reaching that they expect to ½nd ex-
actly three objects there. If the infant
has a mental representation of a set of
two objects (p.ej., object, object) that are
hidden from view, and the infant sees a
new object being added to the set, el
infant creates a mental representation of
a set of three (object, object, object).
Más, computations of one-one corre-
spondence carried out over these models
allow the child to establish numerical
equivalence and number order (p.ej.,
Have I got all the objects out of the box
or are there more?)

Hasta ahora, this is just another demon-
stration that infants represent number.
Sin embargo, an exploration of the limits
on infants’ performance of this task im-
plicates a different system of representa-
tion from the analog magnitude system
sketched above.

Performance breaks down at four
objects. If the infants see four objects
being placed into the box and are al-
lowed to retrieve two of them, or even
just one of them, they do not reach per-
sistently for the remaining objects. Re-
member that in the analog magnitude
system of representation, success at

Cifra 2
Parallel individuation models

Número
of boxes

Image

Abstracto

Speci½c

1 box (cid:1)

obj

box

2 cajas (cid:1)(cid:1)

obj obj box box

3 cajas (cid:1)(cid:1)(cid:1) obj obj obj box box box

Nota: one symbol for each individual; no symbols for
integers.

numerical comparison is a function
of the ratios of the numbers being com-
pared, and that the representations can
handle sets of objects at least as big as
thirty-two. But in this reaching task,
infants succeed at ratios of 2:1 y 3:2,
but fail at 4:2 e incluso 4:1; as soon as
the set exceeds three, infants cannot
hold a model of distinct items in their
short-term memory.2
En suma, human infants (and other pri-

compañeros) are endowed with at least two
distinct systems of representation with
numerical content. Both take sets of in-
dividuals as input. One creates a summa-
ry analog representation that is a linear
function of the number of individuals
in the set. This process is noisy, y el
noise is itself a linear function of the set
tamaño, with the consequence that the rep-
resentations are merely approximate.
For several reasons, this system is too
weak to represent the positive integers.
For one, there is likely an upper bound to
the set sizes that can be represented by
analog magnitudes. More importantly,

2 For evidence of the set-size limits on in-
fants’ representations of small numbers of
objects, see Lisa Feigenson and Susan Carey,
“Tracking Individuals Via Object Files: Evi-
dence from Infants’ Manual Search,” Develop-
mental Science 6 (2003): 568–584.

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animals and infants cannot discriminate
adjacent integer values once the sets
contain more than three or four individ-
uals; eso es, they cannot represent exact-
ly ½fteen or twenty-½ve or forty-nine, o
any other large exact integer. Finalmente,
analog magnitude representations
obscure one of the foundational rela-
tions among successive integers–that
each one is exactly one more than the
one before. It is this relation, called the
successor relation, that underlies how
counting algorithms work and provides
the mathematical foundation of integer
conceptos. Since discriminabilty of ana-
log magnitudes is a function of the ratio
between them, the relation between two
and three is not experienced as the same
as that between twenty-four and twenty-
½ve; en efecto, the latter two values cannot
really even be discriminated within this
system of representation.

The second system–one symbol for
each individual–falls even shorter as a
representation of integers. There are no
symbols for number in this system at all;
the symbols in ½gure 2 each represent an
individual object, unlike those in ½gure
1, which represent an approximate cardi-
nal value. Además, what can be rep-
resented in this system is limited in
number to sets of one, two, and three.

The count list (‘one, two, tres . . .') es

a system of representation that has the
power to represent the positive integers,
so long as it contains a generative sys-
tem for creating an in½nite list. Cuando
deployed in counting, it provides a rep-
resentation of exact integer values based
on the successor function. Eso es, cuando
applied in order, in one-one correspon-
dence with the individuals in a set, el
ordinal position of the last number word
in the count provides a representation
of the cardinal value of the set–of how
many individuals it contains. Successive

symbols in the list refer to cardinal val-
ues exactly one apart: 5 es 4 plus 1, 6 es 5
plus 1, etcétera.

I have argued so far that the count-list

representation of number transcends
the representational power of both of
the representational systems with nu-
merical content that are available to pre-
verbal infants, for these precursors lack
the capacity to represent integers. If this
is so, it should be dif½cult for children to
come to understand the numerical func-
tion of counting.

And so, en efecto, it is dif½cult for chil-
dren to learn how counting represents
number, and details about the partial
understanding they achieve along the
way constrain our theories of the learn-
ing process. En los Estados Unidos (y
every other place where early counting
has been studied, including Western
Europa, Russia, Porcelana, y japon)
children learn to recite the count list
as young two-year-olds, and at this age
can even engage in the routine of count-
ing–touching objects in a set one by one
as they recite the list. But it takes anoth-
er year and a half before they work out
how counting represents number, y
in every culture yet studied, children go
through similar stages in working out
the meanings of the number words in
the count list.

Primero, children learn what ‘one’ means

and take all other words in the list to
contrast with ‘one,’ meaning ‘more than
one’ or ‘some.’ The behaviors that dem-
onstrate this are quite striking. Si usted
present young two-year-olds with a pile
of pennies and ask them to give you one
penny, they comply. If you ask for two
pennies or three pennies or ½ve pennies,
they grab a bunch, always more than
uno, and hand them over. They do
not create a larger set for ‘½ve’ than for
‘two.’ You might suppose that the plural
in ‘pennies’ is doing the work here, pero

Bootstrapping
& the origin
of concepts

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carey
en
aprendiendo

the same phenomenon is observed in
China and Japan, even though Chinese
and Japanese do not have a singular-
plural distinction, and also in the
United States when the contrast is
between ‘one ½sh,’ ‘two ½sh,’ and
‘½ve ½sh.’

Let us call children at this stage of
working out the meanings of number
words ‘one-knowers.’ Many other tasks
provide additional evidence that one-
knowers truly know only the meaning
of the word ‘one’ among all the words in
their count list. Por ejemplo, if you ask a
one-knower to tell you what’s on a series
of cards that contain one, two, o tres
½sh (up to eight ½sh), they say ‘a ½sh’ or
‘one ½sh’ for the card with one, and ‘two
½sh’ or ‘two ½shes’ or ‘two ½shies’ for
all of the other cards. This again indi-
cates a single cut between the meaning
of ‘one,’ which they grasp, and words for
the number of individuals in larger sets,
which they do not.

After having been one-knowers for
about six to nine months, children learn
what ‘two’ means. At this point they can
correctly give you two objects if you ask
for ‘two,’ but they still just grab a bunch
(always greater than two), if you ask for
‘three,’ ‘four,’ ‘½ve,’ or ‘six.’ After some
months as two-knowers, they become
three-knowers, and some months later
induce how counting works.

The performance of children who have
worked out how counting works is quali-
tatively different from that of the one-,
two-, and three-knowers in a variety of
ways that reflect the conceptual under-
standing of counting.

To give just one example, in the task in
which children are asked to give the ex-
perimenter a certain number of items,
say four, one-, two-, and three-knowers
usually give the wrong number, y el
young counters also sometimes make an
error. When asked to check by counting

and then to ½x the set, counters invari-
ably adjust the set in the right direction,
taking an object away if the set is too
large or adding one if it is too small.
One-, two-, and three-knowers, en con-
contraste, almost always add more to the
set–even if they had counted to ½ve or
six or seven when they were checking
whether it had four–con½rming that
they really do not understand how
counting determines the meaning of
number words.

These data suggest that the partial
meanings of number words seem to be
organized initially by the semantics of
quanti½ers–the singular-plural distinc-
tion and the meanings of words like
‘some’ and ‘a.’ If this is right, then we
might expect that children learning lan-
guages with quanti½er systems that
mark numerical contrasts differently
from English would entertain different
hypotheses concerning the partial mean-
ings of number words. They might break
into the system differently. And indeed
they do.

Consider ½rst classi½er languages
such as Chinese and Japanese that do
not mark the distinction between singu-
lar and plural in nouns, verbos, or adjec-
tives. Two independent studies have
found that although children in China
and Japan learn the count list as young
as English-speaking children do, ellos
become one-knowers several months
later and are relatively delayed at each
stage of the process. En cambio, Russian
has a complex plural system in which the
morphological markers for sets of two,
tres, and four differ from those for ½ve
through ten. Two independent studies
have shown that even Russian one- y
two-knowers distinguish between the
meanings of the number words ‘two,'
‘three,’ and ‘four,’ on the one hand, y
‘½ve,’ ‘six,’ ‘seven,’ and ‘eight,’ on the
otro. Unlike the one- and two-knowers

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described above, Russian children in the
early stages of working out how count-
ing works grab smaller sets when asked
to give the experimenter ‘two,’ ‘three,'
or ‘four’ than when asked to give the ex-
perimenter ‘½ve’ or more, and use larger
numbers for larger sets in the what’s-on-
this-card task.3

These phenomena concerning young
children’s partial understanding of the
meanings of number words support
three interrelated conclusions. Primero,
that it is so dif½cult for children to learn
what ‘two’ means, let alone what ‘½ve’
and ‘eight’ mean, lends support to the
claims that preverbal number represen-
tations are not representations of inte-
gers, at least not in the format of an inte-
ger list. Young children–for a full six to
nine months before they work out what
‘two’ means, and a full year and a half
before they work out how the count
list represents integers–know how to
count, know what ‘one’ means, y
know that ‘two,’ ‘three,’ ‘four,’ ‘½ve,'
‘six,’ ‘seven,’ and ‘eight’ represent num-
bers larger than ‘one.’ Second, próximo
to understand how the count list repre-
sents numbers reflects a qualitative
change in the child’s representational
capacities; I would argue that it does
nothing less than create a representa-
tion of the positive integers where none
was available before. Finalmente, a third pos-
sible developmental source of natural
number representations, in addition to
the preverbal systems described above,
may be the representations of numbers
within natural language quanti½er se-
mantics. Por supuesto, natural language
quanti½ers, other than the number
words in the count list itself, do not

3 For a characterization of the early stages of
counting in English, see Karen Wynn, “Chil-
dren’s Acquisition of the Number Words and
the Counting System,” Cognitive Psychology 24
(2) (1992): 220–257.

have the power to represent natural
numbers either.

The problem of the origin of the posi-

tive integers arises at two different time
scales–historical and ontogenetic. En
the dawn of modern anthropology,
when colonial of½cers went out into
the French and English colonial worlds,
they discovered many systems of explicit
number representation that fell short of
a full representation of natural number.
They described languages that marked
number on nouns, adjectives, and verbs,
and which had quanti½ers like the Eng-
lish ‘one,’ ‘two,’ ‘many,’ ‘some,’ ‘each,'
‘every,’ and ‘more,’ but which had no
count list. In this vein, the psychologist
Peter Gordon has described the language
of the Piraha, an isolated Amazonian
gente. He has shown that in addition
to linguistic quanti½ers meaning ‘one,'
‘two,’ and ‘many,’ the Piraha also have
access to the nonverbal systems de-
scribed above (parallel individuation
of small sets and analog magnitude rep-
resentations of large numerosities). Gor-
don con½rms that they have no repre-
sentations of large exact numerical val-
ues.

Anthropologists and archeologists
have described intermediate systems of
integer representation, short of integer
liza, and these intermediate systems
provide evidence for a process of cultur-
al construction over generations and
centuries of historical time.4 Here I con-
centrate on ontogenetic time. ¿Cómo
three-year-olds do it? How do they cre-
ate a representational system with more

4 I would recommend the linguist James Hur-
ford’s review of this literature to any reader in-
terested in this process. James Hurford, Lan-
guage and Number (Oxford: Basil Blackwell,
1987). For work on the Piraha, see Peter Gor-
don, “The Role of Language in Numerical Cog-
nition: Evidence from Amazonia” (under re-
vista).

Bootstrapping
& the origin
of concepts

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Dédalo Invierno 2004

Susan
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power than any on which it is built?5
In answering this question, me gustaría
appeal to bootstrapping processes.

Bootstrapping processes make essen-
tial use of the human capacity for creat-
ing and using external symbols such as
words and icons. Bootstrapping capital-
izes on our ability to learn sets of sym-
bols and the relations among them di-
rectly, independently of any meaning
assigned to them in terms of anteced-
ently interpreted mental representa-
ciones. These external symbols then
serve as placeholders, to be ½lled in
with richer and richer meanings. El
processes that ½ll the placeholders
create mappings between previously
separate systems of representation,
drawing on the human capacity for
analogical reasoning and inductive in-
ference. The power of the resulting sys-
tem of concepts derives from the com-
bination and integration of previously
distinct representational systems.

Let’s see how this might work in the
present case. We must allow the child
one more prenumerical capacity–that
of representing serial order. This is no
problem–young children learn a vari-
ety of meaningless ordered lists, semejante
as ‘eeny, meeny, miney, mo.’

We seek to explain how the child
learns the meanings of the number
words–what ‘two’ means, what ‘seven’
means–and how the child learns how
the list itself represents number–that
the cardinal value of a set enumerated
by counting is determined by the order
on the list, and that successive numbers
on the list are related by the arithmetic
successor relation.

5 In a forthcoming book I argue that the same
bootstrapping process underlying this marvel-
ous feat in childhood also accounts for the de-
velopment in historical time, but that argument
is beyond the scope of this brief paper.

As described above, the child learns
the meanings of the ½rst number words
as natural language quanti½ers. Children
learn the meaning of ‘one’ just as they
learn the meaning of the singular deter-
miner ‘a’ (en efecto, in many languages,
such as French, they are the same lexi-
cal item).

Algunos meses después, ‘two’ is learned,
just as dual markers are in languages that
have singular/dual/plural morphology.
Languages with dual markers have a dif-
ferent plural af½x for sets of two than the
af½x for sets greater than two. It is as if
English nouns were declined ‘box’ (sin-
gular), ‘boxesh’ (dual), ‘boxeesh’ (plu-
ral). In this system, the suf½x ‘esh’
would apply just when the set referred to
contained exactly two items. By hypoth-
esis, children would learn the meaning
of the word ‘two’ just as they would
learn the morphological marker ‘esh’–
if English plural markers worked that
way. By extension, some months later,
‘three’ is learned just as trial markers
are in the rare languages that have sin-
gular/dual/trial/plural morphology.
In the early stages of being a one-,
two-, or three-knower, the child repre-
sents other number words as quanti½ers,
meaning ‘many,’ where ‘many’ is more
than any known number word. As I will
argue below, it is likely that the nonver-
bal number representations that support
the meanings of the known words is the
system of parallel individuation (½gure
2), with natural language quanti½cation
articulation in terms of notions like ‘set’
and ‘individual.’

Mientras tanto, the child has learned the
count list, which initially has no seman-
tic content other than its order. El
child knows one must recite ‘one, two,
tres, four, ½ve,’ not ‘two, tres, uno,
½ve, four,’ just as one must say ‘a, b, C, d,
mi,’ not ‘c, a, mi, d, b.’

Dédalo Invierno 2004

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The stage is now set for a series of
mappings between representations.
Children may here make a wild analo-
gy–that between the order of a particu-
lar quantity within an ordered list, y
that between this quantity’s order in a
series of sets related by additional indi-
viduals. These are two quite different
bases of ordering–but if the child recog-
nizes this analogy, she is in the position
to make the crucial induction: For any
word on the list whose quanti½cational
meaning is known, the next word on the
list refers to a set with another individ-
ual added. Since the quanti½er for single
individuals is ‘one,’ this is the equivalent
to the following induction: If number
word X refers to a set with cardinal value
norte, the next number word in the list re-
fers to a set with cardinal value n + 1.

This bootstrapping story provides dif-

ferent answers for how the child learns
the meaning of the word ‘two’ than for
how she learns the meaning of ‘½ve.’
According to the proposal, the child
ascertains the meaning of ‘two’ from
the resources that underlie natural lan-
guage quanti½ers, and from the system
of parallel individuation, whereas she
comes to know the meaning of ‘½ve’
through the bootstrapping process–
es decir., that ‘½ve’ means ‘one more than
four, which is one more than three . . .'
–by integrating representations of natu-
ral language quanti½ers with the exter-
nal serial ordered count list.

I began by sketching two systems of

preverbal representation with numerical
contenido: the analog magnitude system
and the system of parallel individuation.
You may have noticed that the analog
magnitude system played no role in my
bootstrapping story. It would be quite
possible to imagine a role for this system
in a slightly different bootstrapping pro-
posal, and it may be that such a proposal

would be empirically correct, at least for
some children. We do know that chil-
dren come to integrate their integer list
with analog magnitudes, such that ‘½ve’
comes to mean both ‘one more than
four, which is one more than three. . .'
and ‘––––––––––,’ the analog magnitude
symbol for the cardinality of a set of ½ve
individuals. This integration is undoubt-
edly very important; bootstrapping pro-
vides richer representations precisely
through integration of previously dis-
tinct systems of representation.

As important as the integration of the

integer list representation with analog
magnitude representations may be,
there is good reason to believe that this
integration is not part of the bootstrap-
ping process through which the concept
of positive integers is ½rst understood.
Research suggests that it is not until after
children have worked out how the count
list represents number–in fact some six
months later–that they know which
analog magnitudes correspond to which
numbers above ½ve in their count list.
That ½nding–along with the fact that
the precise meanings of number words
are learned in the order ‘one,’ then ‘two,'
then ‘three,’ followed by the induction
of how the count list works–leads me to
favor the bootstrapping proposal above.
I doubt that anybody would deny that
language helps us occupy the distinctive
cognitive niche that we human beings
enjoy. It is obvious that culturally con-
structed knowledge is encoded in lan-
guage and can then be passed on to new
generations through verbal communica-
tion–you can tell your children some-
thing, saving them from having to dis-
cover it themselves. Still, this account
misses the equally obvious point that
children are often unable to understand
what we tell them, because they lack the
concepts that underlie our words. El
problem then becomes accounting for

Bootstrapping
& the origin
of concepts

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Dédalo Invierno 2004

Susan
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how they acquire the relevant concepts
they need to understand what we are
telling them.

I have argued that bootstrapping

mechanisms provide part of the solution
to this problem. In thinking about how
bootstrapping might work, we are led
to a fuller appreciation of the role of
language in supporting the cultural
transmission of knowledge. We cannot
just teach our children to count and ex-
pect that they will then know what ‘two’
or ‘½ve’ means. Learning such words,
even without fully understanding them,
creates a new structure, a structure that
can then be ½lled in by mapping rela-
tions between these novel words and
otro, familiar concepts. And so eventu-
ally our children do know what ‘½ve’
medio: through the medium of language
and the bootstrapping process sketched
here they have acquired a new concept.

Dédalo Invierno 2004

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