Download Causation and Theory Formation

Fodor’s Problem: The Creation
of New Representational
Resources
Descriptive Problem—C1-C2, what’s
qualitatively new?
Explanatory Problem—learning
mechanism?
Fodor’s 2 line argument
• Hypothesis testing the only learning
mechanism we know.
• Can’t test hypotheses we can’t represent;
thus hypothesis testing cannot lead to new
representational resources.
Meeting Fodor’s 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.
Case 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.
Dehaene’s “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, Triesman’s
object-files.)
The 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
Transcending 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
Interim conclusion
• 1) “infants represent number”—yes, but not
natural number. Specify representational
systems, computations they support, can be
precise what numerical content they include
• 2) Descriptive part of Fodor’s challenge—
characterized 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.
Descriptive 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 learned—a
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.
Wynn’s Difficulty of Learning
Argument
Give a number
Point to x
What’s 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.
“What’s On This Card?” : Procedure
“What’s on this card?”
“That’s right! It’s one apple.”
“What’s on this card?”
(No model)
“What’s on this card?”
“What’s on this card?”
(No model)
“That’s right! It’s one bear.”
“1 knowers.” Use “two” for all
numbers > 1.
(N = 7; mean age = 30 months)
Slope from 2 to 8
2
1.5
1
0.5
0
-0.5
-1
“Two knowers:” Have mapped “one” and “two”.
Use “three” to “five” for all numbers > 2.
(N = 4; mean age = 39 months)
Average of
slopes (3 to 8)
0.2
0
-0.2
-0.4
-0.6
-0.8
LeCorre’s studies
• Within-child consistency in knower-levels
on give-a-number and what’s on this card
• “Two” used as a generalized plural marker
by many “one knowers.”
• Partial knowledge of “one, two, threeknowers” does not include mapping to
analog magnitudes.
Interim 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,
three…eight” contrast numerically with “one”
(more than one), but don’t know what “two”
means.
• Constrain learning story, because tell us
intermediate steps.
Descriptive Challenge
• Systems of representations not part of
core knowledge might not be crossculturally universal.
• Peter Gordon’s Piraha, Dehaene et al.’s
Munduruku. Same issue of Science
Cultures 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.
Is 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,
okasa,
okasa urapun,
okasa okasa urapun
• Counting Taboos
1
2
21 (= 3)
221 (= 5)
The Pirahã
Peter Gordon, Columbia
 Hunter-gatherersUniversity








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 …)
The Pirahã
Quantifiers in Pirahã
• hói (falling tone) = “one”
• hoí (rising tone) = “two”
• baagi = “many”
Pirahã 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
Eliciting Number Representations
Lemons
1
2
3
4
5
6
7
8
9
10
Number word
hói
hoí
baagi
hoí
hoí
baagi
baagi
baagi
hói
baagi
Fingers
2
3
5-3
5
6-7
1-8
5-8-9
5 - 10
5
Non-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?
Peter Gordon’s 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
Limitations
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
· Don’t annoy your subjects, they might kill
you
One-to-One Line Match
1-1 Line M atch
Proportion Correct
1
0.75
0.5
0.25
0
Targe t
Line Copying
. Line Draw Copy

Line -draw Copy
Proportion Correct
1
0.75
0.5
0.25
0
Targe t
1
2
3
4
5
6
7
Cluster-Line Match
Clus te r-Line M atch
Proportion Correct
1
0.75
0.5
0.25
0
Targe t 2
3
4
5
6
7
8
9 10
Orthogonal Line Match
Orthogonal Line Match
Proportion Correct
1
0.75
0.5
0.25
0
Target 1
2
3
4
5
6
7
8
9
Brief Presentation (Subitizing)
Brie f Pre s e ntation
Proportion Correct
1
0.75
0.5
0.25
0
Targe t 1
2
3
4
5
6
7
8
9
Nuts-in-Can Task
Nuts -in-Can Tas k
Proportion Correct
1
0.75
0.5
0.25
0
Targe t 2
3
4
5
6
7
8
9
Averaged Responses Across Tasks
9
2
8
7
1.5
5
1
4
SD
Mean
6
Mean
SD
3
0.5
2
1
0
0
1
2
3
4
5
6
7
8
9
Target
Coeff Var
0.2
0.15
0.1
Coeff Var
0.05
0
1
2
3
4
5
Target
6
7
8
9
Evidence for Analogue Estimation
Mean Responses track target values perfectly
(rules out performance explanations)
Coefficient of variability constant over 3.
Estimation follows Weber’s Law
Comparable to studies with larger n,
human adults without counting, and with
animals and infants
Summary of Number Studies
• Small numbers: Parallel
Individuation (accurate)
• Large Numbers: Analog
Estimation (inaccurate)
Conclusions, Gordon’s 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
Intermediate 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
Explanatory 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
Bootstrapping 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).
Planks 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)
A 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.
…continued
• “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.
Tests
• 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)
…continued
• 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” relations—next in the count list, and
next in series of sets related by “open a new
object file.”
…continued
• Induction—If “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.
Surprising Conclusion
• One of the evolutionarily and
ontogenetically ancient systems of core
knowledge that underlie mature number
representations (Core System 1—analog
magnitudes) seems to play no role in the
construction of natural number.
• Becomes integrated about 6 months later
(LeCorre) and greatly enriches children’s
number representation.
Quinian 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
New representational power
• Obtained by integrating representations
from distinct constructed and core
systems.
Conclusions
• 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.