Cognitive perspectives on counting
Marcin Wągiel
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Introduction
Counting
common ⇒ everyday experience
cognitive ∼ linguistic perspectives
three different though related concepts
count list ⇒ recitation
arithmetic ⇒ abstract operations
quantification ⇒ cardinality of a set
(1) a. one, two, three, four, five, six,…
b. Three times two equals six.
c. three cats
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Number sense
Two cognitive systems
Hyde (2011)
OTS ⇒ object tracking system
ANS ⇒ approximate number system
Figure 1: Object tracking Figure 2: Approximate number
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Number sense
Object tracking system
Carey (1998, 2009), Piazza (2010)
mental ability to immediately enumarate small sets
no counting via individuation
manifests in infants
Figure 3: How many marks?
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Number sense
Object tracking system
Carey (1998, 2009), Piazza (2010)
mental ability to immediately enumarate small sets
no counting via individuation
manifests in infants
Figure 4: How many marks?
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Number sense
Approximate number system
Feigenson et al. (2004), Nieder & Dehaene (2009), Cantlon et al. (2006)
estimation of the magnitude of a collection
no reliance on symbolic representation
manifests in infants ⇒ develops with age
Figure 5: Compare
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Number sense
Number sense in non-human animals
Davis & Pérusse (1998), Gallistel (1989), Dehaene (1997)
primates ⇒ operations on quantities
apprehension
comparison
approximate addition
other mammals: dolphins, cats, rats
also: birds, fish
botanics ⇒ plant arithmetic
however, no evidence for symbolic addition except for the
chimpanzee after long training
https://www.youtube.com/watch?v=t-SQisIYPh4
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Psychology of counting
Implicit knowledge of counting in children
Gelman & Gallistel (1978)
intuitive understanding of the cardinality of a set
and its conservation under changes not affecting quantity
each entity must be count once and once only
1 number cannot be associated with more than 1 entity
no explicit formulation ⇒ children are never taught that
Figure 6: Enumerating sets
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Psychology of counting
Innate principles of counting
Gelman & Gallistel (1978)
stable order ⇒ ordered list of symbols
1-1 correspondence ⇒ symbols related to objects
cardinality ⇒ determined by the last symbol
Figure 7: Counting and order
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Psychology of counting
Acquisition of counting
Wynn (1990)
children 6–18 months
stable order and 1-1 correspondence observed
fail when asked to give ‘two’ or ‘three’ objects
2,5 years
understanding that counting is an abstract procedure
applicable to different kinds of objects
3,5 years
order of recitation ⇒ crucial
order of pointing at objects ⇒ irrelevant
children indicate and correct subtle errors
4 years
counting can be generalized to novel situations
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Psychology of counting
Quinean bootstrapping ⇒ crucial linguistic component
Carey (2009)
learning the ordered list ⇒ relative order
learning the meaning of symbols
learning how the list represents number
(2) a. eeny, meeny, miny, mo,…
b. one, two, three, four, five, six,…
(3) three = 3
Figure 8: Cardinality
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Spatial integrity in counting
Object/substance distinction
Soja et al. (1991), Hauser & Carey (2003), Hauser & Spaulding (2006)
innate ontological commitments
manifested in infants
assumptions ⇒ nature of objects
boundedness ⇒ natural boundaries
cohesion ⇒ parts stick together
movement across space along continuous paths
substances ⇒ not expected to have those properties
also in non-human animals
https://www.youtube.com/watch?v=hwgo2O5Vk_g&t=2s
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Spatial integrity in counting
Broken object experiments
Shipley & Shepperson (1990), Dehaene (1997), Melgoza et al. (2008)
children between 3 and 4 years
count only discrete integrated objects
Figure 9: Relevance of integrity in counting
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Spatial integrity in counting
Broken object experiments
Shipley & Shepperson (1990), Dehaene (1997), Melgoza et al. (2008)
other forms of linguistic quantification
comparative constructions and pluralization
Figure 10: Integrity in quantity comparison and pluralization
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Part-whole structures
Ontological intuition
Varzi (2016), Priest (2014)
Pre-Socratics ⇒ roots of mereology
entities ⇒ made up of smaller entities (parts)
Plato ⇒ Parmenides and Theaetetus
unity ∼ arbitrary sum of parts
structure ⇒ arrangement of parts
Figure 11: Material parthood Figure 12: Individual parthood
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Part-whole structures
Part-whole perception
Elkind et al. (1964), Kimchi (1993), Boisvert et al. (1999)
simultaneous perception ⇒ wholes ∼ collections of parts
manifests in young children
Figure 13: Part-whole perception
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Mass/count distinction
Countability ⇒ mass nouns ∼ count nouns
Jespersen (1913) among many others
uncountable ∼ countable nouns
grammatical category
pluralization, compatibility with numerals
intuition ⇒ object/substance distinction
(4) a. cat
b. cats
c. two cats
(5) a. mud
b. *muds
c. *two mud/muds
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Mass/count distinction
Object mass nouns
Barner & Snedeker (2005), Chierchia (2010), Landman (2011)
grammatical category ⇒ mass nouns
denote discrete objects
clash ⇒ grammar ∼ perception
(6) a. furniture
b. silverware
c. footwear
(7) a. nábytek
b. bižuterie
c. obuv Czech
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Mass/count distinction
Object mass nouns
Barner & Snedeker (2005), Chierchia (2010), Landman (2011)
quantity comparison task
object mass nouns pattern with count nouns
attested in several typologically distinct languages
Figure 14: Object mass – count – mass
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Counting and measuring
Counting and measuring are independent operations
Rothstein (2017), Wągiel (2018)
distinct syntax and semantics
counting indicates integrity ⇒ measuring does not
ml1 ml2 ml3
Figure 15: Inegrity in measuring and counting
(8) a. There are three mililiters of liquid on the table.
b. #There are three objects on the table.
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Counting and measuring
Measuring is not sensitive to integrity
Wągiel (2018)
numeral phrases ⇒ counting/measuring ambiguity
counting ⇒ measuring shift (possible but restricted)
(9) context: John is cooking with his child. They put
three whole apples on a table. John says:
a. There are three apples on the table…
b. Let’s count them together: one, two, three.
(10) context: John is cooking with his child. They sliced
three apples and put the slices into a bowl. John says:
a. There are three apples in the bowl…
b. #Let’s count them together: one, two, three.
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