Title: Fodor
1Fodors Problem The Creation of New
Representational Resources
- Descriptive ProblemC1-C2, whats qualitatively
new? - Explanatory Problemlearning mechanism?
2Fodors 2 line argument
- Hypothesis testing the only learning mechanism we
know. - Cant test hypotheses we cant represent thus
hypothesis testing cannot lead to new
representational resources.
3Meeting Fodors Challenge
- 1) Descriptive Characterize conceptual system
1 (CS1) at time 1 and CS2 at time 2,
demonstrating sense in which CS2 transcends, is
qualitatively more powerful than CS1. - 2)Explanatory Characterize the learning
mechanism that gets us from CS1 to CS2.
4Case Study for Today
- Number. Two core systems described in
- Feigenson, Spelke and Dehaene in the TICS in your
package. - --1) Analog magnitude representations of number.
Dehaenes number sense. - --2) Parallel representation of small sets of
individuals. When individuals are objects,
object indexing and short term memory system of
mid-level vision (Pylyshyn FINSTs, Triesmans
object-files.)
5The Descriptive Challenge
- CS1 the three core systems with numerical
content described last time. - Analog magnitude representations
- Parallel Individuation
- Set-based quantification of natural language
semantics - CS2 the count list representation of the
positive integers
6Transcending Core Knowledge
- Parallel Individuation
- --No symbols for integers
- --Set size limit of 3 or 4
- Analog Magnitude Representations
- --Cannot represent exactly 5, or 15, or 32
- --Obscures successor relation
- Natural Language Quantifiers
- --Singular (1), Dual (2) sometimes Paucal or
Triple (3 or few), many, some - --No representations of exact numbers above 3
7Interim conclusion
- 1) infants represent numberyes, but not
natural number. Specify representational
systems, computations they support, can be
precise what numerical content they include - 2) Descriptive part of Fodors
challengecharacterized how natural number
transcends (qualitatively) input (the three core
systems) - --Infants, toddlers less than 3 ½, people with no
explicit integer list representation of number
(e.g., Piraha, Gordon, in press, Science), cannot
think thoughts formulated over the concept seven.
8Descriptive Challenge
- Met by positive characterization of CS1, CS2
(format, content of representational systems,
computations they enter into) - Also important evidence for difficulty of
learning. (a seems understand with adult
semantic force as soon as it is learneda blicket
vs. a blickish one, a ball vs. some balls, a dax
vs. Dax) in constrast children know the words
two and six, know they are quantifiers
referring to pluralities for 9 to 18 months,
respectively, before they work out what they mean.
9Wynns Difficulty of Learning Argument
- Give a number
- Point to x
- Whats on this card
- Can count up six or ten, even apply counting
routine to objects, know what one means for 6
to 9 months before learn what two means, takes
3 or 4 months to learn what three, and still
more months to learn four/induce the successor
function.
10 Whats On This Card? Procedure
111 knowers. Use two for all numbers gt 1. (N
7 mean age 30 months)
12Two knowers Have mapped one and two. Use
three to five for all numbers gt 2. (N 4
mean age 39 months)
13LeCorres studies
- Within-child consistency in knower-levels on
give-a-number and whats on this card - Two used as a generalized plural marker by
many one knowers. - Partial knowledge of one, two, three- knowers
does not include mapping to analog magnitudes.
14Interim conclusions
- Further evidence for discontinuity. If integer
list representation of natural number were part
of core knowledge, then would not expect have
identified the English list as encoding number,
know what one means and that two, threeeight
contrast numerically with one (more than one),
but dont know what two means. - Constrain learning story, because tell us
intermediate steps.
15Descriptive Challenge
- Systems of representations not part of core
knowledge might not be cross-culturally
universal. - Peter Gordons Piraha, Dehaene et al.s
Munduruku. Same issue of Science
16Cultures with no representations of natural
number?
- First generation of anthropologists
- 19th century colonial officers
- Many cultures with natural language quantifiers
only (1, 2, many, or 1, 2, 3, many) - Much variety in systems that could represent
exact larger numbers, intermediate steps to
integer lists with recursive powers to represent
arbitrarily large exact numbers.
17Is existence of 1-2-many systems a myth?
(Zaslavsky, 1974 Gelman Gallistel, 1978)
- Innumerate societies or alternative counting
- Systems?
- Finger Gestures, Sand Marking, Body Counting
System - Non-Decimal Systems (e. g., Gumulgal, Australia)
- urapun, 1
- okasa, 2
- okasa urapun, 21 ( 3)
- okasa okasa urapun 221 ( 5)
- Counting Taboos
18The Pirahã Peter Gordon, Columbia University
- Hunter-gatherers
- Semi Nomadic
- Maici River (lowland Amazonia)
- Pop about 160 - 200
- Villages 10 to 20 people
- Monolingual in Pirahã
- Resist assimilation to Brazilian culture
- Limited trading (no money)
- No external representations (writing, art, toys )
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21The Pirahã
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25Quantifiers in Pirahã
- hói (falling tone) one
- hoí (rising tone) two
- baagi many
26Pirahã Numbers
- No evidence of taboos or base-3 recursive
counting - Pirahã directly name numerosities rather than
counting them - Number words are not consistent, but are
approximations - Finger Counting?
- Yes, finger representation of number but not
counting
27Eliciting Number Representations
- Lemons Number word Fingers
- 1 hói
- 2 hoí 2
- baagi
- 3 hoí 3
- 4 hoí 5 - 3
- baagi
- 5 baagi 5
- 6 baagi 6 - 7
- 7 hói 1 - 8
- 8 5 - 8 - 9
- 9 baagi 5 - 10
- 10 5
28Non-linguistic Number Representation Tasks
- Core knowledge evidence for
- --Small, exact, number of objects. Parallel
individuation of 3 or 4 object files? - --Large approximate number. Analog magnitude
representations? - Any evidence for representation of large exact
number, even in terms of 1-1 correspondence with
external set?
29Peter Gordons Studies
- Can the Piraha perceive exact numerosities
despite the lack of linguistic labels? - Developed tasks that required creation of
numerosity. Could be solved without counting if
used 1-1 correspondence with fingers, or between
objects. - Progressively more difficult
- One-to-one mapping
- Different configurations
- Memory representations of number
30Limitations
- Carried out in 2 villages in 6 weeks
- Very limited language skills
- Total of 7 subjects, most tasks only have 4
to 5 subjects - Payment for participation (food, beads
etc.), but easily bored - Dont annoy your subjects, they might kill
you
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32One-to-One Line Match
33Line Copying
34. Line Draw Copy
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36Cluster-Line Match
37Orthogonal Line Match
38 Brief Presentation (Subitizing)
39Nuts-in-Can Task
40Averaged Responses Across Tasks
41Evidence for Analogue Estimation
- Mean Responses track target values perfectly
- (rules out performance explanations)
- Coefficient of variability constant over 3.
- Estimation follows Webers Law
- Comparable to studies with larger n, human
adults without counting, and with animals and
infants
42Summary of Number Studies
- Small numbers Parallel Individuation (accurate)
- Large Numbers Analog Estimation (inaccurate)
43Conclusions, Gordons Studies
- Pirana have only core knowledge of number
- Natural language quantifiers,
- Analog Magnitude Representations,
- Parallel Individuation of Small sets of objects
- Further evidence that positive integers not part
of core knowledge, require cultural construction
44Intermediate Systems
- 1) External individual files. (Fingers, pebbles,
notches on bark or clay, lines in sand).
Represents as do object files, 1-1
correspondence. Exceeds limit on parallel
individuation by making symbols for individuals
external. - 2) External individual files with base system
- 3) Finite integer list, no base system
- 4) Potentially infinite integer list, base
system - ALL THIS IN ILLITERATE SOCIETIES. SEPARATE
QUESTION FROM WRITTEN REPRESENTATIONS OF NUMBER
45Explanatory Challenge Quinian Bootstrapping
- Relations among symbols learned directly
- Symbols initially partially interpreted
- Symbols serve as placeholders
- Analogy, inductive leaps, inference to best
explanation - Combine and integrate separate representations
from distinct core systems
46Bootstrapping the Integer List Representation of
Integers
- How do children learn
- The list itself?
- The meanings of each word? (that three has
cardinal meaning three that seven means
seven)? - How the list represents number (for any word X
on the list whose cardinal meaning, n, is known,
the next word on the list has a cardinal meaning
n 1).
47Planks of the Bootstrapping Process
- Object file representations
- Analog magnitude representation
- (Capacity to represent serial order)
- Natural language quantificational semantics
- (set, individual, discrete/continuous more,
singular/plural)
48A Bootstrapping Proposal
- Number words learned directly as quantifiers, not
in the context of the counting routine - One is learned just as the singular determiner
a is. An explicit marker of sets containing
one individual - The plural marker -s is learned as an explicit
marker of sets containing more than one
individual.
49continued
- Two, three, four are analyzed as quantifiers
that mark sets containing more than one
individual. Some children analyze two as a
generalized plural quantifier, like some. - Two is analyzed as a dual marker, referring to
sets consisting of pairs of individuals. Three,
four, contrast with two. - Three is analyzed as a trial marker.
50Tests
- Role of natural language quantifier systems in
earliest partial meanings. One-knowers.
Chinese (Li, LeCorre et al) and Japanese
(Sarnecka) toddlers become one-knowers 6 months
later than English toddlers, in spite of equal
number word input (counting routine, CHILDES data
base) - Russian one-knowers make a distinction between
small sets (2, 3 and 4) and large sets (5 and
more), as does their plural system (Sarnecka)
51continued
- Meanwhile, the child has learned the counting
routine. - Child notices the identity of the first three
words in the counting routine and the singular,
dual, and trial markers one, two, three. - Child notices analogy between two distinct
follows relationsnext in the count list, and
next in series of sets related by open a new
object file.
52continued
- InductionIf X is followed by Y in the
counting sequence, adding an individual to an X
collection results in what is called a Y
collection. - Adding an individual is equivalent to adding
one, because one represents sets containing a
single individual.
53Surprising Conclusion
- One of the evolutionarily and ontogenetically
ancient systems of core knowledge that underlie
mature number representations (Core System
1analog magnitudes) seems to play no role in the
construction of natural number. - Becomes integrated about 6 months later (LeCorre)
and greatly enriches childrens number
representation.
54Quinian Bootstrapping
- Relations among symbols learned directly
- (one role for explicit symbols in language
general to all Quinian Bootstrapping) - Symbols initially partially interpreted (second
role for language in this case, idiosyncratic,
quantifier meanings as source of meaning). - Symbols serve as placeholders
- Analogy, inductive leaps, inference to best
explanation - Combine and integrate separate representations
from distinct core systems
55New representational power
- Obtained by integrating representations from
distinct constructed and core systems.
56Conclusions
- Other case studies rational number, theory
changes in childhood and in history of science - Parts of this overall process have been formally
modelled (e.g., structure mapping models of
analogical reasoning) others could be. - Proposal can be tested short of that however
(e.g., training studies, cross-linguistic
studies). - Uniquely human learning mechanism (because of
role for external symbols, serving as
placeholders.