Arithmetic is Object Collection

[coberst]

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

[skasian]

Personally, I love mathematics and the logics that links with it. I believe that numbers itself is encoded in every single aspects of our being,life,world and universe. I believe it is numbers that God used to construct a code during Creation. Measurement, time, movement, music, properties of matter I believe are governed by numbers.

Thank you for sharing the insights, I agree the theories that have been made here.

[0=2]

Too bad the full potential of mathematics are simplified by our, ironically, "science" obsessed culture. We love to "rationalize" but forget that every quantitive function only exists because it has a QUALITIVE COEFFICIENT.

In other words, numbers don't create the objects they grasp, they simply quantify a qualitive association. This is why, in real life, 1-1 does NOT equal zero, because the residue remains and there are too many qualitive functions that keep such residue around. An absolute "zero" is only functional when looked at SUBJECTIVELY. It's relative to the object/subject you're quantifying.

This is why mathematics is commonly misunderstood as an "object" interpretation, when the things we choose to SUBJECTify as a "1" and as not a "1" determine what is and isn't, and the isness of a such is the foundation on which we build our subjective consciousness.

In this way we even limit mathematics itself. For those mathematically attuned I strongly recommend the integral or holistic approach to... approaching mathematical application, and hte refining of this, especially on or in psychological terms. Here's a link http://indigo.ie/~peter/integral.html

This, however, is quite interesting. I just thought I'd tote in and remind those collegial godhead academia that, at the foundation of every one, in an inexpressible zero, and that according to the weapon they choose to subjectify in hte name of thier "absolute", there is no observer, there is only participant.

[coberst]

How does a hen recognize a missing chick?

I live in the mountains and often hike down country roads. One morning while hiking down a road I noticed an unfamiliar rustle in the bushes beside the road. I stepped off the road and into the brush and up popped a flurry of wings as a turkey hen went flapping about my head and settling finally in a tree across the road. I recognized that the rustle in the undergrowth that I saw was the movement of a turkey hen with her brood of chicks.

Later on the return hike down that road heading home I heard what was obviously the turkey hen clucking about in the underbrush trying to gather up her brood. It occurred to me that many animals that have more than progeny at a time must have some way of recognizing that one or more is missing. What makes that possible?

I later discovered the fact of subitizing. Which might correctly we called the innate idea of arithmetic.

[skasian]

How does a hen recognize a missing chick?

I live in the mountains and often hike down country roads. One morning while hiking down a road I noticed an unfamiliar rustle in the bushes beside the road. I stepped off the road and into the brush and up popped a flurry of wings as a turkey hen went flapping about my head and settling finally in a tree across the road. I recognized that the rustle in the undergrowth that I saw was the movement of a turkey hen with her brood of chicks.

Later on the return hike down that road heading home I heard what was obviously the turkey hen clucking about in the underbrush trying to gather up her brood. It occurred to me that many animals that have more than progeny at a time must have some way of recognizing that one or more is missing. What makes that possible?

I later discovered the fact of subitizing. Which might correctly we called the innate idea of arithmetic.

I agree with this as animals and ourselves are born with sense and appreciation to arithmetic. We are created by the laws of arithmetic and numbers therefore it is in our natures to have basic knowledge and the effects of arithmetic.

I have to argue with the idea that "arithmetic is object collecting" where object collecting is not the only category that arithmetic belongs to.
Take the example of the golden ratio, where it links with Fibonacci sequence, and is an algebraic irrational number that is obtained by an equation. Golden ratio exists in nature, and the components that makes it up is "beautiful" or "appealing" for us. We are an example of this and so is A4 paper.
The central point I am getting into is that we forget that irrational things in life such as the golden ratio and E=mc2 are also composed by arithmetic and also not "object collection."
In a matter of fact, this statement can be altered into "arithmetic is manipulation of numbers" as it is simply changing of numbers.

[skasian]

Too bad the full potential of mathematics are simplified by our, ironically, "science" obsessed culture. We love to "rationalize" but forget that every quantitive function only exists because it has a QUALITIVE COEFFICIENT.

In other words, numbers don't create the objects they grasp, they simply quantify a qualitive association. This is why, in real life, 1-1 does NOT equal zero, because the residue remains and there are too many qualitive functions that keep such residue around. An absolute "zero" is only functional when looked at SUBJECTIVELY. It's relative to the object/subject you're quantifying.

This is why mathematics is commonly misunderstood as an "object" interpretation, when the things we choose to SUBJECTify as a "1" and as not a "1" determine what is and isn't, and the isness of a such is the foundation on which we build our subjective consciousness.

In this way we even limit mathematics itself. For those mathematically attuned I strongly recommend the integral or holistic approach to... approaching mathematical application, and hte refining of this, especially on or in psychological terms. Here's a link http://indigo.ie/~peter/integral.html

This, however, is quite interesting. I just thought I'd tote in and remind those collegial godhead academia that, at the foundation of every one, in an inexpressible zero, and that according to the weapon they choose to subjectify in hte name of thier "absolute", there is no observer, there is only participant.

I have to argue, 1-1 does equal 0 and that there is no remainder from the equation. You may be accounting the remainders that exists outside the equation however, as 0 accounts in this equation, after the equation, there is none of the 1 that may represent a component. If you account this in reality, this equation would not be 1-1 as in real life it would be "infinity - 1" as the universe = infinity. In this way the equation will not equal zero, making your point correct.

I agree with the absolute zero not zero as in reality, it is not zero degrees, but -273degree celsius where it is the temperature where no organism can carry out life function.

[0=2]

Yes, but when applying mathematics to ANYTHING, as i previously stated, you MUST secularize to reach an absolute that is not dependent on the whole, such as... zero.

A zero is in relation to the something you saw and the completion of the something related to the terms you defined the problem with. Thus showing that the objects themselves do not "contain" these numbers indefinitely, because the numbers themselves are relative.

[skasian]

Interesting points, but indeed 1-1=0 indeed is in reality.

[coberst]

skasian


The conceptual metaphor forms the foundation for this description of where arithmetic comes from. The post was an attempt to arouse the curiosity of the reader to this important and revolutionary concept. I read lots of stuff and write as I study because writing helps me learn. I post those ideas that I think are important that everyone could profit from studying. My hope is that some of the readers will become curious enough to go to the books and they might become self-actualizing self-learners.

[skasian]

Ah. Well I would like to thank you for your attmpts and effort to provoke other people to think in a new perspective and ponder areas that are new and interesting to them. I am one of them, and I thank you for that. I found the link you provided is very interesting and I hope other people will think that way as well.
May I ask what is that you are studying?

[coberst]

skasian


I am a retired engineer with a good bit of formal education and twenty five years of self-learning. I began the self-learning experience while in my mid-forties. I had no goal in mind; I was just following my intellectual curiosity in whatever direction it led me. This hobby, self-learning, has become very important to me. I have bounced around from one hobby to another but have always been enticed back by the excitement I have discovered in this learning process. Carl Sagan is quoted as having written; “Understanding is a kind of ecstasy.”

I label myself as a September Scholar because I began the process at mid-life and because my quest is disinterested knowledge.

Disinterested knowledge is an intrinsic value. Disinterested knowledge is not a means but an end. It is knowledge I seek because I desire to know it. I mean the term ‘disinterested knowledge’ as similar to ‘pure research’, as compared to ‘applied research’. Pure research seeks to know truth unconnected to any specific application.

I think of the self-learner of disinterested knowledge as driven by curiosity and imagination to understand. The September Scholar seeks to ‘see’ and then to ‘grasp’ through intellection directed at understanding the self as well as the world. The knowledge and understanding that is sought by the September Scholar are determined only by personal motivations. It is noteworthy that disinterested knowledge is knowledge I am driven to acquire because it is of dominating interest to me. Because I have such an interest in this disinterested knowledge my adrenaline level rises in anticipation of my voyage of discovery.

We often use the metaphors of ‘seeing’ for knowing and ‘grasping’ for understanding. I think these metaphors significantly illuminate the difference between these two forms of intellection. We see much but grasp little. It takes great force to impel us to go beyond seeing to the point of grasping. The force driving us is the strong personal involvement we have to the question that guides our quest. I think it is this inclusion of self-fulfillment, as associated with the question, that makes self-learning so important.

The self-learner of disinterested knowledge is engaged in a single-minded search for understanding. The goal, grasping the ‘truth’, is generally of insignificant consequence in comparison to the single-minded search. Others must judge the value of the ‘truth’ discovered by the autodidactic. I suggest that truth, should it be of any universal value, will evolve in a biological fashion when a significant number of pursuers of disinterested knowledge engage in dialogue.

In the United States our culture compels us to have a purpose. Our culture defines that purpose to be ‘maximize production and consumption’. As a result all good children feel compelled to become a successful producer and consumer. All good children both consciously and unconsciously organize their life for this journey.

At mid-life many citizens begin to analyze their life and often discover a need to reconstitute their purpose. Some of the advantageous of this self-learning experience is that it is virtually free, undeterred by age, not a zero sum game, surprising, exciting and makes each discovery a new eureka moment. The self-learning experience I am suggesting is similar to any other hobby one might undertake; interest will ebb and flow. In my case this was a hobby that I continually came back to after other hobbies lost appeal.

I suggest for your consideration that if we “Get a life—Get an intellectual life” we very well might gain substantially in self-worth and, perhaps, community-worth.

As a popular saying goes ‘there is a season for all things’. We might consider that spring and summer are times for gathering knowledge, maximizing production and consumption, and increasing net-worth; while fall and winter are seasons for gathering understanding, creating wisdom and increasing self-worth.

I have been trying to encourage adults, who in general consider education as a matter only for young people, to give this idea of self-learning a try. It seems to be human nature to do a turtle (close the mind) when encountering a new and unorthodox idea. Generally we seem to need for an idea to face us many times before we can consider it seriously. A common method for brushing aside this idea is to think ‘I’ve been there and done that’, i.e. ‘I have read and been a self-learner all my life’.

It is unlikely that you will encounter this unorthodox suggestion ever again. You must act on this occasion or never act. The first thing is to make a change in attitude about just what is the nature of education. Then one must face the world with a critical outlook. A number of attitude changes are required as a first step. All parents, I guess, recognize the problems inherent in attitude adjustment. We just have to focus that knowledge upon our self as the object needing an attitude adjustment rather than our child.

I am not suggesting a stroll in the park on a Sunday afternoon. I am suggesting a ‘Lewis and Clark Expedition’. I am suggesting the intellectual equivalent of crossing the Mississippi and heading West across unexplored intellectual territory with the intellectual equivalent of the Pacific Ocean as a destination.

[skasian]

Yes, I agree with your views and insights.
I do believe that "Understanding is kind of ecstasy” as one understanding may trigger an interest that will provoke the person to understand more to the understanding they already have. I see it as an insatiable hunger, like when you are reading a book, you must know the whole detail or plot that will satisfy your curiosity that is arose from the beginning. I also believe that in ourlives, it is impossible to be fully educated and learnt as there are infinite facts and informations that are to be learnt.

I have always thought that learning something that interests a being gives them extraordinary physical energy or strength that will help persue the person from learning more. For example one can spend a couple of days without sleeping continuing learning something they are deeply interested and have a passion to. They feel their focus is unhindered and they sense a great energy that will prevent them from feeling tired. So I have to ask you how can you be interested in disinterested knowledge? Correct if my understand is wrong, does disinterested knowledge mean knowledge a person have that is not interesting to them?

At high school, I have been taught in science classes that we must not look with our eyes when we are observing or learning something that can be seen as the most simple. That we must look outside the square and with more insight, attempting to unveil the obvious and discover a new layer. Scientists have a living job of this, seeing or identifying the obvious and trying to perceive in a new perspective or layer in order to use its function in a new, innovative way. I agree that our culture must focus the method of learning to encourage people to perceive a different world in order to alter a world that is better, faster, stronger than the one we are recognising now and one way as you have suggested is changing the bad attitudes that we may have.