1. Core Knowledge System of mental representations, conceptual content Acquisition supported (partially) by innate, domain specific, learning mechanisms Entity identification supported by innate, domain specific input analyzers Evolutionarily ancient (often) Continuous throughout development

2. Case Study for Today Number. Two core systems described in Feigenson, Spelke and Dehaene --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.)

3. Core System 1: Analog Magnitude Representations Representations of approximate cardinal values of large sets of individuals (at least to the 100s)

8. Analog Magnitude Models Number represented by a quantity linearly related to the cardinal value of the set. One: __ Two: ____ Three: ______ Seven: ______________ Eight: ________________

9. Weber’s law Neural quantity a linear function of real world quantity, variance proportional to the mean Neural quantity a logarithmic function of real world quantity, constant variance

10. 8 vs. 16

11. Weber Fraction Signature Seven month-old infants succeed at: 8 vs. 16 (dots and streams of sounds) 16 vs. 32 Seven month-old infants fail at: 8 vs. 12 (both modalities) 16 vs. 24 By nine months of age, the ratio of success is 2:3. In both modalities.

12. Characteristics Amodal (input from different modalities transformed into abstract analog magnitude representation of number) Computations carried out over these representations: --addition, subtraction, numerical comparison, multiplication and division. Including in pre-arithmetic infants and preschoolers.

13. Core System 2 Parallel individuation, attention/short term memory—signature, limited to 3 or 4 items attended to in parallel (Pylyshyn multiple object tracking, subitizing, limits on short- term working memory) Not a dedicated number representational system. Implicitly represents the number of individuals in small sets

14. Converging measures of small-number representations: The locomotor choice task (Feigenson, Carey & Hauser, 2002)

17. Box reach task + 2 in, 1 out, compared to 1 in, 1 out + 3 in, 1 out, compared to 1 in, 1 out + 3 in, 2 out, compared to 2 in, 2 out X 4 in, 2 out, compared to 2 in, 2 out X 4 in, 1 out, compared to 1 in, 1 out

18. Parallel Individuation Models 1 cracker ■ 2 crackers ■ ■ 3 crackers ■ ■ ■ --one symbol for each individual --no symbols for integers

19. Underlies Habituation with small numbers Wynn’s addition and subtraction paradigms with small numbers Cracker choice Reach into box task In adults—parallel individuation in attention, short-term memory (Pylyshyn’s Multiple object tracking, subitizing)

20. Not a dedicated number representation system Computations defined over these representations include: --sum total spatial extent, comparisons of total volume/surface area/contour (simple habituation when individuals have same properties, cracker choice) --1-1 correspondence, establish numerical equality/inequality; numerical more/less (simple habituation when individuals have distinct properties, box reach choice) --chunking/set binding (can do two sets, each subject to limit of 3—e.g., cracker choice)

21. Numerical content Criteria of individuation and numerical identity. (spatio-temporal criteria for numerical identity). Successor function implicit. Operations defined over object files; compose sets, addition, subtraction, 1-1 correspondence to establish numerical equivalence, more less.

22. DISTINCT CORE SYSTEMS Analog Magnitude— Dedicated number representation, symbol for number Limited by Weber fraction Not concerned with properties of individuals, including size, nor with spatial properties of sets, including overall envelope, density of individuals within etc. Not spontaneously deployed for sets under 4.

23. Parallel individuation Limited by absolute set size. Cannot represent sets of 4 or greater. No summary symbol for number. Binds properties of individuals in representations, including size Some number relevant computations defined over object files, but also very sensitive to continuous quantities like total front surface area.

24. Core Knowledge System of mental representations, conceptual content Acquisition supported (partially) by innate, domain specific, learning mechanisms Entity identification supported by innate, domain specific input analyzers Evolutionarily ancient (often) Continuous throughout development

25. A third system Failure at 4 vs 1 --cracker choice --box reach 12, 14, 18, 20 months. When tracking individuals, no representation of number of individuals beyond 3. No singular/plural distinction defined over object file representation or analog magnitude

26. Analog Magnitude Models Number represented by a quantity linearly related to the cardinal value of the set. One: __ _______________no principled break Two: ____ Three: ______ Seven: ______________ Eight: ________________

27. Parallel Individuation Models 1 cracker ■ ________________no principled break 2 crackers ■ ■ 3 crackers ■ ■ ■ --one symbol for each individual --no symbols for integers

28. Third (Core ?) System: Chierchia/Link Atoms, Semi-lattice of sets constructed out of those atoms. Unified treatment of singular/plural, quantifiers (“no, some, all, each”), count/mass distinction, definite determiner, collective nouns, classifier languages. Cross-linguistically universal

29. {a,b,c,d,…} {a,b,c}{a,b,d}{b,c,d}{a,c,d}... {a,b}{a,c} ab {a,d}{b,c}{b,d}{c,d} cd...= At...

30. Relation to language acquisition? No core knowledge of singular/plural, quantifiers? Becomes available to conceptual system only upon learning language? 1)Quinian bootstrapping? 2)Becomes conceptually salient/available only upon mastery in language?

31. Question 1—When plural marking mastered in English? Koudier et al. (on website) Wood et al. (will describe)

32. Koudier et al.

36. Overall pattern of results Failure at 20 months with multiple cues, Failure at 24 months when plurality marked only on noun morphology Success at 24 months with multiple cues Success at 36 months (not tested earlier) when plurality marked only on noun morphology

37. Conclusions English learning infants master singular/plural marking between 20 and 24 months. “Is a” vs “Are some” before “N” vs “Ns” Seems to have adult semantic force from the beginning. (No Quinian scrambling up a bootstrapping ladder to master the semantic force of “a”.

38. Parallel results—box search task Look into box. “Look, this is a ball in the box. I see a ball.” Hand to child “You get the ball.” vs Look into box. “Look, there are some balls in the box. I see some balls.” Hand box to child. “You get the balls.”

39. Dependent Measure There is always only 1 ball in the box. The child reaches in and retrieves it. Dependent measure—search time for additional ball(s). Results—at 24 months, children succeed. At 20 fail. At 24 months, fail if marked by noun morphology alone. “I see my balls in the box.” vs “I see my ball in the box.” Etc.

40. Two convergent measures Exactly the same age trends.

41. Relation between verbal/non-verbal singular/plural marking Non-verbal 1-4 task—infants fail even at 20-months. Infants come to mark singular-plural by 24 months, fail to do so at 24-months. Question—are these developments related? If so, direction of causality?

42. Barner et al.

46. Conclusions English plural marking mastered between 20 and 22 months. Success on box-reach task emerges between 20 and 22 months. Success on two tasks correlated within this age range. I.e., success on box reach task among 22 and 24 month olds carried by those whose parents say are producing plural morphology.

47. Direction of causation? Conceptual development allows learning of singular/plural distinction? Mastery of singular/plural distinction makes conceptual distinction between individual/set of multiple individuals available or more salient?

48. French and Chinese French learning children master singular plural between 16 and 18 months. Succeed at 4-1 box search task by 18 months at least (Kouider data). Chinese (stay tuned). No singular/plural to master. Fail at 4-1 box search task at 24 months. Age of success not yet known

49. EFFECT OF LANGUAGE ON THOUGHT But—MORE SALIENT? Or QUINEAN BOOTSTRAPPING? Non-human primates have singular/plural distinction. Succeed at 1vs. 2, 1 vs. 5 under conditions fail at 2 vs. 4 and 2 vs. 5. Set based quantification part of primate heritage. Not likely to require Quinean bootstrapping.

50. Next time Sarnecka et al replaced with a short paper, Carey in Daedalus. Read that. An argument that none of the core systems of number representation have the power to represent the positive integers. Quinean bootstrapping needed. A proposal for how it really works.