Old 09-12-2009, 06:06 AM   #1
coberst
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Default 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
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Old 09-12-2009, 10:31 AM   #2
devonin
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Default Re: I challenge the student of mathematics

You again? No thank you.
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