Greg Detre
Tuesday, April 08, 2003
MBB series, Fay House
recently moved from MIT???
the problem: how to understand the human capacity for conceptual representations
e.g. atom, belief, dog, immoral, fifteen, steel
concept
unit of thought
mental symbol with
representational content
picks out an extension
plays an inferential role
three crude possibilities for the origin of a concept
1. evolutionary origin
predate hominid evolution
arise with hominid evolution
2. cultural origin
3. individual discovery/learning
two major theses:
1. existence of innate representations (core knowledge (Spelke))
2. human begins transcend core knowledge
description: discontinuities?
explanation: mechanisms?
core knowledge
representational content
their acquisition is supported by innate, domain specific, learning mechanisms
entity identification is supported by innate, domain specific input analysers
evolutionarily ancient (often)
remain constant throuhgout development
natural number
�the integers were crated by God; all else is man-made� � Leopold Kronecker
for our purposes, we could replace God with evolution
she thinks he�s wrong though � even natural numbers are cultural constructions
babies as young as five months old are sensitive to the difference between 1 + 1 and 2 � 1
Barry Mazur � Imagining numbers (particularly the square root of minus fifteen)
three distinct systems with numerical content � core knowledge
1. analogue symbol representations of approximate number
2. parallel individuation of small sets of individuals
3. natural language quantification (singular/plural, some, many) etc.
but none of these have the power to represent the concept of 15 though
data for simple numbers
can distinguish 8 from 16 � habituate them to one, then they show interest when shown a picture of a bunch of dots of a higher number
babies look longer at the unexpected outcomes, if you put two objects behind a screen one at a time, then reveal only 1 behind
(rhesus) monkeys succeed with the same (low) numbers
����
what about the format of representation of numbers? possibilities:
mentally represented list of symbols
e.g. !, @, #, $, %, ^
applied in order to the sets, in 1-1 correspondence
the ordinal position that you get to fixes the cardinal value of set
in the same way that 1, 2, 3, 4, 5 works
learning to count in natural language should then be easy, because once you�ve learned the list you can just apply the different ones in place
toddler�s (c. 24 months) master the count routine (1..9)
i.e. can count on fingers
but apparently they can be in that state for 6-9 months before they know what the word two means
when you ask how many fingers on your hand (�how many was that?�) they can only answer �one-two-three-four-five�
they don�t pick up a bigger handful when you ask for five then when you ask for five � they always pick up a plurality � they know that they contrast, but they don�t know how exactly
this is not accounted for by the Numeron List hypothesis
evidence that we (adults) have them
evidence that so do babies (and monkeys too???)
representations of approximate cardinal values of large sets of individuals (> 100s)
if asked to quickly guess at a single flashed image, the variability is huge, but the mean of many repeated tests would be about right (even with just one person)
the reaction time is not proportional to the number of dots � you�re not counting iteratively
number is being represented by a quantity linearly related to the cardinal value (i.e. one, or the unit value) of the set
at least one of the systems of core knowledge:
in rats, pigeons, non-human primates, children and adults can represent numbers approximately without counting (about 15)
whereas human adults have constructed a mapping between those representations and integer list representations � that is, we can express about 15 as �about 15�
parallel individuation of small sets of objects
subitisation (???)
subitize /"sVbItVIz/ v.i. & t. Also -ise.M20. [f. L subit- (see SUBITANEOUS) + -IZE.] Psychol. Apprehend immediately without counting (the number of items in a small sample).
object tracking
short term memory for distinct individuals
near perfect performance on 1, 2 or 3
these are called �object files�
a symbol for an individual object in the world
we can pay attention and track in parallel up to 3 at once
how high does this go?
above 4, the reaction time to give the exact number is a linear function of the number
but for 1, 2, 3 and 4, you can immediately apprehend the right answer
with the object file representation, there�s no summary representation of �threeness� when tracking 3 objects, only the 3 objects
object file representations small number experiments
something to do with ratios of sets � ???
if you show babies 1 cracker, then 4 crackers, and give them a choice, they do no better than chance! they have no summary representation of fourness, in fact, they don�t even seem to be able to represent the four as plural
although they can choose 1 vs 2 and 2 vs 3
they just can�t represent the fourness
object file representations
natural language quantifiers
analogue magnitude
infants and primates do represent numbers
but not the natural numbers, nor even a finite subset of successive integers
therefore, they definitely can�t represent 15
are there any human adults who only have the three core knowledge systems of number? i.e. cultures
many cultures with natural language quantifiers only (i.e. 1, 2, many or 1, 2, 3, many)
hunter-gatherers, semi-nomadic, amazon basin
Peter Gordon
hoi (falling tone = 1, rising tone = 2), baagi = many
can they perceive numerosities despite the lack of linguistic labels? can they manage 15?
after all, they could use their fingers (�external individual files�), or one-to-one correspondence with fingers
task: match a line of 6 with 6 objects
they appeared to understand the task, because they can do that perfectly for 1, 2, 3
i.e. their parallel individuation with small small numbers was perfect
but they can�t manage 6 exactly correctly very often
and their means responses tracked the target value exactly
after all, all they have to do is one-to-one correspondence, but they can�t
he didn�t find any task where he could elicit performance with exactly 7
were there further studies with chunking?
yeah, there were further, harder tasks, where they get better by using other strategies
so, Piraha have only core knowledge of number
some deaf creoles use hand direction and positions from body to count to 40
do any cultures count other than with base 2???
yes, some cultures count with base 2, 5, 12, 16
and you have to be careful when you think they�re saying 1, 2, many that they aren�t actually counting in base 2
body counting systems
finite list, no base system
relations among symbols learned directly
symbols initially partially interpreted
symbols serve as placeholders
analogy, indivcutive leapers, inference from best explanation
comnine and ingtegarate separate representations from different core systems
how they children learn:
the list itself
the meaning of each word
how the list represents number
Planks of the bootstrapping process:
object file representations
analogue magnitude representation
capacity to represent serial order
natural language quanificational semantics
set, individual, discrete/continuous etc.
Dehaene, Wynn (sp?), Gelman, Gallistel
analogue magnitudes as the evolutionary and ontogenetic source of integers:
�
reasons to doubt this though
Possible bootstrapping story
learns integer list as menaingless ordered list
number words mapped onto approximate analogue magnitude
wild analogy: later in the list is larger analogue magnitude
notices that 2 is 1 more than 1 � that 3 is 1 more than 2
induces counting principles, i.e. that the next in list is 1 more
but this isn�t right J
why learn meaning of �one� 6 months before �two� in turn some months before �three�?
why make induciton after �three� or �four�
success relations (++) are much more transparent in object-file representations than analogue magnitude representations
�
one-knower
if you show them �one�, they tell you �one�
if you show them three or five or ten, they say �two�
two-knower
six months later, they get one and two right, but no larger plurals
evolution/God did not give man the integers
number words are learned directly as quantifiers
�one� is �a�, �two� is a �pair�
plural marker �s� marks plural
so the two-knowers are mis-identifying �two� as a plural marker
learns counting routine - notices the identity of the first three words in the counting routine and the singular, dual and trial markers
notices analooy between two distinct �follows� relations
next in the count list and the sets markers
�
the human capacity to represent number is built from several systems of core knowledge, each of which lacks the power to represent natural number (i.e. 15)
the integer list representation of natural number is a cultural construction that transcends core knowledge
each child transcends core knowledge when mastering it (with difficulty)
one of the evolutionary foundations of number is not part of the historical or ontogenetic origin for the first representations of natural number
Lorraine Daston: what about counting non-homogenous objects?
by the time they�ve learned counting, children can deal with them
some other system has to pick out the system that you�re trying to enumerate
why don�t rats have a concept of natural number?
they don�t need it
it�s expensive
plus, you need an external placeholder system, which needs language
in contrast, analogue magnitude systems are computationally really easy to build and they do the trick
African grey parrots and chimpanzees will never make the leap to natural numbers
why don�t some cultures have natural numbers? after all, you can�t have money without an integer system
that�s why there are so few of such cultures left
the Paraha are culturally committed to staying separate from the rest of Brazilian culture
humans then could survive without numbers
the Paraha may be the only remaining examples
even the Aboriginals seem to use sand instead of pebbles as an external counting system
are the babies in the cracker 1 vs 4 experiment maybe resetting their file systems are 3?
no, they appear to be trying to track all 4 and it�s breaking down in the same way that we break down when we try and track too many
is there any way to map the representations of small sets of numbers onto a neural substrate, maybe as some kind of continuous bubble of activation, or is it too high-level???
where can I read about this???
in Barry Mazur???
Hofstadter speculated that any machine that really understood numbers in the way we do, would probably do arithmetic as slow as we do � though it could cheat by having an internal calculator � would a learning machine have to go through this same process???
what about zero???