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Sat 12 Sep, 2009 05:10 am

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

@coberst,

Sorry, but you are wrong on several fronts.

First, you are completely wrong about child development. A child is unable to add numbers (even 2 + 2) until well past the age of language development (you can test this very easily with the nearest 2 year old who can talk). Children aren't able to do Piagetian conservation tasks (understanding that rearranging a collection of objects doesn't change its number) until they are near school age.

A four year old (including mine) adds by counting two piles... if you move one object from one pile to the other, she counts again (not understanding that the number doesn't change).

@ebrown p,

You seem to be confusing Mathematics with arithmetic. This is like confusing Language with vocabulary.

Take modern mathematics... in my job we use some very abstract ideas of linear algebra and statistics to do speech recognition. It would silly to call this "object collection". The fact is, this mathematics works. It provides a useful service to our customers that has a measurable quality.

As far as a purposeful life...

The Universe has been around for 15 billion years. OK, let me just once write that out: 15,000,000,000 years. It is huge with hundreds of billions of stars all blinking in and out of existance.

Our little speck of dust around a fairly typical little star has been around for about 5,000,000,000 years. In another 5 billion years or so the sun will expand, swallowing up the earth and any trace of humanity.

Of course far before that, in just 500 years or so, we will not only be dead, but there will be no memory of us left. The universe will go on as if we had never existed.

In a New York museum there is a scale version of the history of the Universe, the Big Bang at one end of a long hall... current time at the other end. The amount of time that humans have existed is about the width of this letter "n" (and of course my puny insignificant little life is a small fraction of that).

Thinking that the life of one little human has any purpose, given the brief instant we are alive and the little speck of dust orbiting a speck of dust we live on-- is kind of laughable.

@ebrown p,

ebrown p wrote:

First, you are completely wrong about child development. A child is unable to add numbers (even 2 + 2) until well past the age of language development (you can test this very easily with the nearest 2 year old who can talk).

A total diversion, sorry, but there is a tribe in the amazon whose language prevents them from comprehending addition. They have words for one, two, and many. So they understand quantities of one, two, and everything else. It's quite fascinating. I'll be back with reference, but your post just reminded me.

Edit: here it is.

http://www.sciencedaily.com/releases/2008/07/080714111940.htm
@FreeDuck,

That's cool FreeDuck. I had never heard of them.

@ebrown p,

Quickie from Wiki: "Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic[1] when referring to number theory, but this should not be confused with elementary arithmetic."

@FreeDuck,

That tribe understands even less than that, Free Duck: they can't grasp that 4 is bigger than 3 no matter how often you demonstrate it using shells or whatever. The "1,2, many" concepts are true for all original African languages as well as aboriginal languages from Brazil to Papua-New Guinea, though some tribes have native words for numbers as high as 5.