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Can we know only what we are prepared to kno


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#16    Siara

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Posted 05 April 2009 - 01:38 PM

Meiliken on Apr 5 2009, 01:13 PM, said:

You're misunderstanding what I said.  If someone is raised away from all mankind and raised in the wild, they're not prepared to do for example math.  Or to interact with people, or anything else for that matter.  You added a variable by saying that a person that thinks 1+1=3 would be around others that knew 1+1=2.  I said that they'd never encounter the truth.


Wrong.  They will encounter the truth in the course of their experience as logical beings in the rational, physical world.  They will understand the difference between two and three because they'll notice that if they have 3 coconuts they can't hold each one in a different hand at the same time because they only have two hands and three does not equal two.  They'll encounter the truth as soon as they begin interacting with the physical world because the truth exists outside our heads in the world around us.


#17    Meiliken

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Posted 05 April 2009 - 03:25 PM

Siara on Apr 5 2009, 09:38 AM, said:

Wrong.  They will encounter the truth in the course of their experience as logical beings in the rational, physical world.  They will understand the difference between two and three because they'll notice that if they have 3 coconuts they can't hold each one in a different hand at the same time because they only have two hands and three does not equal two.  They'll encounter the truth as soon as they begin interacting with the physical world because the truth exists outside our heads in the world around us.


You're still missing my point.  If the word "two" didn't exist to everyone, and they equated "two" with "three", and all of mathematics ended up getting based around this concept.  Then no one would question it.  If the concept is not there, they'll never concieve it to ever be wrong.  another example would be if someone doesn't realize that walking across the street was illegal(jaywalking), they'd never worry about it.  Now yes, no one really cares about jaywalking these days, but technically it is illegal.  Very few people even know this, and truly no police officer will issue a ticket for such, but they can issue one if they the police officer is a punk.  I know many many people who don't know that jaywalking is illegal.  If they are never told such, the concept is beyond them to question it.  If the concept is beyond someone, if they are never taught it, they'll never know to ask.  Truth doesn't exist outside our heads, but is a concept thought up by people.  A person raised by animals would never understand 1+1=2 because they are raised by animals.

The more I study religions the more I am convinced that man never worshipped anything but himself
Sir Richard Francis Burton

There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance
Hippocrates

A wise man proportions his belief to the evidence.
David Hume

Facts do not cease to exist because they are ignored.
Aldous Huxley

#18    coberst

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Posted 05 April 2009 - 07:15 PM

It is a hypothesis of  SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child constitute a conceptual metaphor at the neural level leading to a primary metaphor that ‘arithmetic is object collection’.  The arithmetic teacher attempting to teach the child at a later time depends upon this already accumulated knowledge.  Of course, all of this is known to the child without the symbolization or the conscious awareness of the child.

The pile of objects became ‘bigger’ when the child added more objects and became ‘smaller’ when objects were removed.  The child easily recognizes while being taught arithmetic that 5 is bigger than 3 and 3 is littler than 7.  The child knows many entailments, many ‘truths’, resulting from playing with objects.  The teacher has little difficulty convincing the child that two collections A and B are increased when another collection C is added, or that if A is bigger than B then A+C is bigger than B+C.

At birth an infant has a minimal innate arithmetic ability.  This ability to add and subtract small numbers is called subitizing.  (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol).  Many animals display this subitizing ability.

In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment, and independent order.
  

Subitizing ability is limited to quantities 1 to 4.  As a child grows s/he learns to count beyond 4 objects.  This capacity is dependent upon 1) Combinatorial-grouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities).

“Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the part-whole structure of objects, with distances, with movement and location, and so on.”

“Conceptual-blending capacity.  You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.”

Primary metaphors function somewhat like atoms that can be joined into molecules and these into a compound neural network.  On the back cover of “Where Mathematics Comes From” is written “In this acclaimed study of cognitive science of mathematical ideas, renowned linguist George Lakoff pairs with psychologist Rafael Nunez to offer a new understanding of how we conceive and understand mathematical concepts.”

“Abstract ideas, for the most part, arise via conceptual metaphor—a cognitive mechanism that derives abstract thinking from the way we function in the everyday physical world.  Conceptual metaphor plays a central and defining role in the formation of mathematical ideas within the cognitive unconscious—from arithmetic and algebra to sets and logic to infinity in all of its forms.  The brains mathematics is mathematics, the only mathematics we know or can know.”

We are acculturated to recognize that a useful life is a life with purpose.  The complex metaphor ‘A Purposeful Life Is a Journey’ is constructed from primary metaphors: ‘purpose is destination’ and ‘action is motion’; and a cultural belief that ‘people should have a purpose’.

A Purposeful Life Is A Journey Metaphor
A purposeful life is a journey.
A person living a life is a traveler.
Life goals are destinations
A life plan is an itinerary.

This metaphor has strong influence on how we conduct our lives.  This influence arises from the complex metaphor’s entailments: A journey, with its accompanying complications, requires planning, and the necessary means.

Primary metaphors ‘ground’ concepts to sensorimotor experience.  Is this grounding lost in a complex metaphor?  ‘Not by the hair of your chiney-chin-chin’.  Complex metaphors are composed of primary metaphors and the whole is grounded by its parts.  “The grounding of A Purposeful Life Is A Journey is given by individual groundings of each component primary metaphor.”


The ideas for this post come from “Philosophy in the Flesh”.  The quotes are from “Where Mathematics Comes From” by Lakoff and Nunez






#19    Meiliken

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Posted 05 April 2009 - 08:15 PM

coberst on Apr 5 2009, 03:15 PM, said:

It is a hypothesis of  SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child constitute a conceptual metaphor at the neural level leading to a primary metaphor that ‘arithmetic is object collection’.  The arithmetic teacher attempting to teach the child at a later time depends upon this already accumulated knowledge.  Of course, all of this is known to the child without the symbolization or the conscious awareness of the child.

The pile of objects became ‘bigger’ when the child added more objects and became ‘smaller’ when objects were removed.  The child easily recognizes while being taught arithmetic that 5 is bigger than 3 and 3 is littler than 7.  The child knows many entailments, many ‘truths’, resulting from playing with objects.  The teacher has little difficulty convincing the child that two collections A and B are increased when another collection C is added, or that if A is bigger than B then A+C is bigger than B+C.

At birth an infant has a minimal innate arithmetic ability.  This ability to add and subtract small numbers is called subitizing.  (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol).  Many animals display this subitizing ability.

In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment, and independent order.
  

Subitizing ability is limited to quantities 1 to 4.  As a child grows s/he learns to count beyond 4 objects.  This capacity is dependent upon 1) Combinatorial-grouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities).

“Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the part-whole structure of objects, with distances, with movement and location, and so on.”

“Conceptual-blending capacity.  You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.”

Primary metaphors function somewhat like atoms that can be joined into molecules and these into a compound neural network.  On the back cover of “Where Mathematics Comes From” is written “In this acclaimed study of cognitive science of mathematical ideas, renowned linguist George Lakoff pairs with psychologist Rafael Nunez to offer a new understanding of how we conceive and understand mathematical concepts.”

“Abstract ideas, for the most part, arise via conceptual metaphor—a cognitive mechanism that derives abstract thinking from the way we function in the everyday physical world.  Conceptual metaphor plays a central and defining role in the formation of mathematical ideas within the cognitive unconscious—from arithmetic and algebra to sets and logic to infinity in all of its forms.  The brains mathematics is mathematics, the only mathematics we know or can know.”

We are acculturated to recognize that a useful life is a life with purpose.  The complex metaphor ‘A Purposeful Life Is a Journey’ is constructed from primary metaphors: ‘purpose is destination’ and ‘action is motion’; and a cultural belief that ‘people should have a purpose’.

A Purposeful Life Is A Journey Metaphor
A purposeful life is a journey.
A person living a life is a traveler.
Life goals are destinations
A life plan is an itinerary.

This metaphor has strong influence on how we conduct our lives.  This influence arises from the complex metaphor’s entailments: A journey, with its accompanying complications, requires planning, and the necessary means.

Primary metaphors ‘ground’ concepts to sensorimotor experience.  Is this grounding lost in a complex metaphor?  ‘Not by the hair of your chiney-chin-chin’.  Complex metaphors are composed of primary metaphors and the whole is grounded by its parts.  “The grounding of A Purposeful Life Is A Journey is given by individual groundings of each component primary metaphor.”


The ideas for this post come from “Philosophy in the Flesh”.  The quotes are from “Where Mathematics Comes From” by Lakoff and Nunez


Thank you for stating it in a way that was more indepth than I would have.  If the concept is not understood or taught, they won't advance further to question it.  I'm not sure if others here would understand what you wrote, but good job anyway. thumbsup.gif  





The more I study religions the more I am convinced that man never worshipped anything but himself
Sir Richard Francis Burton

There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance
Hippocrates

A wise man proportions his belief to the evidence.
David Hume

Facts do not cease to exist because they are ignored.
Aldous Huxley

#20    Siara

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Posted 06 April 2009 - 01:24 AM

Meiliken on Apr 5 2009, 04:25 PM, said:

You're still missing my point.  If the word "two" didn't exist to everyone, and they equated "two" with "three", and all of mathematics ended up getting based around this concept.  Then no one would question it.


I do not believe that we have to be taught to differentiate between the concepts of "two" and "three".  I think it is innate.  There are a few primitive cultures in the world that don't count, but they have physical symbols which they equate on a one-to-one basis with members of the group they're considering.  Example: a hunter wants to catch a rabbit for each member of his family.  He can't count but he wears a pouch around his neck containing 5 pebbles.  The red pebble is associated with his mate.  The large, dark pebble is associated with his oldest son.  The tiny, white pebble is associated with the new baby, etc.  After he's killed a few rabbits he takes the pebbles out of the pouch and spreads them out on the ground, then places a rabbit next to as many pebbles as possible.  If any of his special pebbles don't have a rabbit, he keeps hunting.  He does this until each pebble has a rabbit, then takes the kill home to his family.  He knows that each family member will have a rabbit because every pebble in his pouch corresponds to a rabbit.

This is the absolute most primitive manifestation of counting in the world.  Every human being has this or something more sophisticated.  The feral children in central European countries that were adopted by dog packs still understand one-to-one correlation.  It's hard wired.

There's quite a bit of evidence that many other mammalian species are hard wired to differentiate between one, two, three, and many as well.  When a dog sees a single ball in a box and an experimenter puts other balls in the box, dogs in psychology experiments will spend more time examining the box if it sees one more ball put in the box from a distance and as it draws near it finds three balls in the box (with an extra ball added via a trap door).  It knows that if there was one ball in the box to begin with and someone added another ball there shouldn't be three balls. The dog reacts the same way if it sees two balls put in the box and one is removed via a trap door.  They realize there's a logic error.

I'm not missing your point.  I disagree with you (I've read quite a bit about the subject too).  There aren't any people for whom "two" and "three" aren't separate concepts- except very severely brain damaged people.

Edited by Siara, 06 April 2009 - 01:53 AM.


#21    Siara

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Posted 06 April 2009 - 01:44 AM

coberst on Apr 5 2009, 08:15 PM, said:

The child knows many entailments, many ‘truths’, resulting from playing with objects.  The teacher has little difficulty convincing the child that two collections A and B are increased when another collection C is added, or that if A is bigger than B then A+C is bigger than B+C.

At birth an infant has a minimal innate arithmetic ability.  This ability to add and subtract small numbers is called subitizing.  (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol).  Many animals display this subitizing ability.


In other words, he doesn't need to learn it from other humans.  All he has to do is manipulate elements in his environment.  Any human who has been allowed to manipulate objects in his/her environment intuitively knows that two doesn't equal three.

This is what George Lakoff means by the "embodied mind".  There is quite a bit of basic knowledge that doesn't come to us via other people and for which there's no such thing as "not being prepared to see it" ( referring to the OP).  Lakoff was a student of Noam Chomsky's.  He didn't go for the deconstruction of physical reality.  

Quote

Meiliken wrote:

If the concept is not understood or taught, they won't advance further to question it. I'm not sure if others here would understand what you wrote, but good job anyway.

I don't see how you can draw the conclusion that the person needs to be TAUGHT when you've just read quite a bit of evidence that in fact the person DOESN'T need to be taught.  The person understands intuitively, on his own, from manipulating his environment.  Being taught is not our only source of understanding.  Though "I'm not sure others here would understand" this original.gif

Edited by Siara, 06 April 2009 - 02:20 AM.


#22    Meiliken

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Posted 06 April 2009 - 09:36 AM

Siara on Apr 5 2009, 09:44 PM, said:

I don't see how you can draw the conclusion that the person needs to be TAUGHT when you've just read quite a bit of evidence that in fact the person DOESN'T need to be taught.  The person understands intuitively, on his own, from manipulating his environment.  Being taught is not our only source of understanding.  Though "I'm not sure others here would understand" this original.gif


A person will innately understand that if they add another object to their collection that they have more than they previously had, but the concept of "I had 2, now I have 3" is beyond their understanding unless they are taught it.

The more I study religions the more I am convinced that man never worshipped anything but himself
Sir Richard Francis Burton

There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance
Hippocrates

A wise man proportions his belief to the evidence.
David Hume

Facts do not cease to exist because they are ignored.
Aldous Huxley

#23    UziStuNNa1

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Posted 06 April 2009 - 02:34 PM

Meiliken on Apr 6 2009, 04:36 AM, said:

A person will innately understand that if they add another object to their collection that they have more than they previously had, but the concept of "I had 2, now I have 3" is beyond their understanding unless they are taught it.


Then, according to your argument, how did the numbers two and three become constants in society?  Somebody must have invented them, right?  But what you are saying is it's impossible for someone to understand it unless they are taught it.  All information on Earth today was created/thought of by humans.


#24    HAJiME

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Posted 06 April 2009 - 02:48 PM

Meiliken on Apr 1 2009, 09:30 AM, said:

If one is born and taught that 1+1=3, and are never taught the truth, they'll never question it because they don't realize to.
If this was the case, our species would never discover anything new. Most individuals probably do not question what authority tells them, but quite a few must do.

This may interest some people. It's about the natural tendancy to believe what you are told.

http://books.google.co.uk/books?id=yq1xDpi...lt&resnum=1




#25    Meiliken

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Posted 08 April 2009 - 07:13 AM

UziStuNNa1 on Apr 6 2009, 10:34 AM, said:

Then, according to your argument, how did the numbers two and three become constants in society?  Somebody must have invented them, right?  But what you are saying is it's impossible for someone to understand it unless they are taught it.  All information on Earth today was created/thought of by humans.



Yes, by people who invented them to teach to others.  A person who is raised by wolves will not magically know the concept of numbers until it is taught to them.  It's not even an argument, it's been shown by people who were raised by animals.

The more I study religions the more I am convinced that man never worshipped anything but himself
Sir Richard Francis Burton

There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance
Hippocrates

A wise man proportions his belief to the evidence.
David Hume

Facts do not cease to exist because they are ignored.
Aldous Huxley

#26    Meiliken

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Posted 08 April 2009 - 07:17 AM

HAJiME on Apr 6 2009, 10:48 AM, said:

If this was the case, our species would never discover anything new. Most individuals probably do not question what authority tells them, but quite a few must do.

This may interest some people. It's about the natural tendancy to believe what you are told.

http://books.google.co.uk/books?id=yq1xDpi...lt&resnum=1


Exactly.  Most believe what they are told, some question.  The key word here is told.  Take a baby out of all civilization and put them with animals.  They'll never learn human speech, human concepts of numbers, human convictions.  Until they are taught them.

The more I study religions the more I am convinced that man never worshipped anything but himself
Sir Richard Francis Burton

There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance
Hippocrates

A wise man proportions his belief to the evidence.
David Hume

Facts do not cease to exist because they are ignored.
Aldous Huxley




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