You explicitly don't demonstrate verbal_symbolic
AND visual_spatial skills as a one comprehensive framework, all along this thread.
Here is a concrete example of your failure (which is supported by scientific material) :
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Here is the abstract taken from Philip J. Davis and James A. Anderson book called “
Nonanalytic Aspects of Mathematics and Their Implication for Research and Education,” SIAM Review 21(1979), 112-117:
Abstract
In this paper we make a distinction between the practice of mathematics as it is usually presented--a logical chain of abstract, analytical reasoning from premises to conclusions--and how mathematics seems to be done in actuality--as a series of nonverbal, analog, often kinesthetic or visual insights. Mathematics in recent years has created a hierarchy with highly abstract,
logical and symbolic material at the peak and with more
geometrical, visual, and analog material held to be of lesser worth. We argue that humans are known to vary widely in their approaches to cognition and that the areas of the human brain specifically related to language and logical analysis seem to comprise only a part of the machinery of our intellect. We suggest that it would be wise for the practitioners of mathematics, and perhaps especially the students of mathematics to be aware of the very important nonverbal elements in mathematics.
We feel that excessive emphasis on the abstract, analytic aspects of thought may have had deleterious effects on the profession and that a more appropriate balance, more in line with our cognitive endowment as humans, is desirable.----------------------------------------------------------
Here is a part taken from H. M. Hubey book “
The diagonal infinity: problems of multiple scales” (
http://books.google.com/books?id=wD...m=6&sqi=2&ved=0CD0Q6AEwBQ#v=onepage&q&f=false ) page 297:
“A large part of the human brain is devoted to spatial computation, which in the language of serial logic becomes mere
intuition. But the only way in which we can actually comprehend the most primitive concepts of all sciences, including mathematics of course, is in terms of these basic
intuitions. If so, then what purpose does it serve to split the formation into syntax and semantics? What is obviously true by what some call intuition is the working of the parallel-visual-spatial system of the brain. Rigor seems to consist of turning these truths into words. Either this is done so that we can then learn from this to solve those problems that cannot be visualized or it’s done to satisfy those that cannot visualize.
In no case is it necessary to stamp and certify only serial symbols as constituting rigor. If anything it is our very capability of spatial and parallel processing that even allowed us to entertain the possibility of a language for expressing truths. If anything language is a tool that allows us to partially reconstruct what we can see or seen to those that haven’t. If it were not so, only animals capable of speech (i.e. humans) would be capable of intelligence, and clearly it is not so.”
I would add that verbal_symbolic
AND visual_spatial skills complement each other into a valuable framework.
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The Real-line and non-local numbers
By using verbal_symbolic
AND visual_spatial skills we get a one comprehensive framework.
For example, by Traditional Mathematics (which is mostly based on verbal_symbolic skills) 0.111...
2 = 0.999...
10 =
1 where
1 is the considered mathematical object (the number itself) and 0.111...
2 or 0.999...
10 are some numerals (out of many representations) that represent number
1.
By using verbal_symbolic and visual_spatial skills as follows:
[qimg]http://farm7.static.flickr.com/6142/5962015728_d2fe37cc5f_z.jpg[/qimg]
one understands that no branch of that tree actually reaches any other branch of that tree "downward" , no matter how many levels
that tree has (in other words, there is no homeomorphism between 0 dimensional space (notated by "0";"1" symbols)
and 1 dimensional space (notated by "_____" spatial non-composed object)).
According to this comprehensive framework
0.111...2 is a number of its own < number
1 by
0.000...12 where the "
...1" part
of that number is the irreducibility of ___ 1 dimensional space into 0 dimensional space (known as a point).
By using verbal_symbolic
AND visual_spatial skills one enables to distinguish between non-local numbers like
0.111...2 or
0.000...12, and local numbers like
1 or
0.
Furthermore, no collection of, for example, 0 dimensional spaces along a 1 dimensional space has the power of the continuum of a 1 dimensional space.
By understanding the power of the continuum in terms of spatial skills, one understands that no collection of sub-objects (for example: "_ _ _" or ".....") of a given space (for example: "_____"), mathematical or physical, has the power of the continuum of that space, or in other words, any given collection of sub-objects is incomplete with respect to the "host" space.
The non-locality of
0.111...2 or
0.000...12 is "naturally vague" in terms of location, and one actually discovers/invents that the
Real-line has a non-empty set of non-local numbers between 0 dimensional space and 1 dimensional space.
(By generalization, given a "host" space, no collection of "hosted" spaces has the power of the "host" space).
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In other words, you can scratch each other backs as much as you like, it will not help you to actually use your verbal_symbolic
AND visual_spatial skills, since your mutual back's scratches are done only by verbal_symbolic skills.
