Fodor’s Problem: The Creation of New Representational Resources

1. Fodor’s Problem: The Creation of New Representational Resources Descriptive Problem—C1-C2, what’s qualitatively new? Explanatory Problem—learning mechanism?

2. 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.

3. 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.

4. 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.)

5. 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

6. 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

7. 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.

8. 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.

9. 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.

10. “What’s on this card?” “What’s on this card?” “That’s right! It’s one apple.” (No model) “What’s on this card?” “What’s on this card?” “That’s right! It’s one bear.” (No model) “What’s On This Card?” : Procedure

11. “1 knowers.” Use “two” for all numbers > 1. (N = 7; mean age = 30 months)

12. “Two knowers:” Have mapped “one” and “two”. Use “three” to “five” for all numbers > 2. (N = 4; mean age = 39 months)

13. 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.

14. 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.

15. 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

16. 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.

17. 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

18. 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 …)

21. The Pirahã

25. Quantifiers in Pirahã • hói (falling tone) = “one” • hoí (rising tone) = “two” • baagi = “many”

26. 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

27. 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

28. 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?

29. 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

30. 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

32. One-to-One Line Match

33. Line Copying

34.  . Line Draw Copy

36. Cluster-Line Match

37. Orthogonal Line Match

38. Brief Presentation (Subitizing)

39. Nuts-in-Can Task

40. Averaged Responses Across Tasks

41. 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

42. Summary of Number Studies • Small numbers: Parallel Individuation (accurate) • Large Numbers: Analog Estimation (inaccurate)

43. 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

44. 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

45. 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

46. 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).

47. 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)

48. 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.

49. …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.

50. 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)