Monday, December 17, 2012

Abstraction (Mathematics)


From wiki:
"Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world; statistics has its origins in the calculation of probabilities in gambling; and algebra started with methods of solving problems in arithmetic.
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract.
The advantages of abstraction are :
  • It reveals deep connections between different areas of mathematics 
  • Known results in one area can suggest conjectures in a related area
  • Techniques and methods from one area can be applied to prove results in a related area.
The main disadvantage of abstraction is that highly abstract concepts are more difficult to learn, and require a degree of mathematical maturity and experience before they can be assimilated.
These persons we have placed in positions of economic authority simply CANNOT DO THIS... they do not possess the necessary degree of this cognitive skill.

I've heard it said that one of the WORST things a so-called "educator" can say to a student is: "How can you not understand this... IT IS SOOO EASY!... ".

An "educator" who would say something like that is perhaps not qualified to be an educator, perhaps politics would be a better field for that person to operate within.

12 comments:

Matt Franko said...

"Pre-emptive/snatched comment" from Calgacus from a thread at NEP that started me thinking of this post here at Mike's:

"The mainstream economists who look like they are doing math – are repelling the man in the street by this appearance – but are doing things that make a mathematician vomit. Good niche for a scam, which is why it has worked so well.

But I have to disagree, just as much, with what you (& Paul) sometimes say about “semantics”. Nobody can not “think semantically”,”think metaphysically”. It’s like resolving to speak without uttering nouns (& verbs for that matter), or nouns and verbs you understand. The point is whether you think about your semantic thinking or not. Do it too much, and you will lose your mind. You have to stop and think “operationally”, unreflectively, abstractly at least once in a while, and often much more than that. But do it too little, and you will begin to talk nonsense about nothing. Mainstream econ manages to talk illogically and avoid reality at the same time. Their amount of crap would never have come out if there weren’t money in it for someone.

People naturally reason abstractly. Cognition goes from the abstract to the concrete, not vice versa. Abstract is easy, to everyone. Concrete is hard, as those of us who have banged their heads against walls know. Look at Hegel’s short essay “Who thinks abstractly?” for a painless intro to this point of view -& remind me what it says – haven’t reread it in decades. Man is a born metaphysician and does metaphysics before he can walk or talk. Like mainstream economics (worldly philosophy), mainstream (of the last century odd, in the Anglosphere especially) philosophy gets everything backwards.

I am not against graphic models and diagrams to help people, as you suggest. Anything that works. But this is part of the main thing, which is always to think it gooder yourself, to always explain it better, more truly, more simply. And to put more of the onus on those of us who think we know what we are talking about, not our students/victims. The main problem of popularization I think is “not what they don’t know, but what they know that ain’t so”, though."

I do not understand everything that Cal writes here... I don't think people can just "turn on" this ability for themselves.

This skill that we all take for granted looks like it just not available to many among us...

rsp,

paul said...

re Calgacus' comment:

Nobody can not “think semantically”,”think metaphysically”. It’s like resolving to speak without uttering nouns (& verbs for that matter), or nouns and verbs you understand.

"People naturally reason abstractly. Cognition goes from the abstract to the concrete, not vice versa. Abstract is easy, to everyone."

Here he goes from nobody can not "think semantically" to "like speaking without uttering nouns and verbs"…thinking and speaking are not the same…then seems to be contradicting himself further as he says "people naturally reason abstractly".

I claim the reverse…no one can not think abstractly…and he is partly right, most of it is easy, but the part that isn't easy is the part many are missing or unskilled at…pattern recognition.

One thinks abstractly and the translation to words when one speaks is where semantics comes in. A thought is abstract, the phrase uttered is a rough approximation of the thought, unless one is having a conversation within one's own mind.

So, a phrase is uttered, that is a rough approximation of one's thoughts, then the phrase is parsed by another, with another rough approximation as it is translated back into an abstract idea. A lot can be lost in the translations.

Remember the game where a phrase is passed within a group from one to the other until, when it reaches the last participant, the phrase has been so badly mangled the original meaning is lost?

Take any phrase describing some abstract concept uttered by ten different people…every one will likely have a different abstract understanding of that phrase. Language is a charicature or cartoon-like expression of real ideas…a rough approximation. That's all it can be. Words and thoughts are not the same.

I think that people who have trouble with math abstractions are unskilled in pattern recognition, that is where the problem lies, not so much with the math skills, and not so much are they "trapped" in a semantic straight-jacket. I believe our education system does not teach the critical thinking skills necessary, so only those that can pick it up on their own learn them. Pattern recognition is the key.

Math skips the language barrier, so there can be only one understanding in normal parlance. There are not multiple understandings of 2+2=4 except maybe in the world of abstract mathematics. It is a perfect picture. Those things that are not based in math still appear as patterns, but are more difficult to quantify or describe, ie human behavior, etc.

The part of economics examined through MMT principles is pure math…there is only one solution or right answer resulting from each arithmetic operation, even though we find nearly everyone describes this reality in different ways. It is the description that is in error, not the system.

It is a simple system and the transactions are simple arithmetic…every operation transfers a number from one place in the system to another, repeated trillions of times, within a container of objects (dollars) the quantity of which are fixed and finite until some external add or subtract is applied. Everything else that happens (as a result of spending) within the economy follows from this simple system relationship.

Jure Jordan said...

paul
you said it right.
The school is exactly about learning that, patern recognition and expressing it in more and more universal form.
School is supposed to provide standardization of abstracts to masses that could not communicate with all different meanings given to words initially.
Are you saying that schools really failed at that, failed so badly?

paul said...

"Are you saying that schools really failed at that, failed so badly?" - Jure Jordan

Yes, I think so. I believe it is deliberate, though not a conspiracy.

TPTB think it is good to taech people only those skills needed to maximize production, we are thus trained to be worker-bees even though everyone isn't wired that way. Critical thinking is probably a detriment to the system of production we have, as participants would be inclined to question things more, and managers would have a lot more competition from below than they have now, so…

I read a study recently, can't recall where (that's another study) that at the kindergarten level nearly 100% of kids score as geniuses on the creativity spectrum.

The scores decline drastically as one progresses through the education system.

Matt Franko said...

I think MMT will never be widely understood unless this cognitive issue is addressed in the "teaching"....

Converting to a "moral" argument while perhaps will not hurt, doesnt help in this regard....

Even the Pope knows that the current arrangements result in "immoral" outcomes... there are PLENTY of people who understand that the outcomes are "immoral"...

not many that can fully understand what all of our options are...

there is still MUCH true "teaching" to do...

rsp,

Tom Hickey said...

I read a study recently, can't recall where (that's another study) that at the kindergarten level nearly 100% of kids score as geniuses on the creativity spectrum.

Creativity is essentially about the ability and willingness to explore options. This is a natural tendency that is largely bred out in the socialization process of learning to conform and institutional education that basically instructs on how to stay in the boundaries and be successful at it.

Out of the box thinking and creative exploration get tossed under the bus, even in supposedly creative subjects like art and music, where one learn traditional forms and improv is an afterthought. It's supposed "to come later." When that later comes the person is already thorough inculcated.

Virtually all spiritual masters also teach that "higher" learning is forgetting.

"Yen Hui said, 'I'm getting better.'

Confucius said, 'What do you mean?'

'I have forgotten kindness and justice' [primary Confucian values].

'Fine, but that's not enough.'

On another occasion, they met again and Yen Hui said, 'I've improved.'

Confucius said, 'What do you mean?'

'I have forgotten rituals and music.' [Also Confucian values]

'Good, but that is still not enough.'

On another occasion they met and Yen Hui said, 'I'm getting better.'

Confucius said, 'What do you mean?'

'I can sit right down and forget everything.'

Confucius was certainly disturbed by this and said, 'What do you mean by sit right down and forget?'

Yen Hui replied, 'My limbs are without feeling and my mind is without light. I have ignored my body and cast aside my wisdom. Thus I am united with the Tao. This is what sitting right down and forgetting is.'

Confucius said, 'If you are one with the great Way, then you no longer have preferences. If you are one with the cosmos then you are transformed. If this is what you have done, then I would like to follow you.'"


Chuang Tzu 
in Martin Palmer with Elizabeth Breuilly, Translators.
_The Book of Chuang Tzu_
NY: Penguin Arkana, 1996, pp. 57-58

BTW, this is a Taoist teaching story and is not meant a historical account about Confucius (romanization of Kung Tzu, Kongzi, and Kongfuzi, meaning Master Kung).

David said...

People naturally reason abstractly. Cognition goes from the abstract to the concrete, not vice versa. Abstract is easy, to everyone. Concrete is hard, as those of us who have banged their heads against walls know.

That's not what modern child psychologists would say. John Piaget found that children spent years at what he termed "sensory-motor" learning laying the groundwork for later abstact thought. His famous example of where children make the transition from sensory-motor to logico-mathematical reasoning was when in the course of play they could tell that there was the same amount of liquid in a short wide cylinder as there was in a tall thin one.

I think MMT is interesting from this standpoint if you consider that Warren Mosler spent many years "playing" with the money system before he ever theorized about it. He developed a "feel" for how things worked before he tried to explain it in words. I think Piaget was pretty much right. Think of what you're good at. Chances are you have a pretty solid sensory-motor foundation for that skill or knowledge.

Matt Franko said...

D,

right like the wiki states: "Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures."

I would say that is the way it works for me but perhaps not others...

Matt Franko said...

Next question is how do we get these vain idiots in policymaking to admit this and seek alternative qualified counsel????

paul said...

"Next question is how do we get these vain idiots in policymaking to admit this and seek alternative qualified counsel????"

Matt, I'm not sure they would want to if they could…it would take away a lot of leverage from TPTB.

David said...

I remember as a child having a great fondness for the number 8. This fondness was based on the "discovery" I had made that you could divide it in half four times and have still have it come out even. I hadn't learned the rules of arithmetic yet nor was I particularly aware of the meaning of "odd" and "even." By the time I learned arithmetic in school, in the usual way, memorizing rules and tables, all that sort of fascination was gone and math became an unpleasant chore.

The ancient Greeks approached mathematics in that childlike way, concerning themselves with the qualities of numbers and the peculiar relationships between numbers. The Pythagorean Theorem wasn't taught in the mundane way that most of us learn it, but as a culmination to a period of preparation. When the pupil was deemed "worthy" it was "revealed" as one of the great secrets of the universe.

I feel that if I had been encouraged as a child to learn mathematics in the way that came naturally to me, I would have gone a lot further in it and, more importantly, would have gained a living understanding of it that I could have used as a creative impulse in my life.

geerussell said...

The ancient Greeks approached mathematics in that childlike way, concerning themselves with the qualities of numbers and the peculiar relationships between numbers. The Pythagorean Theorem wasn't taught in the mundane way that most of us learn it, but as a culmination to a period of preparation.

Speaking of childlike sense of wonder at math, What was up with Pythagoras?