I challenge the student of mathematics

[coberst]

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

[Preno]

Give it a rest.

[Christina]

Help me here, Preno. It's too early for me to read this kind of thing. Is this philosophy or just miscellaneous rambling?

[Preno]

Is this philosophy or just miscellaneous rambling?It's a coberst post, i.e. a bit of both. I imagine coberst thinks it's philosophy.

[Christina]

Thanks. I'll leave it here for now and see if anyone is interested or if he ever comes back.

[Bane]

Whisky Tango Foxtrot?

[BioBeing]

I do not feel I understand mathematics any better from reading that word salad. How, exactly, is that going to help someone understand anything?

[coberst]

Sorry, evidently I posted this in the wrong place. This is obviously over your head.

[Christina]

Things that are floating somewhere out in space usually are. Off to Philosophy...

[munnki]

What the hell I'll take up the gauntlet

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.


According to who? I know that some people experience difficulties with mathematics but as to its supernatural powers I haven't found too many who regard the subject/category in that way. As to what areas people can and should comprehend - I'm not sure - it's possible to ask the question what areas of mathematics people understand without formal education and that tends to encompass everything from arithmetic etc... - it's also possible to state that people understand seemingly complex areas of mathematics without being able to 'announce' them in the formal symbolic language of the subject. To give an example people playing sports might predict how a ball might curve and its destination without understanding the complex formulae and symbolic language of calculus etc...


What follows is something that I have posted regarding my idea of what ordinary citizens should know abut this very fundamental domain of knowledge.

Fair enough, you have the right to express your opinions about the field...


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.

There are, I suppose parts of arithmetic you could describe as such. I'd have to think about it a lot more but I don't think you can express all arithmetic as object collection. For example, this rules out negative results for arithmetical expressions.
'7-9', for example, is a valid arithmetical expression but it yields a negative result. We can talk about two objects being 'gone' from the shelf or similar but we cannot talk about negative two objects in a pure sense. This is one of the problems we experience in trying to translate mathematics from the maths of 'common sense' into the mathematics that exists in the symbolic plane.
This type of mathematics constitutes the majority of the subject. And while some of its results occur 'naturally' to people who do not use the language, the formalization of mathematics has tended to result in its abstraction.
I would not have preferred the definition 'object collection' for arithmetic. But, up to you...
As to children having an innate grasp of mathematics. I think that's fairly uncontroversial and kind of trivially true. Much the same holds for language in general.


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.

Again, while this seems reasonable... there's nothing particularly new here...

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.

True, and I'm sure any parent or teacher notices this. Children enjoy building, assembling and grouping things as part of play.

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).

I'm not so sure about this. Research I've read does not start or stop at four nor recognize difficulties beyond this number... I'd love to hear the sources of research that announce this number in particular as a stopping point.

“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.”

Sure, but I would have argued that understanding and manipulating metaphors is fairly rigidly built-in. It has been part of the conversation of how learning takes place for teachers certainly since Schoen formalized it in 'The Displacement of Concepts'... but really, as Schoen readily says, you could trace the formal identification of metaphor all the way back to Aristotle. That is, the defining and placing of the concept. As to how abstract ideas arising via metaphor ... yes, but are we talking about their conception in the mind (in which case it's controversial) or their expression (in which case it's trivial that metaphor places a role in the translation)....

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

In this part a huge disconnect occurs... what is the link between arithmetic being object collection... abstract ideas being linked to the ability to metaphorize... and the notion of a directed or purposeful life... these are not well-connected in your op and this will explain the comments below it. Can you break this down in your own words as I'm not understanding it...
TY

[Febble]

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.


"Subitizing" means knowing how many objects there are in a set without serially counting them. Four is the usual limit. AFAIK, the term isn't used for the "ability to add and substract" numbers. It is simply used for the ability to recognise how many objects there are in a group without serially counting them.

Some rare people can do it for larger numbers.

[Garrett]

Is this philosophy or just miscellaneous rambling?It's a coberst post, i.e. a bit of both. I imagine coberst thinks it's philosophy.
I imagine you think you just gave critique.

[Preno]

Is this philosophy or just miscellaneous rambling?It's a coberst post, i.e. a bit of both. I imagine coberst thinks it's philosophy.
I imagine you think you just gave critique.okd3hLlvvLw

[BioBeing]

Sorry, evidently I posted this in the wrong place. This is obviously over your head.

Fine, then I challenge the student of philosophy to develop and post short comprehensible essays on Internet discussion forums about those fundamental aspects of philosophy that you think people can and should comprehend.

If no-one understands what you are saying, maybe, just maybe, the problem isn't everybody else.

[lpetrich]

It seems to me that coberst is talking about how we come to understand mathematical concepts.

However, it is hard to tell from his discussion, as it is for many of coberst's posts.

[Jobar]

It appears to me that most people look on math as something with supernatural qualities.

"Supernatural"? Scarcely. Or at least, no one that has any slight numeracy looks at math that way.

Are you trying to argue against the abstract nature of math? That would be more interesting.

[Cath B]

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.


"Subitizing" means knowing how many objects there are in a set without serially counting them. Four is the usual limit. AFAIK, the term isn't used for the "ability to add and substract" numbers. It is simply used for the ability to recognise how many objects there are in a group without serially counting them.

Some rare people can do it for larger numbers.

In The Man Who Mistook his Wife for a Hat (http://en.wikipedia.org/wiki/The_Man_Who_Mistook_His_Wife_for_a_Hat)Oliver Sacks cites autistic savant twins who allegedly count 111 matches and divide the into a sum of 3 x 37 (prime numbers) almost instantaneously. Have you read any of Sacks work coberst?

When dots are placed in a familiar form, such as standard reproduction in dominoes or dice, it becomes easy to subitize higher numbers than four.

[Jobar]

http://www.snorgtees.com/images/PiBeRational_Thumbnail.gif

[Cath B]

Sorry, evidently I posted this in the wrong place. This is obviously over your head.

It is?

Maybe a lot of folk are smarter than you give credit for coberst.

And maybe you're not as smart relative to a lot of other folk as you think you are.

Or maybe I think you think you're smarter than a lot of other folk more than you really think you are.

Maybe I'm correct in my gut feeling that posts like the one I'm quoting here are unsubstantiated arrogance.

[Cath B]

http://www.snorgtees.com/images/PiBeRational_Thumbnail.gif

:D

[Yahzi]

We are acculturated to recognize that a useful life is a life with purpose.
What does this have to do with mathematics?

It looks like you've cut&pasted two entirely different discussions into one post. As usual, the results of such Frankenstienan surgery are abominable.

:dunno:

[Christina]

When dots are placed in a familiar form, such as standard reproduction in dominoes or dice, it becomes easy to subitize higher numbers than four.

That makes sense. When I read Febble's post I thought that it must be a mistake because I can easily see up to 12 without specifically counting them but it's really the pattern that I recognize instantaneously. I got as far as six randomly spaced coins before they stopped appearing as one image and became discrete objects that I had to count.

[Rilx]

When dots are placed in a familiar form, such as standard reproduction in dominoes or dice, it becomes easy to subitize higher numbers than four.

That makes sense. When I read Febble's post I thought that it must be a mistake because I can easily see up to 12 without specifically counting them but it's really the pattern that I recognize instantaneously. I got as far as six randomly spaced coins before they stopped appearing as one image and became discrete objects that I had to count.
Please recognize, ladies, that you have learned arithmetics and are familiar with higher numbers than four. You cannot reduce yourselves to the time before.

[Rilx]

Is this philosophy or just miscellaneous rambling?It's a coberst post, i.e. a bit of both. I imagine coberst thinks it's philosophy.
I imagine you think you just gave critique.
The problem with coberst is that he uses philosophy forums as a platform for his blog. He reads good authors like Damasio, Lakoff, etc and posts his comments and thoughts. Nothing bad as such, but philosophy forums are irrelevant sites for personal blogging.

[frazier]

It appears to me that most people look on math as something with supernatural qualities.

"Supernatural"? Scarcely. Or at least, no one that has any slight numeracy looks at math that way.
Actually, I think a lot of people do (notwithstanding the loaded connotations of "supernatural", at least on these forums).

Everybody learns math from a "natural" standpoint: collections of objects, counting, etc. etc. As we move on to algebra, and later calculus, it gets harder and harder to do this. Real Analysis? Galois Theory? Forget about it. But most people stop at algebra, and never get past the "natural" applications, which IRL usually involve dollars and cents.

Remember those long dragged-out arguments about whether 0.999... =1? Those arguing that it doesn't, would invariably employ "natural" arguments, in which they obviously pictured a long chain of stepping-stones, each occupied by a '9', which had to be traversed before the number could be evaluated. The idea of taking a limit is unnatural (not to say supernatural).

eta: Clearly "abstract" is a better term than "supernatural". I wish I had thought of the abstract vs. concrete idea way back when, it might have given the arguments with untermensche a little more clarity. I'm also remembering the "orders of infinity" arguments, which had similar difficulties.

[Cath B]

When dots are placed in a familiar form, such as standard reproduction in dominoes or dice, it becomes easy to subitize higher numbers than four.

That makes sense. When I read Febble's post I thought that it must be a mistake because I can easily see up to 12 without specifically counting them but it's really the pattern that I recognize instantaneously. I got as far as six randomly spaced coins before they stopped appearing as one image and became discrete objects that I had to count.
Please recognize, ladies, that you have learned arithmetics and are familiar with higher numbers than four. You cannot reduce yourselves to the time before.

I can't? :eek:

Well, maybe.

But I'd hunch that even without knowing the names we ascribe to numbers it might still be possible to sus. that six dots on a die is more than 5.

And if humans, generally, can't maybe other species can.

I read some research once about pigeons being better than humans at identifying a whether or not a shape's area is kargder than that of another shape.

A suggested reason for this skill is that a pigeon which can identify a larger food scrap over a smaller one would grab it quick and be more likely to survive to produce offspring