ETT-R&D Publications E. Terrell IT Professional, Author / Researcher February 2002 Internet Draft Category: Informational Document: draft-terrell-math-quant-new-para-redefi-bin-math-00.txt Expires August 22, 2002 The Mathematics of Quantification, and the New Paradigm, which Re-Defines Binary Mathematics Status of this Memo This document is an Internet-Draft, and is in full conformance with all provisions of Section 10 of RFC2026. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet-Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsolete by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress". The list of current Internet-Drafts can be accessed at http://www.ietf.org/ietf/1id-abstracts.txt. The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. Conventions Please note, the font size for the Tables are smaller than the expected 12 pts. However, if you are using the most current Web Browser, the View Section of the Title bar provides you with the option to either increase or decrease the font size for comfort level of viewing. That is, provided that this is the HTML or PDF version. 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 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 upon 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 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. 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. E Terrell [Page 4] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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" 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 MankindĘs 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 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 E Terrell [Page 5] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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, 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." 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? Addition, when dealing with the Binary System, should it be considered different in Arithmetic or Logical consistency from that of the Unary System, which is now the Positive Integers' graphical depiction of a Numerical? 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 looks 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. E Terrell [Page 6] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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 relative 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: 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 E Terrell [Page 7] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 The examination of TABLE 1A, coupled of the Elementary Operations for Addition in Binary Mathematics, the Laws from the Field Postulates, and Set Theory, Cleary it can be 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. And while, I could also say that the Relationship between Columns '2' and '4' has been established has being Logically valid under the Rules and Laws, which govern the Field Postulates and Set Theory, it would be too taxing of a demand, that 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 "Proof by Contradiction" Argument required by 'Mr. Perovic' response to the initial proof of the foundation that established the this New Paradigm for the Binary System. 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 is the Binary Notation that represents, or equals the Integer '3'. 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. However, 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. E Terrell [Page 8] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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. 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. E Terrell [Page 9] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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 00000000 = 01 1 3. 2^1 = 2 00000001 = 10 2 4. 2^F = 3 00000010 = 11 3 5. 2^2 = 4 00000011 = 100 4 6. 2^F = 5 00000100 = 101 5 7. 2^F = 6 00000101 = 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. "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?" E Terrell [Page 10] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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: 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 E Terrell [Page 11] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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 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}. E Terrell [Page 12] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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: Fig 8. -10 + -9 ... -5 +... -2 + -1 + 0 + 1 + 2 + 3 ... 5 +... + 10 E Terrell [Page 13] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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'. Furthermore, any 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', 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 exist only '4' Members or Numbers in the 'BINARY SET'...All 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, an 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 would suffice in not only establishing the necessary proof by Contradiction, 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, Fail the TEST, when one attempts to solve the Equation "1 + 1 = 10"...Which is valid enough, as the proof by contradiction, especially since it does not yield an equivalent integer representation. In other words, it does not represent the integer '3'. 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. E Terrell [Page 15] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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 same presentation. In other words, the conclusion representing the foundation, which Established this New Paradigm for the Binary System, remain unquestionably valid. And Gregor Cantor was very wrong... E Terrell [Page 17] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002 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; 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 And while it should be understood, I definitely have my work cut out for me, and it should be equally clear that time does not always permit an explanation of the Elementary Concepts, which should be understood. 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 frequencies Internet-Drafts Web Page for the latest information regarding the standards governing Computer Technology. And for this, I sincerely apologize. E Terrell [Page 18] 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 E Terrell [Page 19] Quantification, New Paradigm, Re-Defines Binary Math February 22, 2002