Can statistical learning bootstrap the integers?

Can Statistical Learning Bootstrap the Integers?

Lance J. Ripsa, Jennifer Asmuth

b, Amber Bloomfield

c

aPsychology Department, Northwestern University, 2029 Sheridan Road, Evanston, IL 60208 USA

bPsychology Department, Susquehanna University, 514 University Avenue, Selinsgrove, PA 17870 USA

cCenter for Advanced Study of Language, University of Maryland, 7005 52nd Ave., College Park,

MD 20742 USA

Corresponding author:

Lance Rips

Psychology Department

Northwestern University

2029 Sheridan Road

Evanston, IL 60208 USA

847.491.5947

Fax: 847.491.7859

Email: [email protected]

Reply to Piantadosi et al. / 2

Abstract

This paper examines Piantadosi, Tenenbaum, and Goodman’s (2012) model for how children

learn the relation between number words (“one” through “ten”) and cardinalities (sizes of sets with one

through ten elements). This model shows how statistical learning can induce this relation, reorganizing its

procedures as it does so in roughly the way children do. We question, however, Piantadosi et al.’s claim

that the model performs “Quinian bootstrapping,” in the sense of Carey (2009). Unlike bootstrapping, the

concept it learns is not discontinuous with the concepts it starts with. Instead, the model learns by

recombining its primitives into hypotheses and confirming them statistically. As such, it accords better

with earlier claims (Fodor, 1975, 1981) that learning does not increase expressive power. We also

question the relevance of the simulation for children’s learning. The model starts with a preselected set

of 15 primitives, and the procedure it learns differs from children’s method. Finally, the partial knowledge

of the positive integers that the model attains is consistent with an infinite number of nonstandard

meanings—for example, that the integers stop after ten or loop from ten back to one.

Keywords: Bootstrapping, Number knowledge, Number learning, Statistical learning


1. Introduction

According to the now standard theory of number development, children learn to recognize and

produce one object in response to requests, such as “Give me one cup” or “Point to the picture of one

elephant.” They then gradually learn to handle similar requests for two objects and eventually three. At

this point, they rapidly extend their success to larger collections—up to those named by the largest

numeral on their list of number terms, for example, “ten” (Wynn, 1992). (The largest numeral for which

they are successful can vary, but let’s say “ten” for concreteness.) The standard theory sees this last

achievement as the result of the children figuring out how to count objects: They learn a general rule for

how to pair the numerals on their list with the objects in a collection in order to compute the total. We will

refer to this procedure from here on as enumeration rather than counting in order to avoid confusion with

the other meaning of counting—reciting the number sequence “one,” “two,” “three,”….

No one doubts that children in Western cultures learn enumeration as a technique for determining

the cardinality (i.e., the total number of items or set size) of a collection. However, debates exist about

how children make this discovery (see, e.g., Carey, 2009; Leslie, Gelman, & Gallistel, 2008; Piantadosi,

Tenenbaum, & Goodman, 2012; and Spelke, 2000, 2011) and about its significance for their beliefs about

number (e.g., Margolis & Laurence, 2008; Rey, 2011; Rips, Asmuth, & Bloomfield, 2006, 2008; Rips,

Bloomfield, & Asmuth, 2008). Our aim in the present article is to examine a recent theory of how

children learn to enumerate by Piantadosi et al. and to compare it to an earlier proposal by Carey. In doing

so, we are motivated by Piantadosi et al.’s claim that their model solves difficulties we earlier identified

in Carey’s theory (Rips, Asmuth, & Bloomfield, 2006).

1.1. Carey’s bootstrap proposal

Carey (2004, 2009) has given a detailed account of learning to enumerate as an instance of a

process she calls Quinian bootstrapping. In brief, children start with a short memorized list of numerals in

order from “one” to “ten,” but where these numerals are otherwise uninterpreted. Over an approximately


one-year period, children successively attach the numeral “one” to a mental representation consisting of

an arbitrary one-member set (e.g., {o1}), the numeral “two” to a representation consisting of a two-

member set (e.g., {o1, o2}), and the numeral “three” to a representation consisting of a three-member set

(e.g., {o1, o2, o3}). Children next realize that a parallel exists between the order of the numeral list (“one”

then “two” then “three”) and the set representations ordered by the addition of one element ({o1} then {o1,

o2} then {o1, o2, o3}). They infer that the meaning of the next element on the numeral list is the set size

given by adding one to the set size named by the preceding numeral. For example, the meaning of “five”

is the cardinality one greater than that named by “four.” This inference allows them to determine the

correct cardinal value for the remaining items on their count list (up to “ten”). We can refer to the

conclusion of this inference (the italicized proposition above) as the bootstrap conclusion.

According to Carey, Quinian bootstrapping provides a child with new primitive concepts of

number, concepts that the child’s old conceptual vocabulary can’t express, even in principle:

Quinian bootstrapping mechanisms underlie the learning of new primitives, and this

learning does not consist of constructing them from antecedently available concepts (they

are definitional/computational primitives, after all) using the machinery of compositional

semantics alone (Carey, 2009, p. 514, emphasis in the original).

No translation is possible between the old number concepts and the new ones:

To translate is to express a proposition stated in the language of [Conceptual System 2] in

the language one already has ([Conceptual System 1])… In cases of discontinuity in

which Quinian bootstrapping is required, this is impossible. Bootstrapping is not

translation; what is involved is language construction, not translation. That is, drawing on

resources from within CS1 and elsewhere, one constructs an incommensurable CS2 that

is not translatable into CS1 (Carey, 2011, p. 157).

In the last quotation, CS1 is the child’s conceptual system prior to an episode of Quinian bootstrapping,

and CS2 is the conceptual system that results from bootstrapping. As the first of these quotations makes

clear, Quinian bootstrapping is a kind of learning, usually an extended process that takes months or years


to complete. Children don’t acquire the new number concepts by mere maturation. Likewise, external

causal forces don’t merely stamp them in. A challenge in understanding Quinian bootstrapping is how to

reconcile the claim about learning with the claim about discontinuity between old and new concepts.

1.2. Some further issues about the bootstrap inference

Aside from the question about discontinuity, Quinian bootstrapping raises several issues about the

inductive inference that takes children from facts about the first three or four positive integers to a general

conclusion. The bootstrap inference has roughly this form:

“One” precedes “two” precedes “three” precedes … precedes “ten.”

“One” means one.

“Two” means two.

“Three” means three.

___________________________________________

The next numeral on the count list means one more than its predecessor.

A question about this inference, then, is whether a psychologically reasonable and well-defined process

could carry it out. This is a question about Quinian bootstrapping’s internal feasibility—a demand for a

proof of concept. Both Carey’s (2009) original proposal and Piantadosi et al.’s (2012) new one are meant

to reassure us on this point by providing detailed descriptions of how children make this inference. We

briefly described Carey’s proposal in Section 1.1, and we will describe Piantadosi et al.’s in Section 2.2.

Assuming that such a process is feasible, we can also ask whether the process is faithful to the

way children actually do it. Although Carey’s (2009) and Piantadosi et al.’s (2012) proposals could both

be feasible, they can’t both be true, barring big individual differences in the way children carry out the

inference. For one thing, Carey’s proposal includes an analogical inference as its cornerstone, but analogy

plays no part in Piantadosi et al.’s theory. Which proposal (if either) is correct is an obviously empirical

question.


But even if we grant Quinian bootstrapping’s feasibility and faithfulness, we still need to know

how much its conclusion contributes to children’s understanding of the positive integers. The bootstrap

conclusion is certainly true: The next numeral in the list of positive integers stands for a cardinality one

greater than its predecessor. But although this generalization correctly coordinates two infinite sequences,

it says nothing about the structure of the sequences. For example, a child could draw the bootstrap

conclusion while quite consistently believing that the integers stop at 78. He or she would know that

“one” refers to set size one, “two” to set size two, and … and “seventy-eight” to set size 78, but also

believe that this is the end of the story. This issue is one of scope: How much does Quinian bootstrapping

tell children about the integers?

1.3. An overview

Of the four issues about Quinian bootstrapping that we’ve just surveyed, Piantadosi et al.’s (2012)

paper focuses mainly on feasibility. 1 They present an explicit model of how children learn to enumerate

objects, and we believe the model is successful in showing that a mechanism could exist that derives the

bootstrap conclusion from its premises. The same seems true of Carey’s (2009) original proposal, but the

Piantadosi et al. model puts this inference on a firm computational basis.

How well does the model handle the other psychological and logical issues that surround

bootstrapping? First, the relevance of the model depends on establishing that it qualifies as an instance of

Quinian bootstrapping in the sense in which that concept figures in current debate about the topic. Thus,

the model must exhibit discontinuity in the way Quinian bootstrapping is supposed to. Although

discontinuity may not be a central concern for Piantadosi et al., we believe it is the essential feature for

most developmentalists, who are interested in number learning as an example of discontinuous conceptual

change.

1 Unless we otherwise attribute them, all references to Piantadosi et al. are to Piantadosi et al. (2012).


Second, the model needs to be faithful to the method children use in learning the positive

integers. The fact that some computational method or other exists for getting to the bootstrap conclusion

could hardly be in doubt, since adults surely agree that the conclusion is true. If the model’s method of

reaching the conclusion is quite different from that of children’s, the psychological lessons we can derive

from it would be unclear.

Third, questions about the scope of the bootstrap conclusion are also relevant to Piantadosi et al.’s

model. According to these authors (p. 212), their model “was motivated in large part by the critique of

bootstrapping put forth” in our earlier articles (Rips et al., 2006; Rips, Asmuth, & Bloomfield, 2008;

Rips, Bloomfield, & Asmuth, 2008). This criticism focused on whether the bootstrap conclusion pinned

down the meaning of the positive integers—whether it ruled out nonstandard interpretations of the

numerals that are very different from that of the integers’ meaning. (Piantadosi et al. may have read these

criticisms to be about feasibility or rigor, but they were instead about scope.) So if their aim is to meet

these criticisms, Piantadosi et al. have to show that their model yields the right interpretation.

In the following sections, we argue that because the model learns to enumerate by straightforward

concept combination, it does not create a new discontinuous conceptual system. Instead, it illustrates

Fodor’s (1975, 1981) hypothesis that learning elaborates old concepts: It cannot produce a new

representational primitive or construct a new language that increases the child’s expressive power.

Because of this limitation, the model is incapable of bootstrapping in Carey’s sense and does little to clear

up the issues surrounding bootstrapping. Of course, this doesn’t mean that the model is incorrect. It could

provide a correct account of enumeration even if Quinian bootstrapping has no role in its procedure.

However, the model’s method of enumerating differs from that of children, and these differences raise

questions about the faithfulness of the model to children’s actual abilities. Finally, Piantadosi et al.’s

model maps sets of one to ten elements to the terms “one” to “ten,” but has no implications for the

structure of the integers. So the same difficulties about scope that beset Quinian bootstrapping carry over

to this new proposal.


Here’s a summary of our disagreements with Piantadosi et al.: They have tried to rescue Quinian

bootstrapping from the problems that surround it by constructing a computational/statistical model. But

although the model demonstrates one way to get from premises about the meanings of “one” through

“three” to a more general conclusion, we believe the model is not an example of Quinian bootstrapping,

does not accurately model the way children enumerate objects, and does not solve bootstrapping’s central

logical deficiencies.

2. Is the Piantadosi et al. model a form of bootstrapping?

Carey (2004, 2009) introduced “Quinian bootstrapping” as a term for procedures in which people

learn new concepts that are discontinuous from their old ones. Thus, we take the central claims of Quinian

bootstrapping to be these (see Beck, submitted for publication):

Learning: In Quinian bootstrapping, an agent learns a new conceptual system, CS2, in terms of

an old system, CS1.

Discontinuity: After Quinian bootstrapping, CS2 is conceptually discontinuous from CS1.

These properties are explicit in the quotations in Section 1.1 and in many other places in Carey’s (2009)

presentation. Important for the present discussion is the fact that Carey introduces her chapter on how

children acquire representations of the positive integers by setting herself two challenges: “to establish

discontinuities in cognitive development by providing analyses of successive conceptual systems, CS1

and CS2, demonstrating in what sense CS2 is qualitatively more powerful than CS1” and “to characterize

the learning mechanism(s) that get us from CS1 to CS2” (p. 288).

Because of the Discontinuity claim, Quinian bootstrapping opposes the view that all forms of

learning derive new concepts by recombining (or translating from) old ones. According to this alternative

view (Fodor, 1975, 1981), learning is a form of hypothesis formation and confirmation in which the

hypotheses are spelled out in the old conceptual vocabulary (i.e., the concepts that the person possesses

prior to learning). Confirmation merely increases their likelihood without producing anything

fundamentally new. To have a general term for such non-bootstrapping forms of learning, we’ll use


concept recombination. Proponents of bootstrapping agree that mundane forms of learning use

recombination. Bootstrapping occurs only in special cases. So the question that divides theorists is

whether any actual instances of learning are examples of bootstrapping.

When we turn to Piantadosi et al.’s theory of learning enumeration, we find that their approach is

much closer to recombination than to Quinian bootstrapping. Their clearest statement about this issue is

the following passage (p. 214):

One of the basic mysteries of development is how children could get something

fundamentally new… Our answer to this puzzle is that novelty results from

compositionality. Learning may create representations from pieces that the learner has

always possessed, but the pieces may interact in wholly novel ways…This means that the

underlying representational system which supports cognition can remain unchanged

throughout development, though the specific representations learners construct may

change.

We will describe Piantadosi et al.’s model in more detail in Section 2.2, but this excerpt makes apparent

that, whatever its merits, it has nothing to do with creating new primitives or conceptual systems that are

discontinuous with old ones. As such, it jettisons a central part of Carey’s bootstrapping theory, the

Discontinuity claim. “Quinian bootstrapping” is a technical term, so there is limited room for redefining it

while simultaneously claiming to defend it.

We note that developmentalists have used the term “bootstrapping” in ways that differ from

Carey’s “Quinian bootstrapping.” In research on language acquisition, syntactic bootstrapping is a

hypothetical process in which children use syntactic properties of sentences to determine the referents of

component words, and semantic bootstrapping is a process in which children use the referents of words to

determine their syntactic category (see Bloom & Wynn, 1997, for a discussion of these possibilities in the

context of number). Neither of these forms of bootstrapping qualifies as Quinian bootstrapping, according

to Carey (2009, p. 21), since neither creates a conceptual system discontinuous with earlier ones.

Moreover, Piantadosi et al.’s proposal for number learning is not an example of either syntactic or


semantic bootstrapping, since syntactic categories play no role in it. In fact, it might be possible to

contend that their proposal fails to qualify as bootstrapping in an even wider sense, but we won’t argue

for that conclusion here.2 Our concern, instead, is to show that the Piantadosi et al. model is not a type of

Quinian bootstrapping—it does not satisfy both the Learning and Discontinuity criteria. (In the rest of this

article, we will use “bootstrapping” to mean Quinian bootstrapping.)

2.1. Bootstrapping’s central features

To make the distinction between bootstrapping and recombination a little more precise, let c*

represent a new concept that is created in learning, and let c1, c2, …, ck represent old concepts. Proponents

of recombination believe that learning is a function taking the old concepts into the new one:

( ) It matters very much, however, what the function f is like. Advocates of bootstrapping

agree that the input to learning is a set of old concepts and the output a new concept. As Carey (2011,

p. 157) remarks, “Clearly, if we learn or construct new representational resources, we must draw on those

we already have.” But Carey would maintain that in examples of bootstrapping the function is not mere

recombination, as the first quotation in Section 1.1 makes explicit. To distinguish between recombination

and bootstrapping, then, we need some restrictions on f or on its arguments (Rips & Hespos, 2011).

As a first possibility, proponents of bootstrapping could insist that bootstrapping algorithms are

so complex that they go beyond what could reasonably be considered recombination. For example,

Carey’s (2009) theory of how children learn to enumerate includes an analogical inference that maps the

first few numerals (“one,” “two,” “three”) to corresponding representations of cardinalities (see the

description in Section 1.1). If analogical inference is too complicated to be recombination, then learning

to enumerate may be a form of bootstrapping.

2 D. Barner has suggested (personal communication, June 14, 2012) that all prior bootstrapping theories

appear to require that earlier representational stages be psychologically necessary steps in the acquisition

of later ones, whereas earlier representations of number in Piantadosi et al.’s theory (e.g., their Two-

knower function) play no role in producing its later representations (e.g., the CP-knower function). (See

Section 2.2 for a description of this function.)


A second potential way to distinguish bootstrapping and recombination is to hold that the input

concepts (c1, c2, …, ck) in bootstrapping come from a broader domain of knowledge than is possible in

recombination. In the case of number learning, the input concepts to bootstrapping may belong to two or

more distinct cognitive modules. For example, c1, c2, …, ci may come from a module devoted to natural

language quantifiers (e.g., some or all), whereas ci+1, ci+2, …, ck may come from a module for representing

small sets of physical objects. Or the input concepts may include some that don’t appear in the child’s

earlier number representations. For example, the old number representations may include only concepts

c1, c2, …, ci, whereas input to the new representations may also include concepts ci+1, ci+2,…, ck.

It is unclear to us whether either of these strategies suffices to show that bootstrapping and

recombination differ in kind. Sheer complexity of a process doesn’t seem inconsistent with

recombination. The individual steps in learning may be lengthy or difficult without creating anything

fundamentally new. Advocates of bootstrapping owe us an explanation of what aspects of learning cause

it to go beyond recombination. Similarly, why must recombination respect limits on the domain of its

input? Why shouldn’t recombination be allowed to draw on all old concepts in the learner’s repertoire?

Fodor’s (2010) response to Carey’s theory is to deny any such limits. We are not claiming that proponents

of bootstrapping have explicitly adopted either of these strategies. Nor do we claim that the strategies are

exhaustive. 3 In looking at Piantadosi et al.’s proposal, however, let’s keep these options temporarily

open, since they may help us see why these authors believe their model performs a type of bootstrapping.

The underlying issue with bootstrapping is that advocates have to reconcile the Learning and

Discontinuity claims. But doing so is tricky because these claims seem to pull in opposite directions, with

Learning suggesting continuity rather than discontinuity. If Learning and Discontinuity cannot be

reconciled, bootstrapping is incorrect and concept recombination is correct as a theory of human concept

3 The two strategies just described are examples of what Beck (submitted for publication) calls deflationary theories

for reconciling the bootstrapping claims about learning and discontinuity. Neither creates anything totally new to the

child’s conceptual system, but either could bring to light concepts that were only latent within this system. More

radical strategies are also possible for makings sense of the Learning and Discontinuity claims (Beck, submitted for

publication, and Shea, 2011). But these are farther removed from Piantadosi et al.’s theory, and we therefore don’t

discuss them here.


acquisition. A main point of interest, then, in Piantadosi et al.’s model is that it purports to furnish a

working example of bootstrapping and may therefore demonstrate bootstrapping’s viability.

2.2. The Piantadosi et al. model

The Piantadosi et al. model receives as input a series of sets of different sizes, ranging from one

to ten elements, with the frequency of each set size determined by the corpus frequency of the associated

number words (“one” to “ten”). Its goal is to learn which number word applies to each of the ten set sizes.

To do so, it constructs hypotheses from a fixed set of primitive functions by applying syntactic rules.

Among the primitives is a function singleton? that determines whether or not a set contains exactly one

member [e.g., singleton?({a}) = yes], a function next that determines the next numeral on the list of count

terms [e.g., next(“two”) = “three”], a function select that produces a set by picking an element from a

given set [e.g., select({a, b, c}) = {c}], a function set-difference that computes a set containing all the

members of a first set not contained in a second [e.g., set-difference({a, b, c}, {c}) = {a, b}], and a

function L that recursively applies an embedding function to an embedded one (we give an example

below).

The model evaluates the hypotheses it creates from these primitives, increasing the probabilities

of hypotheses that give the right answer (e.g., labeling a set of four items with “four”) and decreasing the

probabilities of hypotheses that give an incorrect or null answer. After sufficient training, the model

converges on a hypothesis—the Cardinal Principle (CP-) knower function—that correctly labels sets of

one to ten elements:

CP-knower function:

λS. (if (singleton? S)

“one”

(next (L (set-difference S (select S)))))

This function tests whether the input set of objects S is a singleton (i.e., one-element set), and if it is,

labels it “one.” If not, it removes an element from S (i.e., set-difference S (select S)) and recursively


applies the same CP-knower function to the reduced set (L accomplishes this recursion). If the reduced set

is a singleton, it labels it next(“one”) or “two.” And so on.

Here, for example, are the steps that CP-knower goes through in enumerating a set of three cups,

S = {cup1, cup2, cup3}:

A1. Is {cup1, cup2, cup3} a singleton?

A2. No, so evaluate the function next on the result of applying CP-knower to the set created by

subtracting an element from {cup1, cup2, cup3}:

B1. Is {cup1, cup2} a singleton?

B2. No, so evaluate the function next on the result of applying CP-knower to the set

created by subtracting an element from {cup1, cup2}.

C1. Is {cup1} a singleton?

C2. Yes, so return the value “one.”

B3. Return the value of next(“one”) = “two.”

A3. Return the value next(“two”) = “three.”

This style of computation will be familiar to Scheme or Lisp programmers.

More interesting is the order in which hypotheses emerge as the most likely candidate. Early in

training, the model’s best hypothesis labels one-element sets with “one” and all other set sizes as

unknown. It then switches to a hypothesis that labels one- and two-element sets correctly (using its

primitive singleton? and doubleton? predicates), then one-, two-, and three-element sets (using

singleton?, doubleton?, and tripleton?), and finally reaches a more complicated rule (the CP-knower

function) that correctly labels one- to ten-element sets.

This behavior is extensionally similar to the progress children make in acquiring words for set

sizes (see Section 3 for qualifications). The learning sequence is a result of several design choices: First,

the model starts with primitive predicates that: (a) directly recognize set sizes of one, two, and three

elements (e.g., the singleton? predicate); (b) carry out logic and set operations (e.g., set-difference);

(c) traverse the sequence of number words “one,” “two,” …, “ten” (the next predicate); and (d) perform


recursion (L). (See Piantadosi et al.’s Table 1 for the full list of 15 primitives.) Second, the model

constructs hypotheses from these primitives in a way that gives lower prior probabilities to lengthier

hypotheses and to hypotheses that include recursion (depending on a free parameter, γ). Thus, the model

starts by considering simple and inaccurate non-recursive hypotheses (e.g., singleton sets are labeled

“one” and all other sets are undefined) and ends with a more complex, but correct, recursive hypothesis as

the result of feedback about the correct labeling.

2.3. Does the Piantadosi et al. model employ bootstrapping?

Piantadosi et al. try to make the case that the discovery of the correct number hypothesis is a form

of bootstrapping, though not quite of the variety Carey described in introducing this term. But on the face

of it, their model looks like a perfect example of recombination. It starts with a small stock of primitives,

and it combines them into hypotheses according to syntactic rules (a probabilistic context-free grammar).

The model learns which of these hypotheses is best by Bayesian adjustment through feedback. Thus, all

the primitive concepts that the model uses to frame its final hypothesis are already present in its initial

repertoire. The only missing element is the correct assembly of these primitives by the grammar. These

restrictions would seem to leave the model with little room for innovation of the sort that bootstrapping

requires. Why should we regard the process as bootstrapping rather than as translating one system into

another?

In setting out the bootstrap idea in Section 2.1, we mentioned two possible strategies to

discriminate it from recombination. Of these possibilities, the first one—that bootstrapping involves a

learning process more complex than standard recombination—is out for the Piantadosi et al. model. The

model’s grammar composes its hypotheses by assembling them from a previously existing base. The

model employs no inference more complex than the Bayesian conditioning that updates the hypotheses’

probability.

Piantadosi et al. have a better chance, then, of defending their bootstrapping claim by adopting

the second strategy. Perhaps the model’s discovery of the pairing between numerals and cardinalities


incorporates concepts that aren’t available in its initial state. Here the obvious candidate is recursion. The

model’s initial hypotheses make no use of recursion, whereas the final CP-knower hypothesis does. The

recursive predicate (L) confers greater computational power on this last hypothesis than is present in the

earlier ones. So perhaps bootstrapping occurs when the model introduces recursion. (The model ensures

that this introduction happens relatively late in learning by handicapping all hypotheses containing the

recursive predicate, as we mentioned earlier.) This accords with Piantadosi et al.’s statement that “the

model bootstraps in the sense that it recursively defines the meaning for each number word in terms of the

previous number word. This is representational change much like Carey’s theory since the CP-knower

uses primitives not used by subset knowers, and in the CP-transition, the computations that support early

number word meanings are fundamentally revised” (p. 212). Similarly, they point out that “the sense in

which our model engages in bootstrapping is that it starts off representing number word meanings with

subitizing primitives (singleton? doubleton? tripleton?) and eventually transitions to a system in which

essentially other primitives (recursion, conditionals, etc.) are used” (personal communication, June 14,

2012). The idea seems to be that the model’s representations for cardinalities before bootstrapping aren’t

extendible to the representations it uses after. Something is missing from the early representations that’s

necessary for a more adult-like understanding.

But in thinking about whether the CP-transition is a form of bootstrapping, we should keep in

mind that in many mundane instances of learning—in discrimination learning, for example—people add

primitives that do not figure in earlier hypotheses. In learning to distinguish poisonous from edible

mushrooms, people may have to take into account new properties like the color of the mushrooms’ spores

that were not parts of their original mushroom representations. No one would claim, though, that

including spore color in the new concept is a discontinuous conceptual change. Likewise, merely

including a previously unused predicate in a new hypothesis about number meaning doesn’t by itself

imply that the hypothesis is discontinuous with old ones. Stretching the concept of Quinian bootstrapping

to include such simple property additions would trivialize this concept. So if adding the recursive


predicate L does produce a big conceptual change that must be because L is special—perhaps because it

significantly increases the hypothesis’s computational power—not because it is new.

However, adding recursion still doesn’t conform to bootstrapping as Carey describes it in the

quotations of Section 1. The model’s grammar prior to adopting the CP-knower hypothesis is identical to

its grammar after adopting it, as is its primitive conceptual vocabulary. So a translation manual could

easily express the new hypothesis in terms of these primitives, precisely as is done in Piantadosi et al.’s

definition of the CP-knower function, which we displayed earlier. This undermines the idea that

bootstrapping does not reduce to translation. For much the same reason, the CP-knower hypothesis in

Piantadosi et al.’s version does not involve the creation of new primitives, and it certainly doesn’t create

them in a way that goes beyond “the machinery of compositional semantics” (see the first of the

quotations from Carey in Section 1.1).

Of course, Piantadosi et al. could position their model as a non-Quinian type of “bootstrapping”

that allows translation and dispenses with the need to create new primitives. But this move would

abandon what we take to be the important and arresting ideas that Carey had in mind in introducing this

concept. Bootstrapping in this revised sense would not create a conceptual system that is discontinuous

with the earlier one, and hence, it would not implement a method that bears the same intellectual interest.

This revision would not simply raise the “semantic” issue of how we should use the term “bootstrapping,”

but it would discard one of bootstrapping’s essential properties. In short, the Piantadosi et al. model could

have made bootstrapping plausible by showing how the Learning and Discontinuity claims can be joined.

Instead, it either jettisons Discontinuity or trivializes it. The reasonable conclusion from the Piantadosi et

al. model is not that it provides a rigorous form of bootstrapping, but that it demonstrates bootstrapping as

unnecessary. Children have no need for bootstrapping, since they can learn a correct method of

enumeration through ordinary recombination.


3. How realistic is the model?

The points raised in the preceding section do not show that the model is incorrect as a theory of

how children learn to label cardinalities. Even if the model doesn’t learn by bootstrapping, it could still be

the right explanation of this learning process. However, three aspects of the model’s behavior deserve

comment and suggest that it does not learn enumeration in the way children do.

3.1. The model’s choice of primitives

First, the model draws on a relatively small set of handpicked primitives—the 15 predicates in

Piantadosi et al.’s Table 1. The model constructs all its hypotheses as combinations of these predicates.

Piantadosi et al. believe these predicates “may be the only ones which are most relevant” to number

learning (p. 202). But this restriction raises the question of whether children also limit their hypotheses in

the same convenient way. We don’t dispute the importance of these predicates to knowledge of number,

but how do children know prior to learning that these predicates are the most relevant ones?

Piantadosi et al. claim that their theory “can be viewed as a partial implementation of the core

knowledge hypothesis (Spelke, 2003),” but also deny that the primitive predicates are part of an

encapsulated core domain devoted to number: “These primitives—especially the set-based and logical

operations—are likely useful much more broadly in cognition and indeed have been argued to be

necessary in other domains” (p. 202). If this particular set of primitives does not come cognitively pre-

packaged, however, then children must search for them among a much larger group of predicates, and we

need an explanation of how they manage to select just those that appear in the model’s hypotheses. This

problem is pressing because many candidates exist in this larger set that carry numerical information but

aren’t included among the model’s primitives. Although the model has a few primitives (e.g., set

intersection) that are not necessary for its hypotheses, the model excludes, by fiat, analog magnitudes,

mental models, and explicit quantifiers (e.g., some), which according to many theories are relevant parts


of children’s beliefs about number prior to (and even after) they master enumeration.4 Consider analog

magnitudes. According to this idea, people have access to a continuous mental measure that varies

positively with the number of physical objects in their perceptual array. People can therefore use this

measure as an approximate guide to cardinality. A model like Piantadosi et al.’s could easily build a

hypothesis that makes use of analog magnitudes to label set sizes, and such a hypothesis would compete

with those of the present version of the model. It could therefore slow or alter the course of number

learning by delaying the success of the CP-knower hypothesis.

On the one hand, if Piantadosi et al.’s predicates are truly the only ones children use in forming

their hypotheses, then we need to know what enables the children to restrict their attention to these items

and exclude information like analog magnitudes. On the other hand, if children consider a wider set of

predicates, what’s the evidence that the model can converge on the right CP-knower procedure and do so

in a realistic amount of time? The quantitative results from Piantadosi’s simulations are not informative

under this second possibility.

3.2. The model’s method of enumeration

A second question about the model’s fidelity is whether the CP-knower procedure is similar

enough to children’s actual enumeration to back the claim that the model learns what children do.

Children match numerals one-one to objects in an iterative way (Gelman & Gallistel, 1978). In counting a

set of three cups {cup1, cup2, cup3}, they label a first object (e.g., cup1) “one” and remove it from further

consideration. They then label the second object (e.g., cup2) “two,” and so on. As Piantadosi et al. point

out, however, “the model makes no reference to the act of counting (pointing to one object after another

while producing successive number words)” (p. 213). As we saw in the example of Section 2.2, what the

4 For analog magnitudes, see, for example, Dehaene (1997), Gallistel and Gelman (1992), and Wynn (1992). For

mental models, Mix, Huttenlocher, & Levine (2002). For quantifiers, Barner and Bachrach (2010), Carey (2009),

and Sarnecka, Kamenskaya, Yamana, Ogura, and Yudovina (2007).


model does instead is recurse through the set of items, taking set differences until it arrives at a singleton,

and then it unwinds through the list of numerals to arrive at the total.

We can put this point about the difference between children’s behavior and the model’s in a

second, semantic way: When older children count “one, two, three…three cups,” the first three number

words do not label the size of sets of cups. Instead, these words refer to ordinal positions in the sequence

of the to-be-counted items. The children then rely on the principle—Gelman and Gallistel’s (1978)

Cardinal Principle—that the final word in the enumeration sequence is the cardinality of the set, and so

they infer that there are “three cups.” Thus, only the second “three” in the earlier phrase denotes a number

of cups. By contrast, the model always uses numerals as labels for set sizes. In their discussion (pp. 214-

215), Piantadosi et al. claim that children’s actual counting behavior is a metacognitive effort to keep their

place in the recursive routine. But what reason could there be for not taking the children’s simpler (and

equally accurate) enumeration algorithm at face value? The model’s inability to arrive at the right

procedure—the procedure children actually use—suggests that something is wrong with its architecture

and calls into question Piantadosi et al.’s claim (p. 200) that “all assumptions made are computationally

and developmentally plausible.”

3.3. The model’s knowledge of the sequence of cardinalities

A third difference between children’s behavior and the model’s behavior is the extent of

children’s beliefs about number at the time they become CP-knowers. The usual test of CP knowledge is

that children can correctly produce sets of up to ten objects when asked to “Give me n.” For example,

when asked to “Give me eight beads,” they can produce eight from a larger pile of beads. Recent evidence

by Davidson, Eng, and Barner (2012), however, shows that children who are able to perform this task are

often unable to say whether a single bead added to a box of five results in six beads rather than seven.

This is the case even though children at the same stage can correctly say that the numeral that follows

“five” is “six” rather than “seven.” Davidson et al. (2012, p. 166) note that their analysis “reveals that

many CP-knowers do not have knowledge of the successor principle for even the smallest numbers…


These data suggest that knowledge of the successor principle does not arise automatically from becoming

a CP-knower, but that this semantic knowledge may be acquired later in development.”

The Piantadosi et al. model is limited to determining the cardinality of a given set of objects. So

no logical inconsistency arises between possessing this skill and not being able to tell that one object

added to a set of five yields a set of six. Still, Piantadosi et al.’s CP-knower function, in the course of

determining that “six” labels a six-item set, also determines that “five” labels a set with one fewer

element. (Steps A3 and B3 in the example of Section 2.2 show the analogous relation between “three”

and “two.”) Given this procedure, children’s difficulty in figuring out that a six-item set is one greater

than a five-item set is mysterious and again suggests that the model’s CP-knower function is more

complex than the routine children actually use at this point in their number development.

Piantadosi et al. (pp. 206) describe their theory as a computational-level model, in the sense of

Marr (1982). So perhaps we should discount these deviations between the model’s behavior and

children’s, since they concern particular methods of pairing number words and cardinalities. The crucial

claims of the Piantadosi et al. paper, however, depend on more than computational description. For

example, whether the model learns by bootstrapping depends on whether the procedures the model

employs before becoming a CP-knower are qualitatively different from the procedure it employs later.

This difference requires a comparison of the algorithms before and after learning, and it limits how

abstractly we can view the model when we come to evaluate it (see Jones & Love, 2011, for general

criticisms along these lines of Bayesian learning theories). We take Piantadosi et al. to be committed to

procedures, such as One-knower, Two-knower, and CP-knower, that are parts of their “language of

thought” theory (but not necessarily to particular implementation details, such as the method of search in

hypothesis space).

4. How much does the model know about the positive integers?

An intriguing aspect of bootstrapping is that it is supposed to produce the child’s first true

representation of the positive integers. According to Carey (2004, p. 65), “coming to understand how the


count list represents numbers reflects a qualitative change in the child’s representational capacities; I

would argue that it does nothing less than create a representation of the positive integers where none was

available before.” Similarly, according to Piantadosi et al. (p. 201):

Bootstrapping explains why children’s understanding of number seems to change so

drastically in the CP-transition and what exactly children acquire that’s “new”: they

discover the simple recursive relationship between their memorized list of words and the

infinite system of numerical concepts.

In Section 2, we examined the issue of whether Piantadosi et al.’s model effects a qualitative change in

representations. But setting that issue aside here, how much does the model know about the positive

integers—the “infinite system of numerical concepts”?

4.1. Does bootstrapping capture the meaning of the first few numerals?

One thing seems clear. The model never learns the full set of positive integers or the key

successor function that generates this set. It learns only the pairing of numerals and cardinalities for the

numerals on its count list.

In another sense, however, the model does possess a general rule for relating number words and

cardinalities: the CP-knower function, shown in Section 2.2. Piantadosi et al. write, “bootstrapping has

been criticized for being incoherent or logically circular, fundamentally unable to solve the critical

problem of inferring a discrete infinity of novel numerical concepts (Rips, Asmuth, & Bloomfield, 2006,

2008; Rips, Bloomfield, & Asmuth, 2008). We show that this critique is unfounded…” (p. 200). But

although we do believe that bootstrapping is unable to solve the “problem of inferring a discrete infinity

of novel numerical concepts,” we did not criticize bootstrapping as inconsistent or circular.5 Moreover,

5 A threat of circularity looms, however, if you read too much into the bootstrap conclusion. You may be tempted to

think that the conclusion fixes the cardinal meaning of the numerals if you understand “next term on the count list”

as involving the full, infinite list for the positive integers. The full list does, of course, fix the numeral’s meaning

since it is isomorphic to the positive integers. But at the time children perform the bootstrap inference, they have no

knowledge of the full list; so assuming this structure as part of the bootstrapping process does lead to circularity.


the bootstrap conclusion (i.e., the next item on the numeral list refers to the set size given by adding one

to that of the preceding numeral) itself is a correct generalization about number word-cardinality pairs, as

we have noted. What is learned is a correlation between advancing one step in the number word sequence

(e.g., from “four” to “five”) and increasing the cardinality of a set by one. (In the Piantadosi et al. model,

this correlation is implicit in the CP-knower procedure rather than declaratively represented, but the effect

is the same.) This is an important discovery for children, and any theory that explains how they do it is

praiseworthy.

The trouble with this principle, however, is that, at the time children learn it, it fails to specify the

meaning of the terms for the positive integers (Rips et al., 2006; Rips, Asmuth, & Bloomfield, 2008; Rips,

Bloomfield, & Asmuth, 2008). After adopting the CP-knower function, the Piantadosi et al. model has a

way to connect the word “one” to cardinality one, “two” to cardinality two,…, and “ten” to cardinality

ten. But the same function is equally extendible to either of the mappings in (1) and (2), as well as an

infinite number of others:

(1) “one” denotes only cardinality one.

“two” denotes only cardinality two.

…

“ten” denotes only cardinality ten.

(2) “one” denotes cardinalities one, eleven, twenty-one,…

“two” denotes cardinalities two, twelve, twenty-two,…

…

“ten” denotes cardinalities ten, twenty, thirty,…

That is, the CP-knower function doesn’t constrain the cardinal meanings of the number words on the

child’s list to their ordinary meanings.

Proponents of bootstrapping now appear to agree with us that the CP-knower function and its

equivalents don’t give children the meanings for numerals beyond those on their list of count terms. But it

doesn’t necessarily give them the correct meanings for numerals on their count lists either, as (1) and (2)


reveal. Knowing that a correlation exists between the numerals and the cardinalities is of no help in

picking out the positive integers from among its rivals unless the child knows either the structure of the

numerals or the structure of the cardinalities. However, the numeral sequence, as given by the next

predicate in Piantadosi et al.’s model, does not continue beyond “ten,” and as Piantadosi et al. emphasize

(p. 212), their model does not build in a successor relation for cardinalities.6 Because the structure of the

positive integers is well understood, we can be quite specific about what the CP-knower function fails to

convey. It does not enforce the ideas that the correct structure is one that has: (a) a unique first element,

(b) a unique immediate successor for each element, (c) a unique immediate predecessor for each element

except the first, and (d) no element apart from those dictated by (a)-(c).

4.2. Can the model exclude rival meanings for the integers?

Results from their simulations show that Piantadosi et al.’s model learns the standard pairing for

the first ten integers rather than an alternative pairing in which “one” is mapped to sets with one or six

elements, “two” to sets with two or seven elements, …, and “five” to sets with five or ten elements. This

latter Mod-5 hypothesis (see their Figure 1) fails for two reasons: First, the model receives feedback that

disconfirms the Mod-5 pairings, and second, the Mod-5 hypothesis is more complex than the correct

alternative, given the choice of primitives. When feedback supports the Mod-5 hypothesis, however, the

model eventually learns it. From these facts, Piantadosi et al. (p. 211) conclude:

This work was motivated in part by an argument that Carey’s formulation of

bootstrapping actually presupposes natural numbers, since children would have to know

the structure of the natural numbers in order to avoid other logically plausible

generalizations of the first few number word meanings. In particular, there are logically

possible modular systems which cannot be ruled out given only a few number word

6 The limit at “ten” is, of course, a computational convenience. The largest numeral on the model’s count list could

have been a larger or smaller value, in accord with the fact, mentioned earlier, that the largest numeral on children’s

count lists also varies. However, the problem in the text holds no matter what the upper limit happens to be.


meanings (Rips et al., 2006; Rips, Asmuth, & Bloomfield, 2008; Rips, Bloomfield, &

Asmuth, 2008). Our model directly addresses one type of modular system along these

lines: in our version of a Mod-N knower, sets of size k are mapped to the k mod Nth

number word. We have shown that these circular systems of meaning are simply less

likely hypotheses for learners. The model therefore demonstrates how learners might

avoid some logically possible generalizations from data…

The problem for theories of number learning, however, is not eliminating hypotheses that the data directly

disconfirm, such as Piantadosi et al.’s Mod-5 hypothesis. Instead, the difficulty lies in selecting from the

infinitely many hypotheses that have not been disconfirmed. For the simulations in Piantadosi et al., these

would include Mod-11, Mod-12, Mod-13, …. The model can’t decide among these hypotheses because

its list of numerals stops at “ten” and because it has no information about cardinalities greater than ten

(see Footnote 6).

Hypotheses like Mod-11 might seem syntactically complex relative to the CP-knower function. If

so, the model would prefer CP-knower to Mod-11, even without training on sets of eleven, due to the

model’s assignment of higher prior probabilities to simpler hypotheses. But this is not the case. How

simple or complex a function must be to capture Mod-11 depends on the structure of the numeral list

beyond “ten” (which we are assuming, without loss of generality, is the child’s highest count term). If the

list continued, “one,” “two,”…, “ten,” “one,” “two,”…, “ten,” “one,” “two,” …, “ten,”…, then the CP-

knower function would respond exactly in accord with Mod-11. Since neither children nor the Piantadosi

et al. model knows how the count list continues, syntactic complexity can’t decide between Mod-11 and

the standard meanings of the numerals; that is, it can’t discriminate between (1) and (2), above. (This is a

variation of Goodman’s, 1955, famous point about the role of syntactic complexity in induction.) In other

words, a model could fail to assign the correct meaning to number words either because it mapped the

standard sequence of number words to the wrong sequence of cardinalities or because it mapped an

incorrect sequence of number words to the right sequence of cardinalities. The literature on this topic

sometimes presupposes that children have access to the correct number word sequence, but they don’t at


the point at which they make the bootstrap inference. They have to induce both the numeral sequence and

the number sequence, and to coordinate them in order to understand the positive integers.

The message in our earlier papers was that the bootstrap conclusion does nothing to settle the

question of whether the cardinal meaning of the first few numerals is given by their usual (adult) meaning

or by Mod-11, Mod-12, and so on. The same is true of Piantadosi et al.’s CP-knower function. As Rey

(2011) has pointed out in connection with Carey’s proposal, this issue is closely related to classic poverty-

of-the-stimulus arguments for learning natural language (e.g., Chomsky, 1965). Proponents of

bootstrapping could contend that children’s beliefs about the meaning of the numerals suffers from the

same problem that the bootstrap conclusion does. Adults clearly know that (1), and not (2), represents the

correct meaning, but children may distinguish them only at a later point in their number development.

However, this conclusion, if it is true, places a stark limit on how much children learn about the numerals

from the bootstrap’s conclusion.

Piantadosi et al. begin to acknowledge this difficulty in noting that “the present work does not

directly address what may be an equally interesting inductive problem relevant to a full natural number

concept: how children learn that next always yields a new number word” (pp. 211-212). They believe that

“similar methods to those that we use to solve the inductive problem of mapping words to functions could

also be applied to learn that next always maps to a new word. It would be surprising if next mapped to a

new word for 50 examples, but not for the 51st” (ibid.). But this conjecture is not obviously correct: Most

lists that children learn—the alphabet, the months of the year, the notes of the musical scale—don’t have

the structure of the natural numbers. Next for the English alphabet ends at the 26th item, and next for the

sequence of U.S. Presidents currently ends at the 44th.

The crucial difficulty, as we’ve emphasized, is that learning the mapping between the numerals

and the cardinalities for one to ten can’t eliminate nonstandard sequences, such as Mod-11, unless

children can somehow induce the correct structure. The structure could come from the cardinalities for the

positive integers, or it could come from the structure of the numerals for these integers, since these

structures are isomorphic. But it has to come from somewhere. Bootstrapping allows children to exploit


the numeral sequence to determine the labels for cardinalities. But this strategy can’t pick out the right

cardinal meanings—it merely passes the buck—unless the problem about how “next always yields a new

number word” is resolved.

5. Conclusions

On our view, the Piantadosi et al. model doesn’t bootstrap. It therefore doesn’t help vindicate

bootstrapping as a cognitive process. What the model does is form hypotheses by recombining its

primitives and confirming them statistically. So should we conclude that children can learn to enumerate

through this (non-bootstrapping) sort of hypothesis formation and confirmation? Perhaps, although

accepting this conclusion depends on ignoring the facts that (a) the model learns by combining a pre-

selected set of primitives but gives no account of how they are selected from the larger set of potential

primitives, (b) finishes with a procedure that differs in important ways from children’s, and (c) has a

firmer grasp of the sequence of cardinalities than children have. But even if the model is a correct

description of how children learn to enumerate, the model still faces the problem that it leaves an

unlimited set of possibilities for the meanings of the first few count terms.


Acknowledgements

We thank David Barner, Jacob Beck, Jacob Dink, Brian Edwards, Emily Morson, James Negen, Steven

Piantadosi, and Barbara Sarnecka for comments on an earlier draft of this article. IES grant

R305A080341 helped support work on this paper.


References

Barner, D., & Bachrach, A. (2010). Inference and exact numerical representation in early language

development. Cognitive Psychology, 60, 40-62. doi: 10.1016/j.cogpsych.2009.06.002

Beck, J. (submitted for publication). Can bootstrapping explain concept learning?

Bloom, P., & Wynn, K. (1997). Linguistic cues in the acquisition of number words. Journal of Child

Language, 24, 511-533. doi: 10.1017/s0305000997003188

Carey, S. (2004). Bootstrapping and the origin of concepts. Daedalus, 133, 59-68.

Carey, S. (2009). The origin of concepts. New York, NY: Oxford University Press.

Carey, S. (2011). Concept innateness, concept continuity, and bootstrapping. Behavioral and Brain

Sciences, 34, 152-161. doi: 10.1017/S0140525x10003092

Chomsky, N. (1965). Aspects of the theory of syntax. Cambridge, MA: M.I.T. Press.

Davidson, K., Eng, K., & Barner, D. (2012). Does learning to count involve a semantic induction?

Cognition, 123, 162-173. doi: 10.1016/j.cognition.2011.12.013

Dehaene, S. (1997). The number sense: How mathematical knowledge is embedded in our brains. New

York: Oxford University Press.

Fodor, J. A. (1975). The language of thought: A philosophical study of cognitive psychology. New York:

Crowell.

Fodor, J. A. (1981). The present status of the innateness controversy. Representations: Philosophical

essays on the foundations of cognitive science (pp. 257-316). Cambridge, MA: MIT Press.

Fodor, J. A. (2010, October 8). Woof, woof [Review of the book The Origin of Concepts, by S. Carey].

Times Literary Supplement, pp. 7-8.

Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43-

74. doi: 10.1016/0010-0277(92)90050-r

Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, Mass.: Harvard

University Press.


Goodman, N. (1955). Fact, fiction and forecast. Cambridge, MA: Harvard University Press.

Jones, M., & Love, B. C. (2011). Bayesian Fundamentalism or Enlightenment? On the explanatory status

and theoretical contributions of Bayesian models of cognition. Behavioral and Brain Sciences,

34, 169-188. doi: 10.1017/s0140525x10003134

Leslie, A. M., Gelman, R., & Gallistel, C. R. (2008). The generative basis of natural number concepts.

Trends in Cognitive Sciences, 12, 213-218. doi: 10.1016/j.tics.2008.03.004

Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the

acquisition of number concepts. Cognition, 106, 924-939. doi: 10.1016/j.cognition.2007.03.003

Marr, D. (1982). Vision: A computational investigation into the human representation and processing of

visual information. San Francisco: W.H. Freeman.

Mix, K. S., Huttenlocher, J., & Levine, S. C. (2002). Quantitative development in infancy and early

childhood. New York, NY: Oxford University Press.

Piantadosi, S. T., Tenenbaum, J. B., & Goodman, N. D. (2012). Bootstrapping in a language of thought:

A formal model of numerical concept learning. Cognition, 123, 199-217. doi:

10.1016/j.cognition.2011.11.005

Rey, G. (2011). Learning, expressive power, and mad dog nativism: The poverty of stimuli (and

analogies), yet again. Paper presented at the Society for Philosophy and Psychology, Montreal.

Rips, L. J., Asmuth, J., & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the

natural numbers. Cognition, 101, B51-B60. doi: 10.1016/j.cognition.2005.12.001

Rips, L. J., Asmuth, J., & Bloomfield, A. (2008). Do children learn the integers by induction? Cognition,

106, 940-951. doi: 10.1016/j.cognition.2007.07.011

Rips, L. J., Bloomfield, A., & Asmuth, J. (2008). From numerical concepts to concepts of number.

Behavioral and Brain Sciences, 31, 623-642. doi: 10.1017/s0140525x08005566

Rips, L. J., & Hespos, S. J. (2011). Rebooting the bootstrap argument: Two puzzles for bootstrap theories

of concept development. Behavioral and Brain Sciences, 34, 145-146.

doi:10.1017/S0140525X10002190


Sarnecka, B. W., Kamenskaya, V. G., Yamana, Y., Ogura, T., & Yudovina, Y. B. (2007). From

grammatical number to exact numbers: Early meanings of 'one', 'two', and 'three' in English,

Russian, and Japanese. Cognitive Psychology, 55, 136-168. doi: 10.1016/j.cogpsych.2006.09.001

Shea, N. (2011). New concepts can be learned. Biology & Philosophy, 26, 129-139. doi: DOI

10.1007/s10539-009-9187-5

Spelke, E. S. (2000). Core knowledge. American Psychologist, 55, 1233-1243. doi: 10.1037/0003-

066x.55.11.1233

Spelke, E. S. (2011). Quinean bootstrapping or Fodorian combination? Core and constructed knowledge

of number. Behavioral and Brain Sciences, 34, 149-150.

Wynn, K. (1992). Children's acquisition of the number words and the counting system. Cognitive

Psychology, 24, 220-251. doi: 10.1016/0010-0285(92)90008-p

Documents

Can statistical learning bootstrap the integers?