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  1. #1
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    I challenge the student of mathematics

    I challenge the student of mathematics

    It appears to me that most people look on math as something with supernatural qualities. I challenge the student of math to develop and post short essays on Internet discussion forums about those fundamental aspects of math that you think people can and should comprehend.

    What follows is something that I have posted regarding my idea of what ordinary citizens should know abut this very fundamental domain of knowledge.

    Arithmetic is object collection

    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

  2. #2
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    Re: I challenge the student of mathematics

    Well, do you want to start with who invented Calculus? Was it Newton or Liebnitz? I can put it in short words if you are interested.

  3. #3
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    Re: I challenge the student of mathematics

    Quote Originally Posted by tweetykiss View Post
    Well, do you want to start with who invented Calculus? Was it Newton or Liebnitz? I can put it in short words if you are interested.
    I hope some one can do better than that.

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    Re: I challenge the student of mathematics

    Quote Originally Posted by coberst View Post
    I hope some one can do better than that.
    Well, I honestly don't know what you are looking for. Many do find that debate very interesting since it is a major part of math history.

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    Re: I challenge the student of mathematics

    Quote Originally Posted by tweetykiss View Post
    Well, do you want to start with who invented Calculus? Was it Newton or Liebnitz? I can put it in short words if you are interested.
    I thought they developed it contemporaneously and independantly. I've never really looked in to it, though.

  6. #6
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    Re: I challenge the student of mathematics

    Quote Originally Posted by kazza View Post
    I thought they developed it contemporaneously and independantly. I've never really looked in to it, though.

    They did it independently and never worked together on it. That was one of the reasons why there was a huge controversy over this subject as to who invented Calculus.

    Newton also hide in the dark for a while since he got 'sick'. Then they accused each other for stealing each other's notes.

  7. #7
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    Re: I challenge the student of mathematics

    Quote Originally Posted by tweetykiss View Post
    Well, I honestly don't know what you are looking for. Many do find that debate very interesting since it is a major part of math history.
    How about explaining why a new math was needed as a result of Newton's theory. Or perhaps why math, the science of pattern, was so necessary for helping us to understand mechanics.

  8. #8
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    Re: I challenge the student of mathematics

    Quote Originally Posted by coberst View Post
    How about explaining why a new math was needed as a result of Newton's theory. Or perhaps why math, the science of pattern, was so necessary for helping us to understand mechanics.
    Maybe you should work on something simple first, like 'punctuation'. You know where you follow a question with something called a 'question mark'.
    Looks like this;
    ?

    Also, If I may ask?

    Why are you posting this same thing all over the internet?
    scienceagogo.com
    thescienceforum.com
    typologycentral.com

    Just a few of the 11, so far, that I found with a quick Google search.
    Ignorance more frequently begets confidence than does knowledge: it is those who know little, and not those who know much, who so positively assert that this or that problem will never be solved by science. -C. Darwin

  9. #9
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    Re: I challenge the student of mathematics

    Quote Originally Posted by coberst View Post
    I challenge the student of mathematics

    It appears to me that most people look on math as something with supernatural qualities. I challenge the student of math to develop and post short essays on Internet discussion forums about those fundamental aspects of math that you think people can and should comprehend.

    What follows is something that I have posted regarding my idea of what ordinary citizens should know abut this very fundamental domain of knowledge.

    Arithmetic is object collection

    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
    i'm not connecting the dots here!? you always seem to have a strange way of changing horses in mid stream and not noticing it!? maybe your trying to mix references in a kind of short hand that only you actually understand!? it's pretty common these days!? at least it pretends to be coherent!? my stuff never seemed to reach that kind of polished appearance!? it wasn't supposed to i can assume!? words without a telepathic connection are mostly complication rather than discovery/solution/release!? you should develop a better "bedside manner" as it's called!? :freak3: :spin2: :
    i do not endorse/recommend any advertising on scam.com associated with my name /posts or otherwise. thank you

  10. #10
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    Re: I challenge the student of mathematics

    Quote Originally Posted by tweetykiss View Post
    They did it independently and never worked together on it. That was one of the reasons why there was a huge controversy over this subject as to who invented Calculus.

    Newton also hide in the dark for a while since he got 'sick'. Then they accused each other for stealing each other's notes.

  11. #11
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    Re: I challenge the student of mathematics

    Quote Originally Posted by coberst View Post
    At birth an infant has a minimal innate arithmetic ability. Many animals display this subitizing ability.
    I know a horse who can solve differential equations but it is his expertise in field propagation theory that is bringing him mounting noteriety.....go figure.

  12. #12
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    Re: I challenge the student of mathematics

    Quote Originally Posted by phlipper View Post
    I know a horse who can solve differential equations but it is his expertise in field propagation theory that is bringing him mounting noteriety.....go figure.
    ROFLOL. :judges::judges:

  13. #13
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    Re: I challenge the student of mathematics

    @ Kazza

    I cannot stop laughing at the Newton/ Liebnitz cartoon, not completely sure why. I've just set it to be my desktop background. Thanks.
    I contend we are both atheists, I just believe in one fewer god than you do. When you understand why you dismiss all the other possible gods, you will understand why I dismiss yours.
    S. F. Roberts

  14. #14

    Re: I challenge the student of mathematics

    This post made no sense whatsoever. It didn't when it was posted in the Randi forum either. Cutting and pasting like this isn't considered good form, is it?

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