Fodor

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Fodors Problem The Creation of New
Representational Resources

• Descriptive ProblemC1-C2, whats qualitatively
  new?
• Explanatory Problemlearning mechanism?

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Fodors 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.

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Meeting 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.

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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.
  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.)

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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

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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

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Interim 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.

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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 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.

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Wynns 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.

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Whats On This Card? Procedure
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1 knowers. Use two for all numbers gt 1. (N
7 mean age 30 months)
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Two knowers Have mapped one and two. Use
three to five for all numbers gt 2. (N 4
mean age 39 months)
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LeCorres 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.

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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, threeeight
  contrast numerically with one (more than one),
  but dont know what two means.
• Constrain learning story, because tell us
  intermediate steps.

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Descriptive 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

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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.

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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

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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 )

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The Pirahã
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Quantifiers in Pirahã

• hói (falling tone) one
• hoí (rising tone) two
• baagi many

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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

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Eliciting 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

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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?

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Peter 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

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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
•      Dont annoy your subjects, they might kill
  you

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One-to-One Line Match
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Line Copying
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. Line Draw Copy
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Cluster-Line Match
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Orthogonal Line Match
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Brief Presentation (Subitizing)
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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 Webers Law
• Comparable to studies with larger n, human
  adults without counting, and with animals and
  infants

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Summary of Number Studies

• Small numbers Parallel Individuation (accurate)
• Large Numbers Analog Estimation (inaccurate)

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Conclusions, 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

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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

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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

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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).

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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)

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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.

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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.

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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)

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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 relationsnext in the count list, and
  next in series of sets related by open a new
  object file.

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continued

• 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.

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Surprising 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.

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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

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New representational power

• Obtained by integrating representations from
  distinct constructed and core systems.

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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.