# Thread: I challenge the student of mathematics

1. ## 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. Originally Posted by coberst
The ideas for this post come from Philosophy in the Flesh. The quotes are from Where Mathematics Comes From by Lakoff and Nunez
These are good books. Readable philosophy. That rare thing! I also agree with the non-Platonic stance they are taking. They need some perseverance, though. As they are trying to lay the foundations of "embodied" philosophy there is quite a lot of nitty-gritty cognitive detail. But, as I say, they are, at least, readable. Where Mathematics Comes From by Lakoff and Nunez can be read "standalone" without ploughing through Philosophy in the Flesh. There's a lot of repetiotion between the two, so pick the one of most direct interest to read.

Does anyone know of a "more popular", perhaps better written, certainly shorter , account of these ideas? Maybe you need to write it coberst?

3. Originally Posted by mal4mac
These are good books. Readable philosophy. That rare thing! I also agree with the non-Platonic stance they are taking. They need some perseverance, though. As they are trying to lay the foundations of "embodied" philosophy there is quite a lot of nitty-gritty cognitive detail. But, as I say, they are, at least, readable. Where Mathematics Comes From by Lakoff and Nunez can be read "standalone" without ploughing through Philosophy in the Flesh. There's a lot of repetiotion between the two, so pick the one of most direct interest to read.

Does anyone know of a "more popular", perhaps better written, certainly shorter , account of these ideas? Maybe you need to write it coberst?
Metaphors We Live By is shorter.

4. You lost me at "it is a hypothesis of..."

5. It reads OK for someone who has read "Where Mathematics Comes From by Lakoff and Nunez" and has had some immersion in cognitive science - Howard Gardner has written a good overview. I doubt many mathematicians would bother taking the challenge, though! "Second generation what...." they will say, throw it down, and continue doing their mathematical research...

Why challenge mathematicians on LitNet, anyway? Try "physics forum"...

To challenge mathematicians you need to speak their language, and plain everyday English.

6. Originally Posted by Neely
You lost me at "it is a hypothesis of..."
I know, reading without the help of a teacher can be demanding.

Originally Posted by mal4mac
It reads OK for someone who has read "Where Mathematics Comes From by Lakoff and Nunez" and has had some immersion in cognitive science - Howard Gardner has written a good overview. I doubt many mathematicians would bother taking the challenge, though! "Second generation what...." they will say, throw it down, and continue doing their mathematical research...

Why challenge mathematicians on LitNet, anyway? Try "physics forum"...

To challenge mathematicians you need to speak their language, and plain everyday English.
I seek to help the American citizen to become more intellectually sophisticated. To do this we must not be afraid to reach beyond our present zone of intellectual comfort.

7. Originally Posted by coberst
I know, reading without the help of a teacher can be demanding.
I am a teacher...

Besides we can all do this:
1. It is of considerable importance to ensure that under no circumstances should anyone fail to deactivate the overhead luminescent function at its local activation point on their departure to their place of residence, most notably immediately preceding the two day period at the termination of the standard working week.
When we mean this:
2. Always turn the lights out when you go home, especially on a Friday.
But it's not good English.

8. Note to moderators: An *ignore thread* function would be a nice addition to the ignore list feature. I could see if this was about math in literary narratives, but the contentions here make no sense to me.

9. Originally Posted by Neely
But it's not good English.
Shouldn't that be "well"?

10. Originally Posted by Jozanny
Note to moderators: An *ignore thread* function would be a nice addition to the ignore list feature. I could see if this was about math in literary narratives, but the contentions here make no sense to me.
Oh, we have that function and it is rather easy to activate.

Oh, we have that function and it is rather easy to activate.

Ohhhh. The Lit Net efficiency model! I see. I am not objecting to the topic. If mathematics is worth discussion in a lit forum it is fine by me, and calculus served as the foil in the film, Stand and Deliver, which is responsible for every super teacher film since--or maybe that was Mr. Chips--, but I am just not sure how this serves as a literary discussion....

[Jozanny now ducks under the desk and refuses to budge!]

12. I found "Where Mathematics Comes From" by Lakoff and Nunez to be one of the most readable mathematics/physics books I've ever encountered. So you might ask if it counts as literature. If we take literature as writing whose "value lies in beauty of form or emotional effect", then surely it counts? It certainly induced a feeling of awe in me, and also tranquility, as he managed to allay some nagging doubts I had about some areas of philosophy of mathematics (while also introducing other nagging doubts - still an emotional impact!)

If we accept a mathematics books as literature, then we have to ask if it is good literature. Serious literary critics tend to avoid this area (Bloom mentions no science books!) Are they frightened about looking foolish in the eyes of the Dukes of Science? The Faber Book of Science by John Carey might be worth a look to see how a noted literary critic approaches science writing. Melvyn Bragg also has a lot to say in this area, but I'm not sure if he qualifies as "noted literary critic" :-)

13. For those literary types who hated maths at school and just want to dip a toe in to get a feel for life on the other side of the cultural divide:

Mathematics: A Very Short Introduction by Timothy Gowers

Also good for prospective mathematicians, and mathematicians who want to back off for an afternoon and take a birds-eye view of the whole field...

What do you think of Gowers' "abstract" approach coberst?

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