I still haven't received an explanation of what Doron's definition of "visual-spatial reasoning" is, and why it matters.
N = {1,2,3,...}
Code:
|{1,2,3,4,5,...}|
↓ ↕ ↕ ↕ ↕
|{1,2,3,4,...}|
is an example of visual_spatial reasoning that can be translated into verbal_symbolic reasoning |N| > |N| and vise versa.
As you can see, we are dealing with one and only one set, and by using also visual_spatial reasoning on that set, we are able to understand that since it has unbounded amount of members, this property prevents the accurate value of its size (unlike the case of sets with bounded amount of members, which their size is accurate).
The definition of visual_spatial AND verbal_symbolic reasoning, is given in
http://www.internationalskeptics.com/forums/showpost.php?p=9821061&postcount=3534.
You can't define visual_spatial AND verbal_symbolic reasoning by using only verbal_symbolic reasoning or only visual_spatial reasoning.
People like jsfisher wish to define visual_spatial AND verbal_symbolic reasoning only in terms of verbal_symbolic reasoning, and as a result they simply miss definitions of visual_spatial AND verbal_symbolic reasoning, as given, for example, in
http://www.internationalskeptics.com/forums/showpost.php?p=9821061&postcount=3534.
In
http://www.internationalskeptics.com/forums/showpost.php?p=9814916&postcount=3488 BenjaminTR thinks that we are unable to deal with the properties of one and only one set, exactly because he does not use visual_spatial AND verbal_symbolic reasoning in its abstract sense (
http://www.internationalskeptics.com/forums/showpost.php?p=9821061&postcount=3534 is an example of how to use visual_spatial AND verbal_symbolic reasoning in its abstract sense).
For example:
BenjaminTR said:
It has not split or reproduced itself into a domain and a separate codomain.
In this
abstract expression
Code:
|{1,2,3,4,5,...}|
↓ ↕ ↕ ↕ ↕
|{1,2,3,4,...}|
there is no separation between domain and codomain
exactly as there is no separation between domain and codomain in the following
abstract expression
Code:
|{1,2,3,4,...}|
↕ ↕ ↕ ↕
|{1,2,3,4,...}|
but BenjaminTR simply misses it in
http://www.internationalskeptics.com/forums/showpost.php?p=9814916&postcount=3488.
Again, the new thing here (that is not used by traditional mathematics) is a function without inverse, whether it is used on the same set, or not on the same set.