UDOREU

In deriving the concepts we call number and mathematic it is crucial to keep in mind that these are ideas. In ODODU all words beginning with U, I, E, or A are considered to represent the real constituents of the universe. They include consciousness, relation, language, communication, and things. We call such words Real Words. Words beginning with O, Q, Y, or H represent ideas in and of language and hence as Idea Words they can only represent aspects of the Real Words. They do this representation by generating ideas of such Real Words, but not as the actual signs, meanings, images or creators themselves which these Real Words comprise. This distinction is crucial when we think about numbers and mathematics.

These four concepts can perhaps best be viewed as representing the four dimensions of the concept of number. Later on we will associate these concepts with the fundamental nature or basis elements of quaternions, where 1 is unity and i, j, and k, all represent different forms of the square root of minus one. Thus i, j, and k all equal the square root of minus one but they are not equal to each other. However, for the present they will simply be viewed as the four dimensions of the general concept of number itself.

In ODODU the counting numbers or positive integers (words beginning with Y, containing one consonant J, and ending with U, YJU format) are represented as single consonants, J. The firsts 16 counting numbers as represented with single J letters, and their correspondence with base ten and base four digits are listed here.

The expansion of the YJU numbers to the YJV*U format (V* can be any one of the first four vowels; U, I, E, or A) continues the interpretation that all ideas and things have four common attributes or fundamental components. To do this properly in the case of numbers we will need to develop some basic mathematics but to get a sense of where this is going consider an initial assignment as follows:

We will revisit these concepts after we have begun our derivation of basic mathematics.

In The ODODU Language the initial mathematical concepts were presented in HVU form as;

HUU, HIU, HEU, and HAU are all geometrical concepts that are usually presented as axiomatic assumptions. We all have an intuitive understanding of what these concepts are and how they have developed from common experience. In ODODU we will not view these concepts as being subsumptively related. Thus a line is not composed of an infinite number of points. A surface is not composed of an infinite number of lines. A space or volume is not composed of an infinite number of surfaces.

Like all mathematical concepts HUU, HIU, HEU, and HAU are ideas. As the initiating ideas for mathematics we consider that they generate a context or an idea space within which all other mathematical ideas can be conceived or considered. Call this the Mathematical Space. This is similar to the way that UIEA generates a concept of a real space which is and contains the universe and ourselves.

HOU: Once we have the Mathematical Space, the first defining event is the construction of the idea of a distinction or boundary. This we shall call a set. In its purest form it is a closed boundary within the Mathematical Space. In its first and simplest form it does not contain anything or do anything, it is just a boundary. Because it does not contain anything it will be designated as an empty set or the null set. Show it as;

HQU: Given one null set, HOU, and then constructing a second null set doesn't change what we have. The two null sets are equivalent and having two or more null sets doesn't generate a new idea. All null sets are equivalent. Show this as;

HYU: Establishing that all null sets are equivalent generates the additional concept of redundancy. Thus if one null set is equivalent to another null set, then two null sets are also equivalent to another null set. Call this the law of redundancy and show it as;

HHU: We now introduce the concept of combining null sets. This actually doesn't make any sense since putting anything into a null set means that it no longer is a null set. So putting a null set into a null set can be viewed as eliminating or cancelling the null set, or both null sets since they are equivalent. This can also be viewed as superimposing one null set over another null set. Call this the law of combination/cancellation and show it as;

The HVU words can now be seen to be similar to the initial development that George Spencer Brown presented in "Laws of Form". The difference is that here we have first derived a Mathematical Space which is an idea space within language and then introduced an initial formation of a boundary as distinguishing nothing. This we have called the null or empty set. Relative to the null set we have constructed concepts of equivalence, redundancy and cancellation.

The next step is to start putting something (ideas) into a boundary which has been derived as a null set. When we do this it is no longer an empty set but becomes what we shall call a Regular Set. It now becomes a boundary which contains or separates something from the rest of the Mathematical Space which contained the initial null set. The defining boundaries of these Regular Sets (just sets from now on) will themselves be subject to the same rules of equivalency, redundancy, and cancellation that we applied to the null set. In general the contents of sets will not follow these rules.

As we derive the HJU words we will do so within the context of the Mathematical Space, sets, and their rules. The result will generate a structure that will resemble, and perhaps actually be, what we usually call mathematics. The HJU words will provide an outline of this argument and the HJV*U words will then fill out the derivation. (V* are the U, I, E, and A vowels.) First the HJU words as they were set forth in The ODODU Lamguage :

The derivations of these HJU words are presented and illustrated in conjunction with the corresponding HJV*U words as follows:

HDU: The first ideas we can put into a null set boundary will be the initiating relations which began this whole derivation. In keeping with the understanding that relation is really four relations, or four dimensional, or has four integral components, it is necessary to start with the whole of relation. This means that all four of the initiating UIEA relations become the first entry into the set concept boundary as we start the derivation of mathematics. Thus HDU is:

Here the rectangular boundary denotes the relational set, which is a set within Mathematical Space, and the UIEA pictographs denote the relations of the relational set. Hence;

HPU: The Relational Cross. This describes how relations in the Relational Set can cross each other within the Relational Set. The cross is just what it implies. Given the symbols for two relations (we represent the relations by their symbols) we perform a cross by moving one of the symbols through the other symbol until it is on the other side of that symbol. This occurs within the Mathematical Space we have just derived.

HPUU: the U or unitary cross. This includes all crosses which involve the U relation as follows;

WE use the = sign in these representations to mean that the meaning of the pair of relations, say U and E, when presented as UE, is the same as presenting it as EU.

HPIU: This is the I cross and it involves the case when E crosses H or H crosses E. The I cross does not actually involve I itself in the cross. It is represented as;

Note that in this case we are saying that for the relations E and A that the presentation EA does not mean the same as AE. This is symbolized with the use of the "not equals" sign.

HPEU: This is the E cross and it involves the case when I crosses H or when H crosses I. It is represented as;

HPAU: This is the A cross and it involves the case where I crosses H or when H crosses I. It is represented as;

Note that in the three crosses HPIU, HPEU, and HPAU that the result is not commutative. Thus if you start with an I and E relational pair, and the I then crosses the E, that the meaning of the resulting E and I pair is not the same as the meaning of the original I and E pair. This contrasts with the situation in HPUU where the U relation can cross with any other relation and the resulting pair will have the same meaning as the original pair. The U relation crosses commutatively with any other relation whereas the other relations do not cross commutatively with each other.

HRU: The Relational Mark. This describes how we can mark or distinguish when a Relational Cross has occurred. The Relational Mark makes a distinction and it therefore behaves like our initial distinction in the Mathematical Space which was the set. The Relational Mark will be symbolized with a # sign. This # sign is the one additional feature within a set which will follow the rules which we have derived for the set defining boundary itself. See HIU, HEU, and HAU above. As was the case for sets, to make a mark, and then make another mark, is the same as making only one mark. Thus;

But when two marks are combined they superimpose themselves over each other and cancel each other out. We are left with nothing or no mark at all. In this case we do a cross and mark it, and then do another cross and mark that. The second mark marks the first mark and they cancel each other out. Thus;

HRUU: Since all HPUU unitary crosses are equal there is no reason to make a mark. Therefore the Relational Mark HRUU is not used for unitary crosses.

This contrasts with the situation in HPIU where the E and A relational cross was not commutative. By introducing the mark # to designate a cross the meaning of the two pairs (E & A) and (#A & E) is the same. Thus the meaning of #AE is the same as the meaning of EA after the E has crossed the A. This applies to the following two marked crosses as well.

HBU: When two relations within the Relational Set cross each other an interaction can occur and this is called the combination/cancellation interaction. This interaction acts like a mathematical binary operation and must result in a valid relation of the Relational Set. The combination/cancellation interaction will be represented by the symbol * and it is defined as a binary operation as follows:

HBUU: This defines how the U relation interacts with the relations of the Relational Set under the * operation;

HBIU: This defines how the * operation acts when a relation is operated on itself:

In this case when the I, E, and A relations are combined with the binary operation * they superimpose on each other and cancel each other out. This is shown in the pictograph as first making a darker, larger symbol, and then setting this equal to a marked U relation. The interpretation that the relational superposition and cancellation does not lead to nothing, as was the case for sets and marks, is one of the defining characteristics of the * operation. Thus a relational cross leads to a marked result, and a relational combination/cancellation under the binary operation * leads to a marked unitary relation, U.

HBEU: This defines how the * binary operation functions when one or both of the relations involved is marked. Thus;

When the * binary operation operates on a marked and an unmarked relation, the result is always a marked relation.

When the * binary operation operates on two marked relations the result is always an unmarked relation.

HBAU: This defines how the * binary operation functions when the two interacting relations are different from each other and neither is the U relation.

In the first of these three operations the mark is due to the cross. In the last two of these operations the mark is due to the superposition of one of the relations on or over part of the other relation and the cancellation of the superimposed part (represented by the darker lines above) resulting in a valid relation.

It is interesting and relevant to note that if we rotate the four relational pictographs by 180 degrees that the rotated pictographs for U, I, and A are indistinguishable from the non rotated symbols. However, this is not the case with E which therefore has a right handed representation or a left handed representation. This actually has meaning in the HBAU derivations. If we call the E relation pictograph we have been using so far as right handed, then we can represent a left handed E as E' and illustrate it with the alternative pictographic symbol of;

In the first case of this alternative interpretation there is no cross and no cancellation so the result is simply that I * E' = A.

In the second case E' first crosses A and this generates a mark #. After this cross all of E' superimposes on part of A and this leads to a second mark and the cancellation of the superimposed part of A. The result is a double marked I and this is equivalent to an unmarked I.

In the third case H crosses I and this generates a mark. Note that H cannot just interact with I to produce E, as I and E' interacted to produce A in the first case, because the result would be a right handed E and this is an HBAU with only left handed E' relations. So H has to fully cross I and thus generate a mark. The I then superimposes on the left hand part of the A and this generates a cancellation of the superimposed part, and generates another mark. Thus we have a double marked left handed E' and this in turn is equal to an unmarked E'.

Using a left handed E' does not change the results of HBUU, HBIU and HBEU in any way other than the E' is left handed instead of right handed. The results of the operations are the same.

Thus there are two possibilities for HBU, one which has a marked E' in HBAU and one which has an unmarked E in HBAU. The difference is significant and will be discussed later.

HCU: An element or member of a set. This concept applies when we consider putting anything other than relation into an empty set. This will now be the meaning of the word set from here on and thus is distinguished from the empty set and the Relational Set. We will derive it as a set containing elements as follows. Remember the derivation of UFU in The Derivation of Archetypal Meaning in ODODU. We can label the components of this derivation as;

HCU: An element or member of a set. This element is derived by the preceding pictograph, where;

HCAU: Is the membership relation by which the element of the set is established or derived as an element of the set.

HLIU: A connecting relation for all elements of an association, relating each of them to all the other elements of the association and not to any other elements of the set.

HLEU: An identifying mark, such as ( ). which defines or illustrates an association within a set such that elements of the association can be clearly distinguished from all other elements of the set.

HLAU: The inclusion of an association of elements in a set as being itself a member of the set. This will now be defined as a subset of the set. Thus an association of elements of a set, which is by definition a subset of the set, is itself a member or element of the set. By also including all associations of a single element (an element associated with itself) as subsets of the set, as well as the empty set and the set itself, we have completed the derivation of the set concept.

HTU: This is a binary operation called intersection which is defined for any two subsets of a set such that the operation meets the following criteria.

HTIU: Relative to any two subsets of a set the intersection operation will define a new or additional subset which contains only those elements which are in both of the given two subsets.

Where the green identifies the subset resulting from the intersection operation on the blue and yellow subsets.

HTAU: The inclusion of the intersection operation as an essential part, process, or law defined on all sets.

HKU: This is a binary operation called union which is defined for any two subsets of a set such that the operation meets the following criteria.

HKIU: Relative to any two subsets of a set the intersection operation will define a new or additional subset which contains any element which is in either of the given two subsets, or is in both of the given two subsets.

Where the green identifies the subset resulting from the union operation on the blue and yellow subsets.

HKAU: The inclusion of the union operation as an essential part, process, or law defined on all sets.

HSEU: Measure of a limit. A quantity approaches a limit as some other measure increases or decreases.

HGUU: A variable relation defined between two subsets. A variable relation is one which can be assigned a number.

HGIU: The change in a variable relation between two subsets as its numerical representation decreases. This may involve redefinition or choice of the relevant subsets.

HGEU: The limit of the change in a variable relation between two subsets as its numerical representation decreases.

HGAU: The definition of the limit of the change in a variable relation between two subsets as its numerical representation decreases, as an element of the set containing the subsets.

HFIU: The symbol represents a relation between two numbers or subsets of the given set.

HFEU: A third number or subset of the given set will be associated with the numbers or subsets which are related by the binary operation.

HXIU: Zero as the limit for the fractions represented as 1/n where n becomes large.

HWUU: The basis which can be raised to a power. The number e, the basis of the Napierian logarithms.

HWEU: The zero notation form of e^x . If y = e^x , then y can be expressed as 0JJJ....where the Js represent fractions (similar to decimal notation with the decimal point being replaced with 0. More on this in the discussion.

The preceding presentation serves to outline a rough foundational structure to derive numbers and mathematics. It needs to be noted that at this point this derivation is extremely limited because we are only considering words with a single consonant. As we progress to include words with multiple consonants the precision and detail of these concepts will increase and will then approach and subsume most of what we understand today as mathematics and number theory. However, even at this simplistic stage of development there are some interesting aspects that deserve some commentary.

The first of these is that the concept of number presented here is that of an idea that exists in the four dimensional or four component idea space of ODODU. Thus the most fundamental concept of number itself has four dimensions or components. See the pictograph of UFU in The Derivation of Archetypal Meaning of ODODU. Our normal sense of what we mean by number is usually tied to the idea of counting numbers, 1, 2, 3, 4, 5, ....and we do not normally think of these counting number concepts as having themselves components or dimensions.

In ODODU this is not the case and we have derived the concept of number in much the same way as we have derived the concept of relation, that is, it has four different forms or expressions. We symbolized this with the words YUU, YIU, YEU, and YAU, which mirrors our fundamental UIEA structure, and initially exemplified these concepts as being similar to, or representative of, the basis elements of a quaternion. Actually the basis elements of a quaternion are viewed in ODODU as deriving from the fundamental four component nature of number and not the other way around. To get a better understanding of this difference in perspective let us first look at the nature of quaternions in conventional mathematics.

Where a, b, c, and d are real numbers and 1 is the unitary concept of one, and v, j, and k are non equivalent imaginary numbers each equal to the square root of minus one. Thus

Although this is the way that quaternions are usually introduced it is somewhat confusing and so we go to a 2 by 2 matrix representation for both the quaternions themselves and their basis elements. Thus we define i as the square root of minus one such that

where minus one is a real number, and derive the basis elements for a quaternion as follows;

The conventional definition for matrix multiplication for a two by two matrix is;

It is easy to show that the other criteria for the quaternion basis elements mentioned above are also satisfied with the two by two matrix interpretation.

What this means is that when we consider a counting number or a digit we are only looking at a number from a one dimensional or one component view. This view corresponds to what we represent as the "real" or unitary part of the quaternion, that part which is a function of (multiplied by) the unitary basis element 1. Thus we only look at the sign or symbol, the USU, of the counting number and don't consider the consciousness which is counting, the UKU, and the derivation, the UGU, and the image, UFU, of the counting number. These are the additional components as shown in the general UFU pictograph.

We can now show what the first four digits as numbers actually represent as follows;

In these four pictographs the digits 1, 2, 3, and 4 are the number symbols which appear in the USU loop. The colored circles in the UGU loop represent things which are considered as singles, doubles, triples, or quadruples, that is, are considered in groups of one, two three, or four. The circles in the UFU loop represent subsets, each one in correspondence with one of the things designated in the UGU loop. The total concept of a number must thus include the USU, UGU, UFU, and UKU components.

Given this general interpretation that the most fundamental nature of number is more like a quaternion than a counting number, it is interesting that the derivation of the first consonant containing mathematical words, HDU, HPU, HRU, and HBU, and their HJV*U extensions, generates a structure that is the same as the criteria for a quaternion presented above. If we say that HDUU is 1, HDIU is v, HDEU is j, and HDAU is k, and we let the mark # be the same as a minus sign, then the relationships presented in HPU, HRU, and HBU are the same relationships used to define a set of quaternion basis elements. In mathematics these relationships are those that represent a mathematical group which is conventionally defined by;

for an association of elements of the collection it does not matter what order the elements are operated on by the operation, and

The quaternion basis elements and the operation of multiplication constitute a quaternion group. We now interpret the ODODU words of HDU, HPU, HRU, and HBU and their associated HJV*U extensions as deriving and constituting the most fundamental mathematic. This primordial mathematic is identical to that associated with the quaternion group and is in fact the reason that the quaternion group, and the definition of groups in general, even exists in the first place.

The continuation of the HJU words then derives set theory with the HCU, HLU, HTU, and HKU words and their associated HJV*U words. This then constitutes the second fundamental mathematic. Once we have group and set theory all other forms of conventional mathematics can be subsequently derived. The remaining HJU words serve to introduce additional concepts which can be incorporated with set and group theory to derive calculus and other forms of mathematics.

Several additional points of interest concern the concept of zero (HXU), various conventions for representing different kinds of numbers, and the derivation of the everyday mathematics we all use involving the real numbers and the operations of addition, division, multiplication, and subtraction.The concept of zero in ODODU is one of a boundary and not as a number per se. It is not included in the counting numbers except as an origin and it occurs in the other number systems as a distinction between positive and negative numbers.

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