Fodor’s Problem: The Creation of New Representational

ResourcesDescriptive 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 inFeigenson, 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 numberPoint to xWhat’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?”

(No model)

“What’s on this card?”

“That’s right! It’s one apple.”

“What’s on this card?”

(No model)

“What’s on this card?”

“That’s right! It’s one bear.”

“1 knowers.” Use “two” for all numbers > 1.

(N = 7; mean age = 30 months)

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Number of stickers on card

Slope from 2 to 8

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“Two knowers:” Have mapped “one” and “two”. Use “three” to “five” for all numbers > 2.

(N = 4; mean age = 39 months)

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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, three- knowers” 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 cross-culturally 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, 1– okasa, 2– okasa urapun, 21 (= 3)– okasa okasa urapun 221 (= 5)

• Counting Taboos

The 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 …)

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 RepresentationsLemons Number word Fingers1 hói2 hoí 2

baagi3 hoí 34 hoí 5 - 3

baagi5 baagi 56 baagi 6 - 77 hói 1 - 88 5 - 8 - 99 baagi 5 - 1010 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

LimitationsCarried 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 Match

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Line Copying

. Line Draw Copy

Line-draw Copy

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Cluster-Line Match

Cluster-Line Match

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Orthogonal Line Match

Orthogonal Line Match

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Brief Presentation (Subitizing)

Brief Presentation

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Nuts-in-Can Task

Nuts-in-Can Task

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Averaged Responses Across Tasks

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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.