# Thread: Arithmetic is Object Collection

1. 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.

3. Originally Posted by coberst
It is a hypothesis of SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child...
For the benefit of anyone who has not actually read any textbooks on second generation cognitive science, could you possibly help us out with an explanation of what an "object" is?

Cheers.

4. Originally Posted by numbers
Originally Posted by coberst
It is a hypothesis of SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child...
For the benefit of anyone who has not actually read any textbooks on second generation cognitive science, could you possibly help us out with an explanation of what an "object" is?

Cheers.
An object for the child is most likely rocks, marbels, candy, etc.

5. Originally Posted by coberst
An object for the child is most likely rocks, marbels, candy, etc.
I obviously did not make my question sufficiently clear.

When you regard "arithmetic as object collection" you are referring to a metaphorical or hypothetical object, yes?

6. Originally Posted by numbers
Originally Posted by coberst
An object for the child is most likely rocks, marbels, candy, etc.
I obviously did not make my question sufficiently clear.

When you regard "arithmetic as object collection" you are referring to a metaphorical or hypothetical object, yes?
'Arithmetic is object collection' is a linguistic metaphor for what is another form of metaphor called a conceptual metaphor. We unconsciously move these conceptual metaphors about in our brain and these concepts become part of other concepts.

I am refering to a child playing on the ground placing rocks on top of one another or playing marbles, or sharing pieces of candy with his brother, or complaining that his brotther got three pieces and he got only two. Children at play are always collecting things or giving things etc. This is why a child knows a lot about adding and subtracting small numbers of things before s/he is three years old.

The purpose of this post is to detail how we all learn things and many of these things we learn become, throgh the process of conceptual metaphor, the foundation of other concepts.

Many years ago, before ‘self-service’, it was common to pull into a gas station and when the attendant came to the car the motorist would say “Fillerup”.

“More is up” is a common metaphor. I think of it every time I pour milk into a measuring cup when baking cornbread. The subjective judgment is quantity, the sensorimotor domain is vertical orientation, and the primary experience is the rise and fall of vertical levels as fluid is added or subtracted and objects are piled on top of or removed from a collection.

We can see (know is see) by this mechanism that we equate vertical motion in the spatial domain with quantity; we use the vertical domain to reason about quantity. We have a vast experience in vertical space domain reasoning and thus we derive this great experience to help us in reasoning about quantity; no doubt a very useful thing when first learning arithmetic. Teachers of mathematics, I suspect, depend upon this storehouse of knowledge to make abstract mathematical reasoning for children more comprehensible.

In a metaphor the source domain, ‘up’, is mapped onto the target domain ‘more’. The neural structure of the sensorimotor domain, the primary metaphor, is mapped onto the subjective domain ‘more’. Reasoning about the vertical motion in the spatial domain is mapped onto reasoning about the quantity domain. This is a one-way movement; reasoning about quantity is not mapped onto spatial domain reasoning. The direction of inference indicates which the source is and which the target domain is.

Physical experiences of all kinds lead to conceptual metaphors from which perhaps hundreds of ‘primary metaphors’, which are neural structures resulting from sensorimotor experiences, are created. These primary metaphors provide the ‘seed bed’ for the judgments and subjective experiences in life. “Conceptual metaphor is pervasive in both thought and language. It is hard to think of a common subjective experience that is not conventionally conceptualized in terms of metaphor.

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