doronshadmi
Penultimate Amazing
- Joined
- Mar 15, 2008
- Messages
- 13,320
Let us explore the traditional point of view about numbers (the values) and numerals (their representations).
According to the traditional point of view, any given R member (some real-number) is a unique value along the real-line (and so is some complex number (C member) in the complex plane).
This unique value can be represented by a unique and single symbol (for example let us use "@" for the ratio 1/4), but it is not useful to give a unique symbol for each given number, so during the years we developed several methods that produce numerals (representations of values) by systematic methods (actually this is similar to the informal way, which uses a finite amount of symbols in order to represent words and sentences according to certain rules).
The most useful one is the place value method that systematically uses a finite amount of basic symbols (their quantity is changed by the base value, for example: base-two has two basic symbols "0","1". Base-three has three basic symbols "0","1","2". Base-n has n basic symbols, ... etc.)
It is well known that the number of symbols that represent some value (some R member, in this case) can be changed if we use different bases to represent this value, and it does not matter if this value is a fraction or a whole number.
For example: in order to represent number five in base ten we need a single symbol, which is "5", but in order to represent number five in base three we need at least three symbols "101".
So, according to the traditional point of view "five", "5" or "101" are no more than different representations of the same value (where the value is the number itself).
Furthermore, it does not matter if we need infinitely many symbols or finitely many symbols in order to represent some value (some number) for example:
0.7[base 10] = 0.10110011001100110011001100110011001100110011... [base 2]
1 = 0.111...[base 2] = 0.222...[base 3] = ... = 0.999...[base 10] = ...
This is the standard mathematical framework.
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My mathematical framework is a complementation between VSL and ASL, where Mathematics of the past 2500 years is mostly based on ASL.
Here is a comparison between ASL and VSL, based on Dr. Linda Kreger Silverman's research (please look at the difference between what is called Visual Special Learning and Auditory Sequential Learning ( http://www.visualspatial.org/Articles/intro.pdf , http://www.visualspatial.org/Articles/idvsls.pdf ). ):
Things are changed if Symmetry is used as the universal principle of logic.
Let a 2-valued framework be represented by A B.
A B relation is:
A B
A A
A B
B A
B B
These relations can be reduced (without a loss of generality) to
symmetric (AA or BB) \ asymmetric (AB or BA) relations, represented as:
AB
XX is AB symmetry --> A=B (AB is the same)
XY is AB asymmetry --> A≠B (AB is not the same)
A=B can be reduced (without a loss of generality) to a single value X.
A≠B cannot be reduced to a single value without a loss of detail X or Y.
Furthermore, in order to conclude that A≠B, they must share the same realm.
So 2-valued framework is at least X (symmetry or sameness) \ XY (asymmetry or difference) relation.
Let us use SA (Symmetry\Asymmetry relation) on 2-valued Logic:
T F NXOR
F F considered
F T not considered
T F not considered
T T considered
T F XOR
F F not considered
F T considered
T F considered
T T not considered
T F NXOR\XOR
F F considered
F T considered
T F considered
T T considered
T F generalization of NXOR\XOR
X X considered
X Y considered
Let us examine 3-valued logic:
Both cases are reduced to SA (Symmetry\Asymmetry relation) without a loss of generality:
3-valued logic can be extended to any x-valued logic where x is any standard or non-standard value of [0,1]. By using Symmetry\Asymmetry relation, we define the universal principle of any given logical system, standard or non-standard.
By Symmetry, Symmetry is Asymmetry.
By Asymmetry, Asymmetry is_not Symmetry.
If there is no common basis, then anything is totally isolated, and nothing can be compared in order to define some difference. If there is only a common thing, then anything is totally connected, and nothing can be found beyond total unity. In both cases we do not get the researchable. So, the researchable is at least a unified isolation. Let us represent this notion by using an ASL\VSL representation method.
Let "=" be is
Let "≠" be is_not
Let "S" be Symmetry
Let "A" be Asymmetry
S=A from S point of view.
A≠S from A point of view, but in order to compare and conclude that A≠S, there must be a common basis that enables the comparison. So, the researchable is at least a unified isolation, represented as:
A particular value is defined only by NXOR(unification)\XOR(isolation) relation.
Since any particular value is at least NXOR\XOR relation, then any number is a particular path along the symmetric ur-element real-line.
For example:
In this case 0.01[base 2] is a particular path and 0.25[base 2] is another particular path.
By the traditional point of view a particular value (a number) is only the particular case of zenith-path along the symmetric ur-element real-line.
Because of this arbitrary limitation the non-zenith path 0.111...[base 2] is equal to the non-zenith path
0.222...[base 3] and both of them are defined as some representations of the zenith-path 1.000...
But when path is a general concept, then 0.111...[base 2] < 0.222...[base 3] < 1.000... as can be clearly seen in:
According to the traditional point of view, any given R member (some real-number) is a unique value along the real-line (and so is some complex number (C member) in the complex plane).
This unique value can be represented by a unique and single symbol (for example let us use "@" for the ratio 1/4), but it is not useful to give a unique symbol for each given number, so during the years we developed several methods that produce numerals (representations of values) by systematic methods (actually this is similar to the informal way, which uses a finite amount of symbols in order to represent words and sentences according to certain rules).
The most useful one is the place value method that systematically uses a finite amount of basic symbols (their quantity is changed by the base value, for example: base-two has two basic symbols "0","1". Base-three has three basic symbols "0","1","2". Base-n has n basic symbols, ... etc.)
It is well known that the number of symbols that represent some value (some R member, in this case) can be changed if we use different bases to represent this value, and it does not matter if this value is a fraction or a whole number.
For example: in order to represent number five in base ten we need a single symbol, which is "5", but in order to represent number five in base three we need at least three symbols "101".
So, according to the traditional point of view "five", "5" or "101" are no more than different representations of the same value (where the value is the number itself).
Furthermore, it does not matter if we need infinitely many symbols or finitely many symbols in order to represent some value (some number) for example:
0.7[base 10] = 0.10110011001100110011001100110011001100110011... [base 2]
1 = 0.111...[base 2] = 0.222...[base 3] = ... = 0.999...[base 10] = ...
This is the standard mathematical framework.
-----------------------------------------------------------------------
My mathematical framework is a complementation between VSL and ASL, where Mathematics of the past 2500 years is mostly based on ASL.
Here is a comparison between ASL and VSL, based on Dr. Linda Kreger Silverman's research (please look at the difference between what is called Visual Special Learning and Auditory Sequential Learning ( http://www.visualspatial.org/Articles/intro.pdf , http://www.visualspatial.org/Articles/idvsls.pdf ). ):
Things are changed if Symmetry is used as the universal principle of logic.
Let a 2-valued framework be represented by A B.
A B relation is:
A B
A A
A B
B A
B B
These relations can be reduced (without a loss of generality) to
symmetric (AA or BB) \ asymmetric (AB or BA) relations, represented as:
AB
XX is AB symmetry --> A=B (AB is the same)
XY is AB asymmetry --> A≠B (AB is not the same)
A=B can be reduced (without a loss of generality) to a single value X.
A≠B cannot be reduced to a single value without a loss of detail X or Y.
Furthermore, in order to conclude that A≠B, they must share the same realm.
So 2-valued framework is at least X (symmetry or sameness) \ XY (asymmetry or difference) relation.
Let us use SA (Symmetry\Asymmetry relation) on 2-valued Logic:
T F NXOR
F F considered
F T not considered
T F not considered
T T considered
T F XOR
F F not considered
F T considered
T F considered
T T not considered
T F NXOR\XOR
F F considered
F T considered
T F considered
T T considered
T F generalization of NXOR\XOR
X X considered
X Y considered
Let us examine 3-valued logic:
Both cases are reduced to SA (Symmetry\Asymmetry relation) without a loss of generality:
Code:
A B C
X X X --> X (symmetry)
X X Y --> X Y
X Y Z --> X Y Z (asymmetry)3-valued logic can be extended to any x-valued logic where x is any standard or non-standard value of [0,1]. By using Symmetry\Asymmetry relation, we define the universal principle of any given logical system, standard or non-standard.
By Symmetry, Symmetry is Asymmetry.
By Asymmetry, Asymmetry is_not Symmetry.
If there is no common basis, then anything is totally isolated, and nothing can be compared in order to define some difference. If there is only a common thing, then anything is totally connected, and nothing can be found beyond total unity. In both cases we do not get the researchable. So, the researchable is at least a unified isolation. Let us represent this notion by using an ASL\VSL representation method.
Let "=" be is
Let "≠" be is_not
Let "S" be Symmetry
Let "A" be Asymmetry
S=A from S point of view.
A≠S from A point of view, but in order to compare and conclude that A≠S, there must be a common basis that enables the comparison. So, the researchable is at least a unified isolation, represented as:
A particular value is defined only by NXOR(unification)\XOR(isolation) relation.
Since any particular value is at least NXOR\XOR relation, then any number is a particular path along the symmetric ur-element real-line.
For example:
In this case 0.01[base 2] is a particular path and 0.25[base 2] is another particular path.
By the traditional point of view a particular value (a number) is only the particular case of zenith-path along the symmetric ur-element real-line.
Because of this arbitrary limitation the non-zenith path 0.111...[base 2] is equal to the non-zenith path
0.222...[base 3] and both of them are defined as some representations of the zenith-path 1.000...
But when path is a general concept, then 0.111...[base 2] < 0.222...[base 3] < 1.000... as can be clearly seen in:
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