Comments on WHERE MATHEMATICS COMES FROM

Shamim Khaliq shamimkhaliq at hotmail.com
Fri Jan 19 09:53:13 EST 2001


Steven Michael Harris <stevenharris at mediaone.net> wrote
> The Brain's Innate Arithmetic?
.....
> The entire argument they present is based on the "proof" that there is a
> basic "hard-wired," genetically programmed ability to perform some simple
> math (language-based math) in some animals and in humans before any such
> ability is taught. Most of this proof is presented in Chapter 1 and the
rest
> of the book is pretty much a synopsis of all mathematics, or the evolution
> of mathematics, constantly referring back to the proof presented in the
> first chapter. So the first chapter is where most of my disagreement will
be
> based as well. It will not take that long.
>
> They claim number discrimination by pre-language babies using some suspect
> logic.
>
> They state as fact that "babies have the following numerical abilities":
>
> At three or four days, a baby can discriminate between collections of two
> and three items (Antell & Keating, 1983). Under certain conditions,
infants
> can even distinguish three items from four (Strauss & Curtis, 1981; van
> Loosbroek & Smitsman, 1990).
> By four and a half months, a baby "can tell" that one plus one is two and
> that two minus one is one (Wynn, 1992a).
> A little later, infants "can tell" that two plus one is three and that
three
> minus one is two (Wynn, 1995).
> These abilities are not restricted to visual arrays. Babies can also
> discriminate numbers of sounds of two or three syllables (Bijeljac-Babic,
> Bertoncini, & Mehler, 1991).
> And at about seven months, babies can recognize the numerical equivalence
> between arrays of objects and drumbeats of the same number (Starkey,
Spelke,
> & Gelman, 1990).
> The assumptions are that the abilities are of a mathematics
(language-based
> mathematics) similar to what we use when we count in our heads "one, two,
> three." but these assumptions come from understandable language-bias in
> interpreting observations from various studies. There are other ways to
> interpret the observations in these studies. I'll need to quote
extensively
> from this chapter in order to present a fair argument.
>
> "Slides were projected on a screen in front of babies sitting on their
> mother's lap. The time a baby spent looking at each slide before turning
> away was carefully monitored. When the bay started looking elsewhere, a
new
> slide appeared on the screen. At first, the slides contained two large
black
> dots. During the trials, the baby was shown the same numbers of dots,
though
> separated horizontally by different distances. After a while, the baby
would
> start looking at the slides for shorter and shorter periods of time. This
is
> technically called habituation; nontechnically, the baby got bored."
> "The slides were then changed without warning to three black dots.
> Immediately the baby started to stare longer, exhibiting what
psychologists
> call a longer fixation time. The consistent difference of fixation times
> informs psychologists that the baby could tell the difference between two
> and three dots. The experiment was repeated with the three dots first,
then
> the two dots. The results were the same. These experiments were first
tried
> with babies between four and five months of age, but later it was shown
that
> newborn babies at three or four days showed the same results (Antell &
> Keating, 1983). These findings have been replicated not just with dots but
> with slides showing objects of different shapes, sizes, and alignments
> (Strauss & Curtis, 1981). Such experiments suggest that the ability to
> distinguish small numbers is present in newborns, and thus that there is
at
> least some innate numerical capacity."
> They later state that the innate ability to count to three and sometimes
to
> four is present at a very early age. But none of these studies has proved
> that counting (language-based math) has occurred. There are a variety of
> ways to approach this. It is more complicated to explain because innate
> biological math is immensely complicated in all animals with enough of a
> nervous system to be animated in complicated ways or to have any evolved
> sense of sight or hearing or to have a liver or.
>
habituation and preferential looking are standard techniques for
investigating infant abilities. it does not say anything about innateness,
as you mention below, because of plasticity of the brain. however, e.g. with
regards to language perception or recognition skills, e.g. an infant can
recognise its mother's voice from birth, this might indicate some kind of
hard-wired sensory input device? however, you're looking for convergence of
evidence from various disciplines, from species-universality to
domain-specific developmental disorders.
>
> One of the problems in this kind of logic is the assumption that the brain
> of any organism is completely the product of genetics, of a hard-wired
> pre-ordained ability like that of a machine. But a nervous system is an
> organization of logic that forms itself based on the mathematical
principles
> it uses to operate and is affected by genetics but really operates as a
> mathematical logic that is constrained by the limits imposed by the
> genetics. Otherwise it would not be possible for a damaged brain to
flexibly
> rearrange its logic in order to repair a function damaged by lesion or
> excessive cerebral fluid or whatever by using another region of the brain
or
> another arrangement of the cells (following the logic of how brain cells
> interact with other cells).
>
> Forget for a moment the fact that any animal that can intercept a moving
> object or that can calculate information based on arrangements of light
> affecting cells in one part of the body and calculate the necessary
> movements of thousands of muscle fibers in order to predict the existence
of
> an object away from the body and reach out and find that object is using
> massive mathematical calculations to perform such feats (even though there
> is no language-based understanding of such math).
>
i'm sure this develops in infants from 6-12 months (guessing) as their
knowledge of rigitity and motion of inanimate objects develops, as evidenced
by the same habituation-dishabituation, preferential looking paradigms you
rejected above. what was your objection? i think you more implied an
objection with, "The entire argument they present is based on the "proof"
that there is a basic "hard-wired," genetically programmed ability.Most of
this proof is presented in Chapter 1 . So the first chapter is where most of
my disagreement will be based as well". You then go on to describe the
evidence, then argue against it with the plasticity of the brain argument,
a -sounds to me- possibly valid argument that the infant may have learned to
respond to intensity or area characteristics of a visual array (though i'm
sure you could control for this; mention it to the experimenters), and some
idea we are born with a concept of ad infinitum?, which doesn't invalidate
the method or the evidence.
>
> When the baby responds to a change from one to two, from two to three,
> rarely from three to four, but never from four to five - something besides
> "counting" is going on.
>
> Think of it this way. The baby is reacting more significantly to
observable
> change. The difference between one and two is a 100% change. The
difference
> between two and three is a 50% change - still a significant change in
amount
> or degree. Beyond three the amount of change is a minority of change. From
> three to four is a change of 33% and from four to five is a 25% change in
> amount, so the response to such change is less likely as there is a much
> smaller percentage of change. (If somebody gave you a glass with milk in
it
> and added 25% to it when you were not looking, you might not notice the
> difference.) So this does not necessarily represent counting ability.
(They
> never said if the study also tried to see a difference in response from
> three to five - a 66% change in amount.)
>
> (Another way of saying the same argument: if you are napping and I
increase
> the light in the room 100%, you are much more likely to respond by waking
> than if I just increased the light in the room by 25% or gradually
increased
> the light by 100%.)
>
innate predisposition, or procedurally embedded learned knowledge? at least
the authors' evidence was backed up by experiment.
>
> Another way of looking at it is that beyond three the brain might be using
> shorthand to assume further repetition so it does not have to continually
> process repeating objects. Remember that it might just take three
> observations of repetition to recognize a predictable pattern and after
that
> an assumption of repetition might be the impulse. (The brain predicts
> objects - especially repeating objects such as the pattern in tile or
> wallpaper when filling in the blind-spot in vision, the reason that you
can
> put a unique object into your blind spot and it disappears when repeating
> visual patterns surround the blind spot.)
>
> You only need three points of observation on the arc of a ball to predict
> where it is going to land.
>
> Any quantity beyond three might be inherently boring or just more than is
> needed when a pattern needs to be perceived.
>
> In comedy, everything is setup in threes. The costume will always have
three
> buttons, never four (unless when created by an amateur). The jokes are
setup
> in threes. There are always three people walking into a bar. The anecdotes
> concerning the first two people setup the pattern. The third anecdote
would
> confirm the pattern but in comedy there is always a switch in the pattern
> with the third anecdote which creates the humor. A joke that has four
people
> walking into a bar would lose the audience if the punchline only came with
> the fourth anecdote (but would work if the switch occurred with the third
> person and then another switch was delivered concerning the fourth
person).
>
> It is our nature to assume that a pattern will go on indefinitely when
> established with three consistent examples of the pattern. (The ellipsis
is
> three dots.) (Infinity is represented by three dots in mathematics.) This
> point is even mentioned by the authors in this book in a later chapter
when
> talking about a different subject (making this essay of mine so much
> easier):
>
"it is our nature". so you do argue that we have innate predispositions that
guide learning. though i think a predisposition to note visual
characteristics is more likely than an innate "theory" that things go on to
infinity.
>
> "Consider a sentence like John jumped and jumped again, and jumped again.
> Here we have an iteration of three jumps. But John jumped and jumped and
> jumped is usually interpreted not as three jumps but as an open-ended,
> indefinite number.".
> "But verbs like swim, fly, and roll are imperfective, with no indicated
> endpoint. Consider sentences indicating iteration via the syntactic device
> of conjunction: John swam and swam and swam. The eagle flew and flew and
> flew. This sentence structure, which would normally indicate indefinite
> iteration with perfective verbs, here indicates a continuous process of
> swimming or flying. The same is true in the case of aspectual particles
like
> on and over. For example, John said the sentence over indicates a single
> iteration of the sentence. But John said the sentence over and over and
over
> indicates ongoing repetition. Similarly, The barrel rolled over and over
> indicates indefinitely continuous rolling, and The eagle flew on and on
> indicates indefinitely continuous flying. In these sentences, the language
> of iteration for perfectives (e.g., verb and verb and verb; over and over
> and over) is used with imperfectives to express something quite
different -
> namely, an indefinitely continuous process.".
> As a writer I would have chosen to assume I'd made my point earlier and
> moved on (because the joke should be over with the third example), but
these
> authors (in the same way they repeated their points in other parts of the
> book) kept going on and on and on.
>
>
p.s.
i heard part of a visiting speaker's address on the same topic. i do feel we
could continue finding further evidence for "innate" modules for everything
from number to taste preference going down the track of double dissociation
and infant studies, but agree with your gist that this may not be useful,
not least by the parsimony criterion. but at least someone is out there
curious and interested enough to research it, which aids our knowledge more
than philosophical debate (shoot me down).







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