ETT-R&D Publications E. Terrell
IT Professional, Author / Researcher February 2002
Internet Draft
Category: Proposed Standard
Document: draft-terrell-math-quant-new-para-redefi-bin-math-03.txt
Expires August 22, 2002
The Mathematics of Quantification, and the New Paradigm,
which Re-Defines Binary Mathematics
Status of this Memo
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E Terrell [Page 1]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
TABLE OF CONTENTS
Contents
Introduction: The Discourse, which Quells the Arguments in Opposition
Chapter I: Another look at the New Binary Paradigm
Chapter II: Developing the Mathematical Foundation for Arithmetic
Operations
Chapter III: The Mathematics of Quantification; Spectacles for Viewing
the Mathematical Possibilities
Chapter IV: Security Considerations
Note: The '^' sign is the Mathematical Symbol used to represent the
Exponential Operation. Where '2^2 = 4', is the same equation
represented by '2 * 2 = 4', which is the Multiplicative
equivalent.
E Terrell [Page 2]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Abstract
This paper provides the Mathematics for the New Paradigm Defining
the Binary System. Furthermore, while the Mathematical foundation
and Logical justification, which established the New Structure for
the BINARY SYSTEM, were derived from The Mathematics of
Quantification. The Mathematics itself, which is used in the New
Binary System however, while providing the viable justification and
the logical reasons that supports the change for the New Binary Model,
is not quite so new. In fact, it can be said that the Mathematics of
Quantification sustains a Cascading Effect, Producing a Profound Change
in the Mathematics for the Entire Mathematical Field. But, the Mathematics
for the New Binary System has a Historical Foundation, which dates to the
beginnings of Mathematics itself.
"This work is Dedicated to my first and only child, 'Yahnay', who is;
the Mover of Dreams, the Maker of Reality, and the 'Princess of the
New Universe'. (E.T.)"
E Terrell [Page 3]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Introduction: The Discourse, which Quells the Arguments in Opposition
It is said: "Arrogance is the Defense using Words, A Pretense, which
is the True Face of Ignorance, Hiding Behind the Mask of Intellectual
Deception."
Whatever the case may, or may not be, I truly attempted without any
doubts, to contact the entire World, and present to everyone, the Gift
from the Beginnings of the Mathematics of Quantification. However,
only one person responded, this time, and their presentation was an
opposition, one that bespeaks of Arrogance...not the anticipated
response from a professional Mathematician or Logician:
"Dear Mr. Terrell,
You are, as anybody else, free to prefer a nonstandard interpretation
(or, rather, enumeration) of the binary system; there is no "true
interpretation", and the ways to map integers to binary numbers is
uncountable (as Cantor proved).
Nonetheless, the standard interpretation which you have chosen to attack
is distinguished by one property which no other enumeration has: a
simple arithmetic well-suited for the computers of our age. Addition,
for example, can in the binary number system simply done as in the
decimal system, except of course, that adding 1 to 1 yields 10, at any
particular place. If you now take two numbers, say 9 and 5, translate
them to their binary representations, and add them according to the rule
mentioned:
00001001 <- 9
00000101 <- 5
++++++++======
00001110 -> 14
and retranslate into the decimal system, you get 14. That means, addition
in the binary system and in the decimal system are _isomorphic_, the same
easy operation yields the same (correct) result in both number systems.
E Terrell [Page 4]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
This is, in short, the reason why the standard interpretation of binary
numbers is the one which computer scientists prefer, as it is easy to
implement in electronic devices and hence forms the basis for modern-day
computer chips.
Your interpretation of the binary numbers, to the contrary, does not have
an arithmetic which is simple, as the zero digit can not function as
neutral element anymore. It is therefore much clumsier to deal with.
Mathematicians do not accept claims at truth of any possible,
non-selfcontradictory (= consistent) mathematical system. The times when
mathematicians were thinking that their axiomatic systems, such as Euclid's
axiomatics of geometry, were obvious truths and the only possible systems,
they went away with the discovery of the consistency of non-Euclidean
geometries in the early nineteenth century. Later on, logicians proved that
mathematical truth is indeed equivalent to mathematical consistency.
To claim that there is a logical fault with the standard binary number
system, you would have to derive a contradiction. This would have the
interesting side effect of destroying the whole of current mathematics and
rendering current computers unusable. I believe that you are right in your
IETF draft which just expired, insofar as "no one has, or is capable" of
deriving such a contradiction. That you make an exception for yourself, is,
in my humble opinion, a sad indication of severe megalomania. I can only
wish you to be healed of it and be able to spare your limited energies for
endeavors not so futile as this one, though my experience with cases such
as yours leaves me with little hope.
Sincerely yours,
Aleksandar Perovic
Chief Executive Administrator
The Electronic Library of Mathematics"
E Terrell [Page 5]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
My work, as a Scientist and a Researcher, speaks for itself, and my
accomplishments ascribes the definition of me and my abilities, which
defies the boundaries imposed by the definitions of the words used in
the many languages denoting MankindÆs diversity. It is sad though, because
I am spending a great deal of time, clarifying Elementary Concepts, once
thought to be Well Understood by the Professionals who populate the Field
of Study for which this Draft represents. And while, I advocate the
necessity regarding the priority for Studying the Historical Documents
comprising the intended Area of Research, prior to any Research
Undertaking. It should be understood however, my advocacy sustains a
Revolution against Dogma, and supports the belief that; 'Regardless of
the Epitome granted by the Historical Documentation, to any individual,
belief or acceptance of their work remains a challenge, which is reserved
for continued Analysis, and the reflection upon the Classical Foundation
from which the Laws, Rules, and Logic that support their work, were
derived.' Needless to say, since Mankind is Not GOD, I stand Poised in
the Ready, and will challenge his perception or interpretation for Reality,
regardless of the underlining subject matter, or the intent his
presentation is said to represent.
Notwithstanding my personal beliefs however, we can make use of the
limited argument provided by 'Mr. Perovic', and derive not only the
supporting Mathematics for the New Binary System, but provide the
"...contradiction", which he claims is necessary to prove that the
Modern Interpretation of the Method for Enumerating in the Binary
System is wrong. Furthermore, what's nice about speaking with Mr. Perovic,
is that, he reveals the Contradiction, unknowing to himself, that we
need, as the focus for this argument, when he said:
"Nonetheless, the standard interpretation which you have chosen to attack
is distinguished by one property which no other enumeration has: a
simple arithmetic well-suited for the computers of our age. Addition,
for example, can in the binary number system simply done as in the
decimal system, except of course, that adding 1 to 1 yields 10, at any
particular place."
E Terrell [Page 6]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Can you see the Foundation, which would allow the presentation of the
Contradiction? In other words, you can not perform the operation of
addition on the equation "1 + 1", because this would equate to "10". But,
isn't this a Numbering System that is Governed by the Elementary Laws of
Mathematics and Logical reasoning, which must ultimately obey the Laws
from the Field Postulates and Set Theory? Furthermore, when dealing with
the Binary System, should it be considered to be governed by slightly
different Arithmetic Operations, and have different Logical consistency
from that of the Unary System? And what about the overall Arithmetic
Operations pertaining to Mathematics itself, isn't this wrong there too?
Well...If it is, then what was Gregor Cantor actually saying? Perhaps,
what he was actually saying, was that; 'If you are wrong, and you are
consistently wrong in what you are saying or doing, then you can make it
look correct, because it is Consistent.' Nevertheless, in any case, the
Argument has been made, and a gradual development of the foundation
supporting the New Paradigm for the Binary Mathematics will be set forth
in the succeeding chapters.
Chapter I: Another look at the New Binary Paradigm
To establish the foundation, which would ultimately lead to the Final
conclusion supporting the New Paradigm for the Binary System, and the
"Contradiction", that would provide the necessary proof that the Modern
Foundation is wrong. I must first provide a Table(s) Listing the related
Numbering Systems, for comparison, and then reiterate parts of the Proof,
which would allow the derivation of the New Paradigm for the Binary System.
Where by, notice the Columns in Table 1A, each is a Representation of the
same object, or each other, differing only in their Graphical Depiction:
E Terrell [Page 7]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
TABLE 1A
1 2 3 4
Modern New Modern Primitive
Binary Binary Positive Unary
System System Integers System
00 0 0 0
01 00 1 1
10 01 2 11
11 10 3 111
100 11 4 1111
101 100 5 11111
110 101 6 111111
111 110 7 1111111
1000 111 8 11111111
1001 1000 9 111111111
1010 1001 10 1111111111
1011 1010 11 11111111111
1100 1011 12 111111111111
1101 1100 13 1111111111111
1110 1101 14 11111111111111
1111 1110 15 111111111111111
10000 1111 16 1111111111111111
The examination of TABLE 1A, coupled with an understanding of the
Elementary Operations for Addition in Binary Mathematics, the Laws from
the Field Postulates, and Set Theory. Where it can be Clearly seen, that
the Operation of Addition in the equation "1 + 1 = 10" is the
"Contradiction", which is Not Violated Under the New Paradigm for the
Binary System. Furthermore, I can also say, from its presentation, the
Relationship between Columns '2' and '4' has been established as being
Logically valid under the Rules and Laws, which govern the Field Postulates
and Set Theory. And further state, it is also valid under the laws
governing the Mathematics of Quantification. However, its proof, would be
too taxing of a demand, which would require the knowledge of the
Mathematics of Quantification. And in this case, it is totally unnecessary,
because the Laws from Elementary Mathematics already has been shown to
suffice for the establishment of the so called, "Proof by Contradiction"
Argument, required by 'Mr. Perovic' response to the initial proof of the
foundation, which established this New Paradigm for the Binary System.
E Terrell [Page 8]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
In other words, 'Mr. Perovic' stated that the flaw in the Modern Method for
Enumerating using Binary Notation, resulted from an Exception to the
Mathematical Law Governing the Operation of Addition. That is, he stated;
"...except of course, that adding 1 to 1 yields 10", which should be the
Binary Notation that represents, or equals the Integer '3', provided at
least one of the addends was a Binary Number. Furthermore, while the
Argument can easily be closed, just from this little example, and of
course, a comparison between Columns '2' and '4' from Table 1A, that would
clearly establish the Method for Elementary Arithmetic Operations for this
New Binary System...Still many would complain, regarding the missing rigor
from the Logical Argument, which would unquestionably rule out any further
opposition.
Nevertheless, prior to beginning the development of the foundation, which
would allow for the derivation of the Methods for the Elementary Arithmetic
Operations, I must first reiterate the conclusions supporting the proof
that established the Foundation for the New Model representing the Binary
System.
"...However, prior to any forthright Construction of Table Ic, following
in sequence from Tables I, Ia, and Ib. It would facilitate the analysis of
the logical argument, if we first reiterate the requirements that were
logically developed, that established the foundational definitions and
requirements, which would be the mandate for any Binary System to exist.
Binary Principles
1. Binary; Consisting of 2 Things, Elements, or Members.
2. Zero and the Null Set are implied by the same definition
3. Zero; Having no Quantity, Size, Members, or elements;
representing a State of Condition of Nothingness.
4. Binary Set; Consisting of 2 and only 2, Elements or Members.
5. Union of Set; Combining the Elements or Members of 2 or more
Sets, resulting in 1 Set containing the total, which represents
the combined total of the Members from the initial Sets.
6. 'Equality': A Relationship, which provides a means to establish
an Identity between 2 or more Objects being compared.
7. Binary Zero is represented by '00', since it is not empty, it
is not equal to either the Zero Integer or the Null Set.
E Terrell [Page 9]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Now if you are satisfied with the list of Principles derived from, and
associated with the Binary System, with the exception of 7. We can
construct Table Ic, which represents another view for the Modern Method
of Binary Enumeration.
TABLE Ic
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only "1's" and "0's"
Depicting the Results from its current Presentation
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
1. 0^0 = 0 00000000 = 0 0
2. 2^0 = 1 00000001 = 01 1
3. 2^1 = 2 00000010 = 10 2
4. 2^F = 3 00000011 = 11 3
5. 2^2 = 4 00000100 = 100 4
6. 2^F = 5 00000101 = 101 5
7. 2^F = 6 00000110 = 110 6
Notice that Table Ic maintains the 'One-to-One' validity as Table IIa.
However, as with Tables I and II, their differences remain the same. In
fact, any comparison with Table IIa maintains the same validity, except
for their different starting points. In other words, Table Ic and Table
IIa are 2 distinct Numbering Systems, that use the Binary Notation in a
'One-to-One Pairing' with the Integers to define and establish equality.
E Terrell [Page 10]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
"Do we now have 2 Binary Systems, establishing a slightly different, and
yet, equal relationship with the Set of Integers? I mean, what do we have
here? Is it possible to have 2 distinct Binary Systems, whose difference
represents a different 'One-to-One Pairing' with the Integers? Or are we
to try once again, and decide, which one of the two Numbering Systems
actually represents a True Binary System?"
TABLE IIa
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
1. 0^0 = 0 0 0
2. 2^0 = 1 00000000 = 00 1
3. 2^1 = 2 00000001 = 01 2
4. 2^F = 3 00000010 = 10 3
5. 2^2 = 4 00000011 = 11 4
6. 2^F = 5 00000100 = 100 5
7. 2^F = 6 00000101 = 101 6
Following the same investigative analysis used in earlier chapters, we can
depict this difference graphically. That is, if we were now to extrapolate
from the respective Binary Notations, as it would be given by the Integers'
additive method of progression, which produces the counting series using
successive additions of 1. We could then generate a number line, denoting
a 'One-to-One Mapping' with the Integers that would more accurately depict
these noted distinctions. Given respectively by figures 3 and 4, we have:
E Terrell [Page 11]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Fig 3.
1 2 3 4 = The Count of Total Number
-+-+-+-+ of Members in the Set
0 1 2 3 = The Elements or Members
Listed in Table Ic's Binary Set
Fig 4.
1 2 3 4 = The Count of Total Number
-+-+-+-+- of Members in the Set
1 2 3 4 = The Elements or Members
Listed in Table IIa's Binary Set
What the bottom row of numbers actually represents, is the total number of
combinations, which will be generated from the Binary Set, {0,1}. However,
these combinations are used in a way similar to the way the '1' is used in
the Integers, which increments from right to left using and changing only
the ' 0 or 1' notations from the Binary Set to generate a series of Binary
Numbers. In other words, they generate a series governed by the operation
of addition. That is, given respectively by figures 5 and 6, we have:
Fig 5.
{01}, {10}, {11}
2 3 4
Fig 6.
{00}, {01}, {10}, {11}
1 2 3 4
E Terrell [Page 12]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Well, how do you begin your count? I mean, if there are 5 objects to be
counted, would your count start with 'Zero' or 'One'? Clearly, the Set of
Integers from which the Counting Numbers were derived, was only a graphical
depiction, to be used in such a way, as to render a picture of the Number
to be represented, which used one or more of these members to achieve the
desired result. And nothing more. In other words, the Set of Integers or
Whole Numbers, maintains the additional distinction of being a short-hand
representation for the Operation of Addition, from which the sequence of
Numbers is derived from the Unary Set {1}.
Furthermore, I am sure you observed from figure 5, that the equating of
Binary Zero to the Integer Zero reduced the number of combinations
resulting from the Binary Set. Which is actually the cause which produces
the SHIFT in the 'One-to-One Pairing' with the Integers. I mean, the
assignment of the Beginning Point for any Numbering Systems is very
important, because it sets the starting point that will be used for
counting.
Moreover, further analysis of the resulting Combinations derived from both
of the respective Binary Sets, using Tables Ic and IIa. Clearly shows the
equality existing between each of these Sets, which is derived from the
'One-to-One Pairing' equating the Points on the Number Line, denoting the
Integers, with the Binary Notations they respectively represent. If
however, we mapped the results indicated by figures 5 and 6, using the
respective mappings given by figures 3 and 4, we would establish the
necessary proof for concluding, that the method derived for Counting using
the Modern Interpretation is wrong. In other words, any 'One-to-One
Mapping' with the Integers and the Combinations resulting from figures 5
and 6, would clearly show that the missing Set, given by the Combination
{00}, would result in a inaccurate mapping denoting an Inequality with
the Sequence of Counting Numbers derived from the Set of Integers; that is,
the Set of Counting Numbers denoted by: {1,2,3,4,5,6,7,8,9,10}. In which
case, the Universal Set " I ", for the Integers, would equal the Set
denoted by:
Fig 7.
x|x is an element of I = Integers
{ {...-10,...-5,-4,-3,-2,-1} {0} {1,2,3,4,5,...,10} }
Where its number line mapping is given by:
E Terrell [Page 13]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Fig 8.
-10 + -9 ... -5 +... -2 + -1 + 0 + 1 + 2 + 3 ... 5 +... + 10
Nevertheless, the System of counting presently being used is a UNARY
System, from which the sequence of Counting begins with the Number '1',
and continues its progression using successive additions of the Number
'1' to derive the next or succeeding numbers. And while it maybe called
or labeled as being something different (i.e. Decimal System), it is
nevertheless Unary. Furthermore, while Zero, '0', is used in every
Numbering System (denoting its' universal application), it is not itself,
a Number. It is only a symbolic notation, which represents emptiness, or
lack of an Object to which it refers. Hence, Binary by definition, means
'2', and nothing more. Therefore, when considering the construction of any
Numbering System that employs or uses Binary Notation, we must first
realize that the first '4' numbers are derived from the Total Number of
Possible Unique Combinations, which are related to and derived from, the
Sequenced Numbers or Elements depicted as being Members of the Binary Set.
And further conclude, that all other succeeding Binary Numbers are derived
from these Combinations. In which case, since the Binary Set equals {0,1},
the total number of Unique Combinations equals the set {00, 01, 10, 11},
which respectively represents the first '4' Binary Numbers whose mapping
with the Set of Integers starts with the Number '1'.
Hence, the Correct Method for Enumeration in the Binary System is given
by the Results displayed in Table IIa, and the Modern Interpretation for
the Method of Enumeration in the Binary System is clearly wrong. But still,
both methods clearly represent a Binary System. Notwithstanding however,
while the conclusions derived with respect to each of these Systems remains
unquestionably valid. It does not stop, nor prevent any decision regarding
choice. In other words, for whatever reason, right or wrong, for now at
least, it does not matter which Binary System is used. Because other than
myself, no one has, or is capable of completing the necessary studies
indicating some out come producing a harm, resulting from the effects for
choosing the wrong System."
E Terrell [Page 14]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Chapter II: Developing the Mathematical Foundation for Arithmetic Operations
First and foremost, it should be pointed out, that while the Numbers in
Binary Notation, as represented in Column '2', from Table 1A, are
derived from the total number of Unique Combinations, which equals the set
{00, 01, 10, 11}, and that they respectively represent the first '4'
Binary Numbers whose mapping with the Set of Integers starts with the
Number '1'. However, any further comparison of Columns '2' and '4' also
reveals, that they are 'Incremented' or 'De-Incremented' using the same
methods as those governing the Unary Set. That is, while the sequence of
Counting does not begin with the Number '1', as such. It uses Number '1' to
derive a progression, which uses successive additions of the Number '1' to
derive the next, and the succeeding numbers in Binary Notation. What this
actually means, or implies, is that, by definition, there can exist only
'4' Numbers, which can be derived from the, and said to members of, the
'BINARY SET'. Everything else is a Synthetic Creation, which facilitates
enumeration beyond a count of '4'. In which case, the 'Unary Set' contains
only '1' Member, and all other numerals results from some combination,
which builds upon, and are related to, the number '1'.
Furthermore, while this process is clearly depicted in Table 1A, any
questions concerning the validity of such an Operation are easily
quelled using the 'Axioms for Equality', which are derived from the Laws
governing the Basic Arithmetic Operations of Elementary Mathematics. And
in this particular case, the Elementary Mathematical Law of Governance, is
the 'Substitution Law for Equality, which states; "If A = B, then A may be
replaced by B, and B by A, in any Mathematical Statement without altering
the Truth or Falsity of the statement." What this means, and is represented
in Table 1A, is that, since {00} = {1}, then {00} may be replaced by {1},
and {1} by {00}, in any Mathematical Statement without changing or altering
the value of the Mathematical Statement itself.
Nevertheless, I will not extend the argument beyond the Elementary
Operations, which deal specifically with Addition and Subtraction, because
these operations completely suffice in not only establishing the necessary
proof, but clearly represents the ease and elegance of the Mathematical
Operations, which represents the New Paradigm for the Binary Set. Not to
mention, that it would be redundant to proceed any further, because the
Modern Interpretation for Representing the Operation of Addition, in the
Current Binary Set Notation, Fails the TEST, when one attempts to solve the
Equation "1 + 1 = 10"... Which is valid enough, to establish the necessary
proof, especially since it does not yield an equivalent integer
representation. In other words, it does not represent the integer '3' from
a Binary Translation, and serves only to raise more questions regarding our
present mathematical and logical concerns.
E Terrell [Page 15]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Nonetheless, if you are satisfied, and I sincerely hope that you are, we
can, by example and comparison using Table 1A, show examples of Addition
and Subtraction using the New Paradigm, which represents the Real Binary
System.
Please note, when observing Table 1A, specifically Column '2', you should
notice that the Progression beyond the Number represented by '00',
'Increments' the next Number by the same amount shown in Column '4',
which represent the Number, or Integer, '1' under Column '3'. Where by,
the Operation of Addition is given in Table 2A, and the Operation of
Subtraction is shown in Table 3A:
Table 2A
Binary Addition Integer Addition Integer Equivalent
1. 00 + 1 = 01 1 + 1 = 2 2
2. 01 + 1 = 10 2 + 1 = 3 3
3. 10 + 1 = 11 3 + 1 = 4 4
4. 11 + 1 = 100 4 + 1 = 5 5
5. 100 + 1 = 101 5 + 1 = 6 6
6. 101 + 1 = 110 6 + 1 = 7 7
7. 110 + 1 = 111 7 + 1 = 8 8
8. 111 + 1 = 1000 8 + 1 = 9 9
E Terrell [Page 16]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Table 3A
Binary Subtraction Integer Subtraction Integer Equivalent
1. 00 - 1 = 0 1 - 1 = 0 0
2. 01 - 1 = 00 2 - 1 = 1 1
3. 10 - 1 = 01 3 - 1 = 2 2
4. 11 - 1 = 10 4 - 1 = 3 3
5. 100 - 1 = 11 5 - 1 = 4 4
6. 101 - 1 = 100 6 - 1 = 5 5
7. 110 - 1 = 101 7 - 1 = 6 6
8. 111 - 1 = 110 8 - 1 = 7 7
Clearly, Tables 2A and 3A provides an adequate representation for the
Elementary Mathematical Operations of Addition and Subtraction, which can
be easily verified using Table 1A, and hence, quells all further doubts
about the Logic, and or Mathematical Operations that encompass the New
Paradigm representing the Binary System. Furthermore, it can be easily
shown, that the even more Complicated Mathematical Operations representing
Multiplication and Division would follow the similar presentation. In other
words, the conclusion representing the foundation, which Established this
New Paradigm for the Binary System, remain unquestionably valid. And
without a doubt, Gregor Cantor was truly wrong, regarding his conclusions.
That is, this New Paradigm represents the True Binary Mathematical
Operations... Where by, in the New Binary Mathematics, the Mathematics for
the Binary Numbers and the Binary Logic is the same; Given by Equations '1'
thru '5', noted below...We have:
E Terrell [Page 17]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
1. 1 + 1 = 10 : In the New Paradigm for the Binary
System, this Equals "00 + 00 = 01",
and "01 + 00 = 10".
2. 00 + 00 = 01 : In the New Binary Mathematics
3. 00 + 01 = 10 : In the New Binary Mathematics
4. 01 + 01 = 11 : In the New Binary Mathematics
5. 10 + 00 = 11 : In the New Binary Mathematics
But, this pattern only follows the Unary Set for Progression, or
Regression, which pertains to the value given by the Unary Set, {1}.
Nevertheless, there is, contrary to the out spoken beliefs, a Binary
Equivalent, which is performed first upon the Right Most Binary Pair;
where {XX} would represent the Right most Binary Digit. Now! Keeping in
mind that this is Pure Binary Mathematics that we will be dealing with. It
should be understood, its' Rules will be somewhat different. Where by, in
Pure Binary Mathematics, whether or not you are working with a Pair of
Columns or a Single Column, something is always Carried to the Next Column,
(or is understood to represent a particular Binary Value) provided that
the Next Column Exist. In other words, in Pure Binary Mathematics, either
a "1" or a "0" will Carry Over to the Next Column. And depending upon the
Binary Value of the Digit in the Next Column, being either a "0" or a "1".
And whether or not you are working with either a Single, Double, or some
Multiple Column Arithmetic, will determine how the Carry will effect the
Mathematics. To be more specific, the Digit being Carried is Governed by
the equations given below (And Note, I will only be performing Single
Column Mathematics);
1. 0 + 1 = 10, where "1" Carry to "0" means use
"0" in the Current Column and
Carry the "1" to the Next Digit.
E Terrell [Page 18]
Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
2. 1 + 1 = 11, where "1" Carry to "1" means use
"1" in the Current Column and
Carry the "1" to the Next Digit.
3. 0 + 0 = 1, where "0" Carry to "0" means use
"0", (0 + 0 = 1) in the Current
Column and Carry the "0" to the
Next Digit; In which case, the
Carry of "0" to "1" equals "10",
and Carry "0" to "0" Equals "0";
given by Table 1A, we have
"00" = "1".
The explanation for these results is given by the Results from the
equations given below, and are respectively labeled as '1a' and '2a'.
1a. 00 + 01 = 10
2a. 01 + 01 = 11
3a. 00 + 00 = 01
4a. 10 + 00 = 11
Now Observe Equations '1a', '2a', "3a' and '4a', when the Right most
Digit is Stripped away, which yields Equations '1b', '2b', '3b', and
'4b', and stripping the Left Most Digit yields equations '1c, '2c,
'3c', and '4c'. These Equations are said to be the Equations
establishing the fundamental Mathematical Operations for Binary Logic,
which would represent the "AND OPERATION"; Given by Table 1B, We have:
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Table 1B
1b. 0 + 0 = 1 1c. 0 + 1 = 0
2b. 0 + 0 = 1 2c. 1 + 1 = 1
3b. 0 + 0 = 0 3c. 0 + 0 = 1
4b. 1 + 0 = 1 4c. 0 + 0 = 1
And these respective Arithmetic examples are representations of the "AND"
Function, the "NOT" function can just as easily be deduced using the same
methods. Nevertheless, the Mathematical Calculations involving the Binary
Numbers, in which the Operation of Addition is performed, is given by
Table 'Ex. 1a': we have:
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Table Ex. 1a (ADDITION)
111 = 8 1111 = 16 11111 = 32
110 = 7 1010 = 11 10110 = 23
____ 15 _____ 27 ______ 55
1110 11010 110110
111111 = 64 100 = 5 1000 = 9
101011 = 44 100 = 5 1000 = 9
______ 108 ____ 10 _____ 18
1101011 1001 10001
10010 = 19 11011 = 28
10010 = 19 11011 = 28
______ 38 ______ 56
100101 110111
Furthermore, it should be understood that the Arithmetic Operation of
Subtraction follows the same Rules Derived for Addition, but Effect is
the Reverse, which yields an Opposite result. Where by, Given by Table
Ex. 2a, we have:
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Table Ex. 2a (SUBTRACTION)
111 = 8 1111 = 16 11111 = 32
110 = 7 1010 = 11 10110 = 23
____ _____ 5 ______ 9
00 = 1 100 1000
111111 = 64 100 = 5 1000 = 9
101011 = 44 100 = 5 1000 = 9
______ 20 ____ 0 _____ 0
10011 0 0
10010 = 19 11011 = 28
10010 = 19 11011 = 28
______ 0 ______ 0
0 0
Note: It should be understood, that when dealing with Subtraction,
'11 - 10 = 00' and '11000 - 10000 = 111', which follows the
Rules provided above.
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
Needless to say, the "CONTRADICTION" now becomes the "CONFLICT",
which is the Difference between the Mathematics pertaining to the Binary
System itself, and the Mathematics for the Binary Logic associated with the
Binary System. In other words, there is No Such Thing as a Derivation of a
"Contradiction", 'Proof or Otherwise', within a Newly Created, or Logically
Derived Numbering System. Because it can only be said to either violate the
Standing Laws which Support it, or it Violates the Newly Derived
Definitions, which are said to Define it. And in this case, the proof is
derived from the conclusion; 'There is No Correlation between the total
number of Unique Combinations which equals or depicts the Numerals
Contained in the Modern Binary Set ({00, 01, 10, 11}), that is used to
develop the Mathematics for, and is derived from, Binary Enumeration, which
was logically derived from the Unary Set'. Hence, Zero once again, regains
its Independence, the inherent Neutrality, which is the Property or Status
belonging only to Zero; æThe Distinction of the Zero Property regarding
itÆs inherent Neutrality, by definition, sets it apart from every Numbering
System, or System of CountingÆ.
Nevertheless, this "Contradiction" with the Mathematics for the Modern
Binary System, is a "CONFLICT" within the Binary System itself, which does
not exist in the New Paradigm, that represents the New Model for the Binary
System. Hence, there is Only One Logically Valid Binary System, and while
anyone can create up to '4' New Binary System Representations, they would
not All be Logically Valid. And Equally True, there is Only One Unary
System, but it can not be Extended in any way, that would provide, or
produce some other Alternatives, as seen in the Binary System.
Chapter III: The Mathematics of Quantification; Spectacles for Viewing
the Mathematical Possibilities
Nevertheless, whether or not you are familiar with Quantification, it
should be clear, since its mention, The power of the Mathematics of
Quantification is indeed daunting, and it should reign over the Entire
Mathematical Field forever, without question. In fact, I am currently
working on more of its promises, which includes the Subjects listed below.
Moreover, it should be an added value to note, accomplishments in these
areas would lead to 'Autonomous Machines', which could actually 'Think'.
(eg.: Computers, Probes, Space Vehicles, Medical Devices for Diagnoses,
Robotics, and Independent, 'Thinking Weapons of Mass Destruction' that
can be used either 'Offensively' or 'Defensively',... etc.)
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
1. Establishing the foundation for Ternary Logic
2. Establishing the Foundation for Multi-Variable Logic
3. The Correction of the Errors in the Logic and
Mathematics in Fuzzy Logic
4. These Results could ultimately lead to the
Development of Hardware for Artificial Intelligence
And while it should be understood, I definitely have my work cut out for
me. It should be equally clear, that time does not always permit an
explanation of the Elementary Concepts, which should be well understood
by the Professionals who populate the intended Area of Study / Research.
Notwithstanding, the joys I derive from my work in the field of
Mathematics, my actual objective is indeed the Natural Sciences, and
perhaps the Engineering Sciences as well. But clearly, it is doubtful,
that any of these works will every find as their home, the postings of
the IETF's Web Page. Needless to say, they would indeed be well beyond
the scope of the audience, who frequents Internet-Draft's Web Pages for the
latest information regarding the standards governing Computer Technology.
And for this, I sincerely apologize.
Chapter IV: Security Considerations
This document, whose only objective was the explanation of the
new foundation for the Binary System, which resulted from the Mathematics
of Quantification, does not directly raise any security issues. Hence,
there are no issues that warrant Security Considerations.
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002
References
1. E Terrell ( not published, notarized 1979 ) " The Proof of
Fermat's Last Theorem: The Revolution in Mathematical
Thought" Outlines the significance of the need for a
thorough understanding of the Concept of Quantification
and the Concept of the Common Coefficient. These
principles, as well many others, were found to maintain
an unyielding importance in the Logical Analysis of
Exponential Equations in Number Theory.
2. E. Terrell ( not published, notarized 1983 ) " The Rudiments
of Finite Algebra: The Results of Quantification
" Demonstrates the use of the Exponent in Logical
Analysis, not only of the Pure Arithmetic Functions
of Number Theory, but Pure Logic as well. Where the
Exponent was utilized in the Logical Expansion of the
underlying concepts of Set Theory and the Field
Postulates. The results yield; another Distributive
Property (i.e. Distributive Law for Exponential Functions)
and emphasized the possibility of an Alternate View of the
Entire Mathematical field.
3. G Boole ( Dover publication, 1958 ) "An Investigation of The
Laws of Thought" On which is founded The Mathematical
Theories of Logic and Probabilities; and the Logic of
Computer Mathematics.
4. R Carnap ( University of Chicago Press, 1947 / 1958 )
"Meaning and Necessity" A study in Semantics and
Modal Logic.
5. R Carnap ( Dover Publications, 1958 ) " Introduction to
Symbolic Logic and its Applications"
Author
Eugene Terrell
24409 Soto Road Apt. 7
Hayward, CA. 94544-1438
Voice: 510-537-2390
E-Mail: eterrell00@netzero.net
{Note: The Multiplication and Division operations
for this New Binary System have been completed
as well. The decision however, was not to
include these Operations in this Draft,
because they are related to other works.}
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Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002