It's an ongoing process, moving further and further away from specific, focused areas of math to "higher" levels of mathematical thought in which conceptual umbrellas are cast over seemingly different subjects, showing common features. Sometimes this results in solutions of long-standing problems, but being more distant from the nitty-gritty of specific areas of thought, like looking at Earth from the space station and not being able to distinguish details, problems, or even opportunities for exploration in those disciplines don't show up.
Higher levels of abstraction are difficult for most students to comprehend. When I entered college in the 1950s I took a course in analytic geometry (AG), drawing figure after figure in the Cartesian plane. With this background, calculus was easier to understand, whereas AG was dropped from most curricula several years later and the subject was quickly and somewhat breezily covered in the first few weeks of calculus. Then, later I came across an introductory calculus text that began with elementary linear algebra in n-dimensional Euclidean space. This coincided roughly with the "New Math" movement.
Richard Feynman (1965):
"If we would like to, we can and do say, 'The answer is a whole number less than 9 and bigger than 6,' but we do not have to say, 'The answer is a member of the set which is the intersection of the set of those numbers which are larger than 6 and the set of numbers which are smaller than 9' ... In the 'new' mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material."
Abstraction and generalization in mathematics also have the effect of opening up potential areas of thought and research topics when the lower levels of mathematics have been pretty much "mined out". So, PhD programs are influential in pushing in these directions. :cool:
and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material."
If I understand it correctly, I can't jump on board with this third point. It's always good to know about the applications and historical context of a subject, but often, the most compelling aspect of a mathematical subject is its elegance and simplicity. If the topic is mathematically elegant, that is usually more than enough to spark interest and curiosity and keep a student interested. Conversely, even the most applicable subjects can be arduous to study due to the clumsy and complicated math.
'Thought abstraction' can be helped too. Learning metacognition is easy and serves a good base: https://tinyurl.com/yawenkr2
But the best route I've found is reading postmodern meta-fiction. The writing is designed to carry reader consciousness beyond realist/modernist (standard) forms using creative meta-persectives, sometimes even making the author part of the story.
John Barth's short story 'Lost in the Funhouse' is one of the most famous: https://www.goodreads.com/book/show/12885.Lost_in_the_Funhouse
Another famous one is 'The Balloon' by Donald Barthelm, which only runs a few pages and can be read here: https://tinyurl.com/ycjcj45p
If you google postmodern novels most of the examples have something meta going on, though sometimes more subtly than the examples above.
A more direct learning approach can be found in the work of Edward de Bono, who was the pioneer of lateral thinking techniques. His short how-to, 'The Use of Lateral Thinking', is excellent: https://www.goodreads.com/book/show/829643.The_Use_of_Lateral_Thinking
Some of these aren't specific to philosophy, but they all train the skill of dimension-hopping, which makes it easier to comprehend abstractions in general.
If I understand it correctly, I can't jump on board with this third point
Feynman, not me. But he had a point about the set theory. I taught college algebra in the era of the New Math, and the first chapter in the book we used was axiomatic structure. No matter who taught the course, the students were not happy.
and the first chapter in the book we used was axiomatic structure. No matter who taught the course, the students were not happy.
Thats interesting because I remember Feynman talking (youtube video) about two methods of math in history, the Greek method and the Babylonian method. The Greek method used axioms, where as the Babylonian method did not.
Reply to Wheatley Well, it helped me answer your post, and the OP. The mooted analysis of analysis - requested by the OP - is that it is patterns of patterns.
I'm literally making this up from my comfy armchair on a quiet Saturday afternoon, while I plan my evening repast and watch Blues videos on FaceTube.
An interesting counter to what I said might involve information; speaking roughly, the stronger the pattern, the less information it contains. SO I suppose that one might argue that the higher the level of abstraction, the lower the information conveyed.
Is that your point?
Roast chicken thigh, herbs and halloumi, I think. A dish I've made a few times before, and am interested in perfecting.
I've modified it for two, using thigh fillets because they are to hand, but the absence of a bone reduces the flavour. It's very oily, which I enjoy too much. I'm planning to reduce the oil tonight and see what happens.
I also use only fresh herbs - straight from the garden, and in larger quantity than suggested here. I keep aside some extra rosemary for a garnish.
No garlic - which seems odd. Might try adding some. I grow my own.
Reply to Wheatley ...the OP was not about mathematics. But I had previously pointed out that one way to understand maths is that it is the setting to of patterns - making patterns explicit.
Comments (26)
https://en.wikipedia.org/wiki/Abstraction_(mathematics)
It's an ongoing process, moving further and further away from specific, focused areas of math to "higher" levels of mathematical thought in which conceptual umbrellas are cast over seemingly different subjects, showing common features. Sometimes this results in solutions of long-standing problems, but being more distant from the nitty-gritty of specific areas of thought, like looking at Earth from the space station and not being able to distinguish details, problems, or even opportunities for exploration in those disciplines don't show up.
Higher levels of abstraction are difficult for most students to comprehend. When I entered college in the 1950s I took a course in analytic geometry (AG), drawing figure after figure in the Cartesian plane. With this background, calculus was easier to understand, whereas AG was dropped from most curricula several years later and the subject was quickly and somewhat breezily covered in the first few weeks of calculus. Then, later I came across an introductory calculus text that began with elementary linear algebra in n-dimensional Euclidean space. This coincided roughly with the "New Math" movement.
Richard Feynman (1965):
"If we would like to, we can and do say, 'The answer is a whole number less than 9 and bigger than 6,' but we do not have to say, 'The answer is a member of the set which is the intersection of the set of those numbers which are larger than 6 and the set of numbers which are smaller than 9' ... In the 'new' mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don't think it is worthwhile teaching such material."
Abstraction and generalization in mathematics also have the effect of opening up potential areas of thought and research topics when the lower levels of mathematics have been pretty much "mined out". So, PhD programs are influential in pushing in these directions. :cool:
If I understand it correctly, I can't jump on board with this third point. It's always good to know about the applications and historical context of a subject, but often, the most compelling aspect of a mathematical subject is its elegance and simplicity. If the topic is mathematically elegant, that is usually more than enough to spark interest and curiosity and keep a student interested. Conversely, even the most applicable subjects can be arduous to study due to the clumsy and complicated math.
Learning how to visualise 4 dimensional space is helpful:
http://www.geom.uiuc.edu/docs/outreach/4-cube/
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/four_dimensions/index.html
'Thought abstraction' can be helped too. Learning metacognition is easy and serves a good base: https://tinyurl.com/yawenkr2
But the best route I've found is reading postmodern meta-fiction. The writing is designed to carry reader consciousness beyond realist/modernist (standard) forms using creative meta-persectives, sometimes even making the author part of the story.
John Barth's short story 'Lost in the Funhouse' is one of the most famous: https://www.goodreads.com/book/show/12885.Lost_in_the_Funhouse
Another famous one is 'The Balloon' by Donald Barthelm, which only runs a few pages and can be read here: https://tinyurl.com/ycjcj45p
If you google postmodern novels most of the examples have something meta going on, though sometimes more subtly than the examples above.
A more direct learning approach can be found in the work of Edward de Bono, who was the pioneer of lateral thinking techniques. His short how-to, 'The Use of Lateral Thinking', is excellent: https://www.goodreads.com/book/show/829643.The_Use_of_Lateral_Thinking
Some of these aren't specific to philosophy, but they all train the skill of dimension-hopping, which makes it easier to comprehend abstractions in general.
Feynman, not me. But he had a point about the set theory. I taught college algebra in the era of the New Math, and the first chapter in the book we used was axiomatic structure. No matter who taught the course, the students were not happy.
Thats interesting because I remember Feynman talking (youtube video) about two methods of math in history, the Greek method and the Babylonian method. The Greek method used axioms, where as the Babylonian method did not.
Patterns.
Then patterns in the patterns.
Then patterns in those patterns.
They say mathematics is the study of patterns. I always wondered what they meant by a 'pattern'.
A pattern is a repetition.
My point is that saying mathematics is the study of patterns doesn't really tell you anything.
That's very rare. That's what random is.
Quoting Wheatley
It tells you what mathematics is...
On the contrary, mathematicians have a lot to say about randomness.
Quoting Banno
A bunch of patterns. Gotcha.
Exactly.
As if this information helps you...
I'm literally making this up from my comfy armchair on a quiet Saturday afternoon, while I plan my evening repast and watch Blues videos on FaceTube.
An interesting counter to what I said might involve information; speaking roughly, the stronger the pattern, the less information it contains. SO I suppose that one might argue that the higher the level of abstraction, the lower the information conveyed.
Is that your point?
Roast chicken thigh, herbs and halloumi, I think. A dish I've made a few times before, and am interested in perfecting.
With steamed green beans.
What herbs? Any garnish?
Baked chicken with haloumi (kotopolulo sto fourno me haloum)
I've modified it for two, using thigh fillets because they are to hand, but the absence of a bone reduces the flavour. It's very oily, which I enjoy too much. I'm planning to reduce the oil tonight and see what happens.
I also use only fresh herbs - straight from the garden, and in larger quantity than suggested here. I keep aside some extra rosemary for a garnish.
No garlic - which seems odd. Might try adding some. I grow my own.
My own lemon, too.
Much more simpler to say mathematics involves levels of abstraction. :sparkle:
I don't think anyone knows what the OP is about.
Quoting Banno
:vomit: We'll leave it at that.
*Cleanup on post 24!*
Salivation achieved.
Yes; this:
Quoting Gurgeh
I would like to discuss the abstract concepts underlined.
I taught college math for a couple of years right out of college. I can tell you, the students still aren't happy. :rofl: