Four Month Old Math Expert Adds & Subtracts
Immortalist
announcement that stunned child psychologists all around the world -
and anybody else who learned of the result for that matter. Wynn
claimed to have shown that babies as young as four months old could do
simple addition and subtraction problems. That's right; there isn't
a typo. Four months. In fact, other experimenters subsequently showed
that babies can do the same math when they are a mere two days old!
One of the first reactions people had when they heard the news -
other than widespread disbelief - was, How had Wynn done it? After
all, four months old babies cannot yet talk, so how can we possibly
discover whether they know that, say, 1 + 1 = 2, to pick one of the
examples Wynn claimed her young subjects could do? And how did Wynn
manage to pose such a question in the first place so that the children
could understand what she was asking?
Before I tell you how Wynn got around these problems, I should make it
clear exactly what Wynn claimed to have discovered.
First, she did not claim that her subjects had any conscious concept of
numbers. As any parent knows, numbers (i.e., the counting numbers, 1,
2, 3, an so on.) have to be taught to young children, and before that
can happen the children have to learn how to use language, something
that does not happen with a four-month old baby. Rather, what Wynn
claimed was that:
1. The children she examined knew the difference between a single
object, a pair of objects, and a collection of more than two objects.
2. They knew that if you take two single objects and put them together,
the resulting collection has exactly two objects in it, not one object
and not three objects.
3. They knew that if you take two objects and remove one of them, you
are left with exactly one object; you don't end up with no objects or
with two objects.
The normal way for an adult to describe these abilities would be to say
that:
1. The children she examined knew the difference between the numbers 1
and 2 and the difference between 2 and any bigger number.
2. They knew that 1 + 1 = 2 and that 1 + 1 is not equal to 1 or 3.
3. They knew that 2 - 1 = 1 and that 2 - 1 is not equal to 0 or 2.
Clearly, to express the ability this way requires an understanding of
numbers, at the very least the numbers 0, 1, 2, and 3. Now, all the
evidence that we have about the way the human brain handles numbers
indicates that our ability to handle numbers only comes after the
individual learns the numbers words "one," "two," "three," and so on.
(Actually, work with chimpanzees and other human-like primates suggests
that learning the number symbols "1", "2", "3", etc. would work just as
well in this regard. The point is that acquisition of the number
concept seems to require first having a word or symbol to refer to that
concept.)
Strictly speaking, then, Wynn's claim was really about numerosity
rather than numbers. What she was saying was that very young children
have a reliable sense of the size of small collections of objects. But
that did not lessen the surprise of Wynn's announcement. After all,
everyone knew that four-month old babies don't know how to use number
words. Most experts assumed that a sense of numerosity developed after
the child learned how to count. Wynn was claiming that the numerosity
sense came first. That meant that either we are born with such a sense,
or at the very least acquire it automatically within at most a few
weeks of birth. (As we'll see presently, subsequent research showed
that, if we are not actually born with a numerosity sense, we acquire
it within, not a few weeks, but at most a few days of birth.)
Here is what Wynn did to obtain her discovery. (Incidentally, Wynn's
experiment has been repeated successfully many times over by different
psychologists around the world, so there is no doubt about the accuracy
of her findings.)
The trick was to make use of the fact that even very young babies have
acquired a fairly well developed sense of "the way things are in the
world." If a baby sees something that runs counter to its expectations,
it will pay attention to it as it attempts to understand what it sees.
By filming the child - particularly its eyes - as it is presented
with various scenes, and then measuring the time the baby spends
attending to each scene, the investigator can determine what runs
counter to the baby's expectations. For example, if a baby is shown a
series of pieces of fruit on plates, and is then shown an apple
suspended in mid air with no apparent means of support, the baby will
stare measurably longer at the suspended fruit than it does at the
fruit on plates.
Wynn sat her young subjects in front of a small puppet theater and set
the (hidden) film camera rolling. (See Figure 1.1) The puppet stage was
initially empty. The experimenter's hand came out from one side and
placed a puppet on the stage. Then a screen came up, hiding the puppet.
The experimenter's hand appeared again holding a second puppet, which
it put behind the screen. Then the screen was lowered to reveal the two
puppets. The child watched attentively throughout.
+ Within a couple of days of being born, human babies know the numbers
1, 2, 3 and can distinguish between a correct addition or subtraction
such as 1 + 2 = 3 and an incorrect one such as 3 - 1 = 1.
+ When a dog runs along a beach and then jumps into the water to
retrieve a ball thrown diagonally into a lake, it instinctively solves
a problem that humans need calculus to solve.
+ When dogs and baseball fielders run to catch a ball thrown (or hit)
high in the air, they run in a curved path that makes (unconscious) use
of the sophisticated mathematics built into the visual system.
+ The Tunisian desert ant finds its way across the featureless desert
sands using the same mathematical technique that sailors in times past
used to navigate the oceans and the Apollo astronauts depended on to
get to the Moon.
+ Lobsters have a built-in positioning system that is the equal of the
hugely expensive and mathematically rich high-tech Global Positioning
System (GPS) human travelers use today.
+ Some bird species migrate up to 18,000 miles each year, navigating by
the stars, the sun, and the Earth's magnetic field-a task that
humans can perform only with the aid of trigonometry.
+ Bats catch prey at night using a sonar system that is far more
accurate than anything human engineers have produced, and which the
U.S. Navy has tried to emulate to develop better minesweeping
technology.
+ When bees build a honeycomb, they instinctively solve a math problem
that took humans over 2,000 years to solve.
+ Chinese and Japanese children have a major advantage over children
growing up in the West, in that their language makes it much easier for
them to learn their numbers and do arithmetic.
Robert J. Kolker
> In 1992, a young American researcher called Karen Wynn made an
> announcement that stunned child psychologists all around the world -
> and anybody else who learned of the result for that matter. Wynn
> claimed to have shown that babies as young as four months old could do
> simple addition and subtraction problems. That's right; there isn't
> a typo. Four months. In fact, other experimenters subsequently showed
> that babies can do the same math when they are a mere two days old!
Have you ever heard of Clever Hans?
Bob Kolker
Mitch
Both Karen Wynn, the researcher, and Keith Devlin, the author of the
book from which that blurb came, certainly do. The experimental
protocol accounted for effects like prompting by the experimenter.
What I found particularly jarring was the juxtaposition of the two
teasers:
"- When a dog runs along a beach and then jumps into the water to
retrieve a ball thrown diagonally into a lake, it instinctively solves
a problem that humans need calculus to solve.
- When dogs and baseball fielders run to catch a ball thrown (or
hit) high in the air, they run in a curved path that makes
(unconscious) use of the sophisticated mathematics built into the
visual system. "
so humans (or to be logically precise, baseball fielders, who are
presumably human) need calculus to catch balls diagonally thrown over
water, but not in a baseball park? Whoever wrote the blurb is suffering
not from innumeracy but ...uh...what's the word... unthinkacy?
Mitch
Immortalist
Mitch wrote:
> Robert J. Kolker wrote:
> > Immortalist wrote:
> >
> > > In 1992, a young American researcher called Karen Wynn made an
> > > announcement that stunned child psychologists all around the world -
> > > and anybody else who learned of the result for that matter. Wynn
> > > claimed to have shown that babies as young as four months old could do
> > > simple addition and subtraction problems. That's right; there isn't
> > > a typo. Four months. In fact, other experimenters subsequently showed
> > > that babies can do the same math when they are a mere two days old!
> >
> > Have you ever heard of Clever Hans?
>
There is a difference when the baby is compared to the horse and Mill's
methods are applied;
Mill's Methods are five methods of induction described by philosopher
John Stuart Mill in his 1843 book A System of Logic. They are intended
to shed light on issues of causation.
1. Method of agreement
This principle states that if a single common factor exists in all
cases where a phenomenon occurs, that we can attribute the phenomenon
to that factor.
2. Method of difference
This principle states that if one set of circumstances leads to a given
phenomenon, and another set of circumstances does not, and the sets
differ only in a single factor that is present in the first set but not
in the second, then the phenomenon can be attributed to that factor.
3. Joint method of agreement and difference
Also called simply the "joint method," this principle simply represents
the application of the methods of agreement and difference.
4. Method of residues
The method of residues states that if a range of factors are believed
to cause a range of phenomena, and we have matched all the factors,
except one, with all the phenomena, except one, then the remaining
phenomenon can be attributed to the remaining factor.
5. Method of concomitant variations
This principle states that if across a range of circumstances leading
to a phenomenon, some property of the phenomenon varies in tandem with
some factor existing in the circumstances, then the phenomenon can be
attributed to that factor. For instance, suppose that various samples
of water, each containing both salt and lead, were found to be toxic.
If the level of toxicity varied in tandem with the level of lead, one
could attribute the toxicity to the presence of lead.
http://en.wikipedia.org/wiki/Mill's_canons
Mill formulates five guiding methods of induction, the method of
agreement, that of difference, the joint or double method of agreement
and difference, the method of residues, and that of concomitant
variations. The common feature of these methods, the one real method of
scientific inquiry, is that of elimination. All the other methods are
thus subordinate to the method of difference. Here we have a case of
the occurrence of the phenomenon under investigation and a case of its
nonoccurrence, these cases having every circumstance in common, save
one, that one occurring only in the former; and we are warranted in
concluding that this circumstance, in which alone the two cases differ,
is either the cause or a necessary part of the cause of the phenomenon.
http://www.utm.edu/research/iep/m/milljs.htm
> Both Karen Wynn, the researcher, and Keith Devlin, the author of the
> book from which that blurb came, certainly do. The experimental
> protocol accounted for effects like prompting by the experimenter.
>
> What I found particularly jarring was the juxtaposition of the two
> teasers:
>
> "- When a dog runs along a beach and then jumps into the water to
> retrieve a ball thrown diagonally into a lake, it instinctively solves
> a problem that humans need calculus to solve.
> - When dogs and baseball fielders run to catch a ball thrown (or
> hit) high in the air, they run in a curved path that makes
> (unconscious) use of the sophisticated mathematics built into the
> visual system. "
>
> so humans (or to be logically precise, baseball fielders, who are
> presumably human) need calculus to catch balls diagonally thrown over
> water, but not in a baseball park? Whoever wrote the blurb is suffering
> not from innumeracy but ...uh...what's the word... unthinkacy?
>
> Mitch
The authors claims for two kinds of math, humans do both but most
animals only do one, from the same page;
There are two kinds of math: the hard kind and the easy kind. The easy
kind, practiced by ants, shrimp, Welsh Corgis - and us - is innate.
But what innate calculating skills do we humans have? Leaving aside
built-in mathematics, such as the visual system, ordinary people do
just fine when faced with mathematical tasks in the course of the day.
Yet when they are confronted with the same tasks presented as
"math," their accuracy often drops. If we have innate mathematical
ability, why do we have to teach math and why do most of us find it so
hard to learn? Are there tricks or strategies that the ordinary person
can do to improve mathematical ability? Can we improve our math skills
by learning from dogs, cats, and other creatures that "do math?"
The answer to each of these questions is a qualified yes. All these
examples of animal math suggest that if we want to do better in the
formal kind of math, we should see how it arises from natural
mathematics.
Patrick
Maybe I'm missing your point but the way I understand it is that
the ball thrown diagonally into the water doesn't require calculus
to catch but to fetch. In order for the fetcher to get the ball
fastest he must plot a path that is partly over land and partly
over water. The dilemma is that the land route is fast but
indirect and the water route is direct but slow. This sort
of thing shows up in differential calculus as a min/max problem.
Needing to plot a path over two types of terrain doesn't
arise when the fielder is in an ordinary ball park.
Immortalist
True, I think in the rest of that chapter the authors talk about the
dog calculating the least amount of travel, within the water, to get to
the ball.
---------------------xpostage
The cerebral substrates of arithmetic:
How and why did human beings evolve the ability to do mathematics?
We all possess the ability to cope with mathematics--if only we
recognize what's required. We all possess, if not literally a gene,
then at least an inherent ability not just for arithmetic but for real
mathematics: algebra, calculus, and the rest. A recent Darwinian
explanation for the origin of this ability, is based on the idea that
being able to handle abstract ideas and relationships confers key
evolutionary advantages.
Human infants have a rudimentary number sense, this sense is as basic
as our perception of color, and that it is wired into the brain. The
invention of symbolic systems of numerals started us on the climb to
higher mathematics. We are now approaching the crossroads where numbers
and neurons intersect. The structure of the brain shapes our
mathematical abilities, and our mathematics opens up a window on the
human mind.
It seems we have a number sense, the human mind seems to have an innate
grasp of mathematics. Place value systems (such as the Arabic numeral
system we use) arose independently in four separate
civilizations--evidence of a universal sense of number. A rudimentary
number sense is wired into our brains at birth. Experiments show that
chimps, like us, use symbols to denote numbers.
In the same mathematical reasoning that inspired Plato with visions of
eternal ideals, we find evidence for a provocative theory of
evolutionary change. The evolution of language is the surest indication
of a new kind of strictly internal brain activity, one neither
stimulated by the environment nor tied to physical activity. Out of
this "off-line thinking" emerged not only the syntax necessary for
speech, but also the symbolic logic essential to mathematics. Enhanced
symbolic abilities let early hominids think in this "off-line" manner,
while asking and answering "what if" questions about tools, predators,
habitats or prey.
Mathematics is a great artistic triumph of the race, one made possible
by an innate human ability. Language evolved in two stages and its main
purpose was not communication. The ability to think mathematically
arose out of the same symbol-manipulating ability that was so crucial
to the very first emergence of true language. Combining a number sense
with symbolic abilities, we use abstractions to manipulate quantities,
leading to arithmetic and potentially to calculus and number theory.
Abstract models describe concrete things--from rotating clock faces to
rattlesnake skins, use higher math abilities. Mathematics is more than
arithmetic. Real mathematics involves making logical arguments about
abstract objects.
Though its deepest structure shares an evolutionary origin shared with
language, math frequently calls upon a neurological number sense,
naturally strong in some, weak in others. Consequently, poets may
command powers of abstraction akin to those of mathematical geniuses,
yet still falter in doing simple algebra. But in any manipulation of
symbols, verbal or mathematical, we can easily see faculties that set
one of the earth's creatures apart from all others. Exploring the
mysterious beginnings of the mind's symbolic powers, takes us a long
way toward understanding what it means to be human.
If people are endowed with a "number instinct" similar to the "language
instinct"-as recent research suggests-then why can't everyone do math?
Why, then, can't we do math as well as we speak? The answer is that we
can and do-we just don't recognize when we're using mathematical
reasoning. Mathematics merely involves a relatively high level of
abstraction--but one we can all cope with, if we work at it. Doing
mathematics is very much like running a marathon. It does not require
any special talent, and 'finishing' is largely a matter of wanting to
succeed."
In a way similar Chomsky's theory that we are all born with
"hard-wired" linguistic ability, the mental process of making logical
connections between abstract objects and the mental process needed to
construct sentences have the identical structure. Thus, we can see that
the genetic heritage that gives us all the ability to communicate by
language also gives us the ability to do mathematics.
The Math Gene: How Mathematical Thinking Evolved & Why Numbers Are Like
Gossip
http://www.amazon.com/exec/obidos/ASIN/0465016197/
The Number Sense: How the Mind Creates Mathematics
http://www.amazon.com/exec/obidos/ASIN/0195132408/
Mitch
Patrick wrote:
> Mitch wrote:
>
> > What I found particularly jarring was the juxtaposition of the two
> > teasers:
> >
> > "- When a dog runs along a beach and then jumps into the water to
> > retrieve a ball thrown diagonally into a lake, it instinctively solves
> > a problem that humans need calculus to solve.
> > - When dogs and baseball fielders run to catch a ball thrown (or
> > hit) high in the air, they run in a curved path that makes
> > (unconscious) use of the sophisticated mathematics built into the
> > visual system. "
> >
> > so humans (or to be logically precise, baseball fielders, who are
> > presumably human) need calculus to catch balls diagonally thrown over
> > water, but not in a baseball park? Whoever wrote the blurb is suffering
> > not from innumeracy but ...uh...what's the word... unthinkacy?
>
> Maybe I'm missing your point
the point involved the comparison between animals and humans, not just
the difference in problems.
> but the way I understand it is that
> the ball thrown diagonally into the water doesn't require calculus
> to catch but to fetch. In order for the fetcher to get the ball
> fastest he must plot a path that is partly over land and partly
> over water. The dilemma is that the land route is fast but
> indirect and the water route is direct but slow. This sort
> of thing shows up in differential calculus as a min/max problem.
>
> Needing to plot a path over two types of terrain doesn't
> arise when the fielder is in an ordinary ball park.
Yes, there are two different mathematical problems here, and both can
be solved innately and formally (using symbolic, rational math). The
presentation for the first one made it sound like animals only use
innate knowledge and humans symbolic, but the second problem had a
human doing it innately. And I thought it obvious that if the human
could do it innately in the second one, certainly they could do so in
the first. "humans -need- calculus to solve"?
Mitch
Patricia Shanahan
...
> Yes, there are two different mathematical problems here, and both can
> be solved innately and formally (using symbolic, rational math). The
> presentation for the first one made it sound like animals only use
> innate knowledge and humans symbolic, but the second problem had a
> human doing it innately. And I thought it obvious that if the human
> could do it innately in the second one, certainly they could do so in
> the first. "humans -need- calculus to solve"?
In each case it is possible that the "innate" approach requires
substantial amounts of practice.
Humans can, and do, practice baseball fielding. The people who do it
really well practice a lot. Many dogs enjoy water retrieving enough to
practice it any time they have a chance.
How many humans have significant practice at the water retrieval problem?
Patricia
Michael Jørgensen
More of the same stuff at http://www.yale.edu/infantlab/Newsletter2004.pdf
Here they talk about other abilities, like proportion.
-Michael.
Mitch
Patricia Shanahan wrote:
>
> In each case it is possible that the "innate" approach requires
> substantial amounts of practice.
>
> Humans can, and do, practice baseball fielding. The people who do it
> really well practice a lot. Many dogs enjoy water retrieving enough to
> practice it any time they have a chance.
>
> How many humans have significant practice at the water retrieval problem?
You don't think humans could get that practice? The thing in the
original that so put me off I suppose was the word "need", as in
"Humans -need- calculus to solve".
Are there are no similarly mathematically modelled problems that
human's can already solve innately?
And to bring in further petty annoyances, wouldn't that be solved by
Snell's law, which is more trig than calculus?
And yet one more... are these innate (learned or not) mechanisms exact
solutions or just approximations?
Mitch
Patricia Shanahan
> Patricia Shanahan wrote:
>
>>In each case it is possible that the "innate" approach requires
>>substantial amounts of practice.
>>
>>Humans can, and do, practice baseball fielding. The people who do it
>>really well practice a lot. Many dogs enjoy water retrieving enough to
>>practice it any time they have a chance.
>>
>>How many humans have significant practice at the water retrieval problem?
>
>
> You don't think humans could get that practice? The thing in the
> original that so put me off I suppose was the word "need", as in
> "Humans -need- calculus to solve".
I strongly suspect that, given sufficient practice, humans could solve
the water retrieval problem using the "innate" method just as well as
dogs. However, I see more dogs than humans practicing water retrieval.
Patricia
Patrick
> Patrick wrote:
>> Mitch wrote:
>>
>>> What I found particularly jarring was the juxtaposition of the two
>>> teasers:
>>>
>>> "- When a dog runs along a beach and then jumps into the water to
>>> retrieve a ball thrown diagonally into a lake, it instinctively solves
>>> a problem that humans need calculus to solve.
>>> - When dogs and baseball fielders run to catch a ball thrown (or
>>> hit) high in the air, they run in a curved path that makes
>>> (unconscious) use of the sophisticated mathematics built into the
>>> visual system. "
>>>
>>> so humans (or to be logically precise, baseball fielders, who are
>>> presumably human) need calculus to catch balls diagonally thrown over
>>> water, but not in a baseball park? Whoever wrote the blurb is suffering
>>> not from innumeracy but ...uh...what's the word... unthinkacy?
>> Maybe I'm missing your point
>
> the point involved the comparison between animals and humans, not just
> the difference in problems.
I don't see the connection. What you wrote was:
"so humans (or to be logically precise, baseball fielders, who are
presumably human) need calculus to catch balls diagonally thrown over
water, but not in a baseball park?"
Seems like you're saying that the blurb-writer is suffering from
"unthinkacy" because there is some contradiction in proposing both
of the following:
(1) A fielder needs calc to to catch balls diagonally thrown over water.
(2) A fielder doesn't need calc to catch balls in a baseball park.
But the blurb writer said nothing that implied either (1) or (2)
so I don't see why, from your argument, we should conclude that
the blurb witter suffers from "unthinkacy".
imagin...@despammed.com
<snip-snop>
Actually quoting from the review cited...
http://www.pgw.com/catalog/catalog.monthly.asp?ShipMonth=62006&Action=View&Index=Title&Book=347842&Order=85
The Math Instinct
Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and
Dogs)
Keith Devlin
1-56025-839-X
"True facts from THE MATH INSTINCT"
Hmm, reference to "True facts" rings a special little bell in my
head...
> + Chinese and Japanese children have a major advantage over children
> growing up in the West, in that their language makes it much easier for
> them to learn their numbers and do arithmetic.
Well I wonder. This little "true?" factoid has been trotted out before,
mostly in pretty flaky articles. The argument appears to be that
because Japanese numerals are (slightly) simpler in surface form than
those of English ("the language spoken in the West"), it should be
easier to "learn numbers". (And since notoriously almost everyone does
better than the Americans at arithmetic tests, this clinches it.)
However, no consideration appeared to have been given to the fact that
many children in countries in the "West" grow up speaking languages
with numerals very much messier than English.
Very roughly, we can say that a numeral is "messy" if it is totally
unrelated to the decimal positional notation form, and "semi-messy" if
it's, well, just a bit unrelated. It's reasonable to say that Japanese
has no messy numerals at all: every numeral contains the (recognisable,
nonzero) decimal digits, in the same order. (e.g. 8945 is
8-1000-9-100-4-10-5) Since zeros are skipped (e.g. 405 is 4-100-5), one
could claim a slight stain, I suppose, but let's give them no
messypoints.
English has two totally messy numerals: eleven and twelve, and quite a
number that are fairly messy: all the teens are reversed (17 is 7-10)
or worse if you don't notice that 'teen' is the same as 'ten' (I
didn't). The rest of the 10s digits are semimessy, but at least all the
bits are in the right order. I award English 35 messypoints.
But now consider French: OK, 30, 40, 50, 60 are OK, but then there's a
major complication that doesn't happen in English, with 6-10-10-7 for
67, and 4-20-12 for 92. (Even if onze-douze is marginally more obvious
than eleven-twelve, all the teens are somewhat messy up to 16, *and*
they get repeated three times in every hundred. So French must get at
least 70, surely?
Then take Danish: well, say 90.
So the same research should also be showing enormous differences across
Europe, with Danish kids trailing very miserably behind even American
kids.
Hmm. I don't believe a word of it. But I willing to shown some serious
evidence.
Brian Chandler
http://imaginatorium.org
Mitch
Yes. That is what I am saying.
> But the blurb writer said nothing that implied either (1) or (2)
Are you saying that "[a dog] instinctively solves a problem that humans
need calculus to solve." does not imply (1) ? (substitute retrieve for
catch)
Mitch
Patrick
To recap I said that nothing the blurb-writer wrote implied
either (1) or (2). Now you've changed (1) to:
(1)* A fielder needs calc to to RETRIEVE balls diagonally thrown
INTO the water.
(1)* and (2) together don't lead to an obvious contradiction
since they talk about two different kinds of tasks. Why
should we suspect that the blurb writer is suffering from
"unthinkacy"?
If your skeptical of the claim that dogs are "doing calculus"
then why not make that point front and center. Why bother
with (1), (1)*, (2) and the alleged ailment of the blurb-writer?
None of those issues has anything to do with the claim that
dogs aren't "doing calculus".
Mitch
Hm... we seem to be assuming different things about -exactly- what the
two items say, but my point about the inconsistency is not particularly
dependent on the two different problems, who's doing the throwing,
catching, or retrieving, and over what kind of terrain (I think I may
have caused confusion by using 'catch' instead of 'retrieve' in my
original post). For the two items I take the two tasks to be different
but requiring "sophisticated mathematics" of comparable complexity...in
order to model it.
> (1)* and (2) together don't lead to an obvious contradiction
> since they talk about two different kinds of tasks. Why
> should we suspect that the blurb writer is suffering from
> "unthinkacy"?
>
> If your skeptical of the claim that dogs are "doing calculus"
> then why not make that point front and center.
That's not my point. But to be explicit, I certainly do assume that
dogs (and other non-humans) cannot "do calculus" -symbolically-. I
don't doubt that certain things they do can only be modelled
symbolically by calculus.
> Why bother
> with (1), (1)*, (2) and the alleged ailment of the blurb-writer?
> None of those issues has anything to do with the claim that
> dogs aren't "doing calculus".
That was never the direction of my disgruntlement. I was just surprised
by the first item when it said that humans -need- calculus to solve
some complicated problem, that is unable to do it by innate means.
So there are a number of dichotomies here. human vs. dog,
symbolic/rational/by thought vs. innate/unconscious/by mechanism (we
haven't even touched on exact vs approximate).
I read the two items as
1) for task X, dogs can solve it innately, humans cannot solve it
innately.
2) for task Y, both dogs and humans can solve it innately.
and since I feel both X and Y can both be 'solved' by neurological
structures of comparable complexity, and I would expect both to be
found in the same organism, I sense a contradiction.
Maybe it would be simpler to say that I think humans can do the same
thing as dogs in the first item.
Does that still not make sense?
Mitch
Patrick
> I hope you don't mind, but I'm taking this off-group.
Sure. This is my last post to this thread.
> I read the two items as
> 1) for task X, dogs can solve it innately, humans cannot solve it
> innately.
> 2) for task Y, both dogs and humans can solve it innately.
>
> and since I feel both X and Y can both be 'solved' by neurological
> structures of comparable complexity, and I would expect both to be
> found in the same organism, I sense a contradiction.
>
> Maybe it would be simpler to say that I think humans can do the same
> thing as dogs in the first item.
I see.
I think the average reader assumes that people don't
practice the sort of thing described in scenario 1.
> Does that still not make sense?
It makes sense.