## Math & Science

## The Derivation of Numbers

In deriving the concepts we call number and mathematics 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 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 involving interrelationships between Real Words, but not as the actual signs, meanings, images or creators themselves that these Real Words comprise. This distinction is crucial when we think about numbers and mathematics.

With this cautionary note as context let us first consider what exactly is meant by the concept of number. Heuristically, it seems reasonable to identify numbers with counting, in that counting is something we all are able to do easily and naturally. Counting is intuitive and seems to have been done in one form or another for almost all of recorded human history. There is even evidence that some animals can count, or at least recognize differences in small numbers of objects. Thus the Natural Numbers or the Counting Numbers provide a reasonable place to start.

By this reasoning it also seems that we can make a good argument that a Fraction is also a reasonable choice for a number. We all have a notion of dividing things into parts so the concepts of one half, one third, two thirds, one fourth, etc. make intuitive sense to us. How do we split up the pie. Fractions have been used by people for much of recorded history (4,000 years or so), and they can be represented exactly as ratios of the Natural Numbers. These features make them understandable as numbers in a somewhat similar manner to the way that we understand the Natural or Counting Numbers.

However, mathematics makes use of several other types of concepts that are also called numbers, but which do not generate the same type of intuitive understanding and comfort in our minds as do Counting and Fractional Numbers. These concepts are used as numbers but, unlike the Fractions, they cannot be exactly represented with the Natural Numbers. Examples include the square root of two, which is the ratio of the hypotenuse of a unit square to one of its sides, or ∏, which is the ratio of the circumference to the diameter of a circle. Neither of these two “numbers” can be exactly represented with the Natural Numbers or the Fractions. Since we still believe that they are a number we define a new type of number, the Real Numbers, or the Transcendental Numbers that are a subset of the Real Numbers. The definitions of these types of numbers depend on mathematics for their precise expression and this will be discussed in subsequent sections (link).

In Ododu numbers are represented by consonants and sequences of consonants, and they form a base sixteen number system. This is in contrast to the base ten number system that is most commonly used today. When the consonants are written with no lead or following vowels they comprise a shorthand for words of the format YJV, where J is any consonant and V is any vowel. In Ododu, when a word begins with Y, and is then followed by a consonant, it signifies that what follows is a number. This is in contrast with words that begin with Y but are followed by a vowel and comprise words of four or more letters. These words signify names that are locally defined and are not part of the core Ododu language. The construction of numbers comprising two or more consonants will be presented after the procedures are developed for single consonant numbers.

Ododu words that comprise three vowels with the first vowel being a Y, a YVV format, are foundational words that represent concepts that are useful in constructing number systems in conjunction with mathematics. They also include words that are used to create a base four number system. This is a useful aid in computation since computation in base four is simple and the results can be easily converted into a base sixteen number system by inspection.

Before deriving the numbers themselves we construct a grammar for the YV words as follows. This will differ from the grammar we have used so far for nouns, verbs, modifiers, and relationals in that the numerical grammar will denote a number system within which a given number will be defined. These number systems will be discussed and eventually derived in the mathematics section.

With this cautionary note as context let us first consider what exactly is meant by the concept of number. Heuristically, it seems reasonable to identify numbers with counting, in that counting is something we all are able to do easily and naturally. Counting is intuitive and seems to have been done in one form or another for almost all of recorded human history. There is even evidence that some animals can count, or at least recognize differences in small numbers of objects. Thus the Natural Numbers or the Counting Numbers provide a reasonable place to start.

By this reasoning it also seems that we can make a good argument that a Fraction is also a reasonable choice for a number. We all have a notion of dividing things into parts so the concepts of one half, one third, two thirds, one fourth, etc. make intuitive sense to us. How do we split up the pie. Fractions have been used by people for much of recorded history (4,000 years or so), and they can be represented exactly as ratios of the Natural Numbers. These features make them understandable as numbers in a somewhat similar manner to the way that we understand the Natural or Counting Numbers.

However, mathematics makes use of several other types of concepts that are also called numbers, but which do not generate the same type of intuitive understanding and comfort in our minds as do Counting and Fractional Numbers. These concepts are used as numbers but, unlike the Fractions, they cannot be exactly represented with the Natural Numbers. Examples include the square root of two, which is the ratio of the hypotenuse of a unit square to one of its sides, or ∏, which is the ratio of the circumference to the diameter of a circle. Neither of these two “numbers” can be exactly represented with the Natural Numbers or the Fractions. Since we still believe that they are a number we define a new type of number, the Real Numbers, or the Transcendental Numbers that are a subset of the Real Numbers. The definitions of these types of numbers depend on mathematics for their precise expression and this will be discussed in subsequent sections (link).

In Ododu numbers are represented by consonants and sequences of consonants, and they form a base sixteen number system. This is in contrast to the base ten number system that is most commonly used today. When the consonants are written with no lead or following vowels they comprise a shorthand for words of the format YJV, where J is any consonant and V is any vowel. In Ododu, when a word begins with Y, and is then followed by a consonant, it signifies that what follows is a number. This is in contrast with words that begin with Y but are followed by a vowel and comprise words of four or more letters. These words signify names that are locally defined and are not part of the core Ododu language. The construction of numbers comprising two or more consonants will be presented after the procedures are developed for single consonant numbers.

Ododu words that comprise three vowels with the first vowel being a Y, a YVV format, are foundational words that represent concepts that are useful in constructing number systems in conjunction with mathematics. They also include words that are used to create a base four number system. This is a useful aid in computation since computation in base four is simple and the results can be easily converted into a base sixteen number system by inspection.

Before deriving the numbers themselves we construct a grammar for the YV words as follows. This will differ from the grammar we have used so far for nouns, verbs, modifiers, and relationals in that the numerical grammar will denote a number system within which a given number will be defined. These number systems will be discussed and eventually derived in the mathematics section.

YU integer

YI rational number

YE real number

YA complex number

YO transfinite number

YQ surreal number

YY hyperreal number

YH hypercomplex number

YI rational number

YE real number

YA complex number

YO transfinite number

YQ surreal number

YY hyperreal number

YH hypercomplex number

Additional discussion about these number systems and why they were selected to represent the eight types of numbers in Ododu is presented in (Link)

The eight numerical words or concepts that are useful and even essential in understanding just what numbers are, and how we can talk about them and use them are the three vowel words that begin with the letter Y. Expressing these words as integers, they are;

The eight numerical words or concepts that are useful and even essential in understanding just what numbers are, and how we can talk about them and use them are the three vowel words that begin with the letter Y. Expressing these words as integers, they are;

YUU base point

YIU 1

YEU 2

YAU 3

YOU 0 This is zero as a boundary or placeholder

YQU - negative sign

YYU base point representation with zero notation

YHU spinor and matrix notation

YIU 1

YEU 2

YAU 3

YOU 0 This is zero as a boundary or placeholder

YQU - negative sign

YYU base point representation with zero notation

YHU spinor and matrix notation

Some exemplification of these concepts follows.

YUU is a start. It serves as a beginning of counting, an intention to count and a boundary that indicates what kind of counting is being preformed. Thus counting with Natural Numbers will be to the left of the starting point, the base point. Fractions will be to the right of the base point. For example, in our common base ten decimal point number system the integers occur to the left of a decimal point, and the fractions are written to the right of the decimal point.

YIU is one. The first counting number. The first thing you do after you decide to start counting. We denote this with a single mark, /. Represent this mark with the digit 1

YEU then is two. After you decided to count you did your first counting action, count one. The next action is to count another, and this is called two. We make a second mark so that our counting now looks like this, //. Represent this additional mark, or both marks together, with the digit 2.

YAU then becomes three. The concept of successor, one count or number after another, begins to form. Three comes after, or is preformed after, two. We make an additional mark and now have ///. Represent this additional mark, or all of them together, with the digit 3.

We could keep going on this way indefinitely but this becomes cumbersome as we get more and more digits defined as successors. This generates longer and longer strings of /////////….. So we develop a placeholder notation that allows for a more efficient accounting of where we are in the counting process. The symbol that has been chosen for this placeholder function is YOU.

YOU is zero. It is a placeholder or boundary that serves to group counts together and thus provides for an efficient representation of larger and larger countings. We could show this as the three counts, ///, with an additional line through them, and then use this as the placeholder, but again this becomes cumbersome. So we use the symbol 0 instead. Now //// or /// with a line through it, becomes 10. The base point at the end of the 10 indicates that this is a base four number system. Note that this is the choice of placeholder that has evolved during the evolutionary development of number and mathematics and is the commonly accepted standard today.

Note now that there is also a real symmetry between this initial development of numerical symbols (or graphemes) and the graphemes that were used in the initial Derivation of Archetypal Meaning in Ododu. This is consistent with the General Principle of Relativity that we are using for the construction of Ododu.

So is 0, zero, a number? It does not represent a collection of things, individual things we can recognize and “count”, but it does act like a number in our useful notation for counting. So we expand our idea of number to include this useful new concept, the zero. It is interesting to note that historically zero was not invented as a number until many years (1,000s?) after the Natural Numbers and Fractions had first been used.

Looking at our next useful numerical word, YQU, we note that, relative to the Derivation of Archetypal Meaning, this is a construct using O with a line through it. It looks like a cross going into or coming out of a circle or boundary. So we will define it as that, a cross of a zero boundary. This becomes an introduction to the idea of negative numbers, numbers that are across the zero from the positive Natural Numbers. We represent this concept with, -, a minus sign, it becomes a sign indicating a negative Natural Number. We now represent the numerical concepts in the YVU base four format as -3, -2, -1, 0, 1, 2, 3. These numbers can also be expanded by using the 0 as a placeholder as well as a boundary. For example; …., -11, -10, -3, -2, -1, 0, 1, 2, 3, 10, 11, …….

Looking at zero as a boundary between negative and positive integers seems to make it look more like a number than it did before. But is it reasonable to let one symbol be both placeholder and boundary? In a sense a placeholder does also represent a boundary between one group of counting numbers and the next group of counting numbers so the argument makes sense.

The development of negative numbers means that once again we have to stretch our understanding of number to include concepts that are more difficult to understand than just the counting Numbers. Minus two things, or two negative things, doesn’t make much sense in terms of real things, but again, like zero, they are very useful concepts in terms of using numbers.

YYU is a concept that combines all of the useful notations that have been presented up to now. This uses the base point period as a boundary between Natural Numbers and Fractions. This is a similar dual use of the base point concept to the one that we encountered when we used 0 as a placeholder and as a boundary between negative and positive numbers. Thus YUU, a period or dot, serves as both a starting point for counting and as a boundary between an integer and a fractional part of that integer. This leads to a very useful convention for representing any real number with an arbitrary level of accuracy. Thus a YYU number will have an integer part to the left of a base point, and a fractional part to the right of the base point. The same convention as we currently use in the decimal point representation of the real numbers.

Because we can add increasingly smaller and smaller fractions to the right, for example, .00000001, or progressively more and more accurate fractions to the right, for example .3856231, we can represent any number to an arbitrary level of resolution. However, this now introduces a concept of indeterminacy into our understanding of number. This exists in the real numbers where many real numbers, such a square root of 2 or ∏ cannot be exactly portrayed in terms of Natural Numbers and Fractions, as discussed earlier. However, they can be represented to whatever level of accuracy that we require for pragmatic or practical application by using increasingly precise fractional parts.

YHU is the final YVU word that is proposed for Ododu. This refers to a general matrix notation that will include one by two and two by one spinor matrices and two by two matrices that are essential for expressing the most comprehensive and fundamental nature of number. This will be the quaternion. It will be argued in the Mathematics section that the quaternion is the numerical representation and meaning of the four Primary Vowels and their Relational structure in Ododu, and that this is also the best representation of the basic space-time structure of the universe itself.

Using these concepts it is now straightforward to construct the true Ododu numbers in terms of consonants. Using J as any one of the sixteen consonants, the words for the numbers in the YJV format are presented in the Grammar Dictionary (link) for YJU, YJI, YJE, YJA, YJO, and YJQ. The number words in the YJY format, and their extensions to words containing more than one consonant, are presented below. This uses the base point notation with zero to generate Ododu number systems in base four and base sixteen that behave in the same way as the base ten number systems that are commonly and almost universally used today.

Base four and base sixteen systems are used because these naturally follow from our General Principle Of Relativity and how it interprets the basic relational structure of Ododu.

First the Integers, words with a YJY format.

YUU is a start. It serves as a beginning of counting, an intention to count and a boundary that indicates what kind of counting is being preformed. Thus counting with Natural Numbers will be to the left of the starting point, the base point. Fractions will be to the right of the base point. For example, in our common base ten decimal point number system the integers occur to the left of a decimal point, and the fractions are written to the right of the decimal point.

YIU is one. The first counting number. The first thing you do after you decide to start counting. We denote this with a single mark, /. Represent this mark with the digit 1

YEU then is two. After you decided to count you did your first counting action, count one. The next action is to count another, and this is called two. We make a second mark so that our counting now looks like this, //. Represent this additional mark, or both marks together, with the digit 2.

YAU then becomes three. The concept of successor, one count or number after another, begins to form. Three comes after, or is preformed after, two. We make an additional mark and now have ///. Represent this additional mark, or all of them together, with the digit 3.

We could keep going on this way indefinitely but this becomes cumbersome as we get more and more digits defined as successors. This generates longer and longer strings of /////////….. So we develop a placeholder notation that allows for a more efficient accounting of where we are in the counting process. The symbol that has been chosen for this placeholder function is YOU.

YOU is zero. It is a placeholder or boundary that serves to group counts together and thus provides for an efficient representation of larger and larger countings. We could show this as the three counts, ///, with an additional line through them, and then use this as the placeholder, but again this becomes cumbersome. So we use the symbol 0 instead. Now //// or /// with a line through it, becomes 10. The base point at the end of the 10 indicates that this is a base four number system. Note that this is the choice of placeholder that has evolved during the evolutionary development of number and mathematics and is the commonly accepted standard today.

Note now that there is also a real symmetry between this initial development of numerical symbols (or graphemes) and the graphemes that were used in the initial Derivation of Archetypal Meaning in Ododu. This is consistent with the General Principle of Relativity that we are using for the construction of Ododu.

So is 0, zero, a number? It does not represent a collection of things, individual things we can recognize and “count”, but it does act like a number in our useful notation for counting. So we expand our idea of number to include this useful new concept, the zero. It is interesting to note that historically zero was not invented as a number until many years (1,000s?) after the Natural Numbers and Fractions had first been used.

Looking at our next useful numerical word, YQU, we note that, relative to the Derivation of Archetypal Meaning, this is a construct using O with a line through it. It looks like a cross going into or coming out of a circle or boundary. So we will define it as that, a cross of a zero boundary. This becomes an introduction to the idea of negative numbers, numbers that are across the zero from the positive Natural Numbers. We represent this concept with, -, a minus sign, it becomes a sign indicating a negative Natural Number. We now represent the numerical concepts in the YVU base four format as -3, -2, -1, 0, 1, 2, 3. These numbers can also be expanded by using the 0 as a placeholder as well as a boundary. For example; …., -11, -10, -3, -2, -1, 0, 1, 2, 3, 10, 11, …….

Looking at zero as a boundary between negative and positive integers seems to make it look more like a number than it did before. But is it reasonable to let one symbol be both placeholder and boundary? In a sense a placeholder does also represent a boundary between one group of counting numbers and the next group of counting numbers so the argument makes sense.

The development of negative numbers means that once again we have to stretch our understanding of number to include concepts that are more difficult to understand than just the counting Numbers. Minus two things, or two negative things, doesn’t make much sense in terms of real things, but again, like zero, they are very useful concepts in terms of using numbers.

YYU is a concept that combines all of the useful notations that have been presented up to now. This uses the base point period as a boundary between Natural Numbers and Fractions. This is a similar dual use of the base point concept to the one that we encountered when we used 0 as a placeholder and as a boundary between negative and positive numbers. Thus YUU, a period or dot, serves as both a starting point for counting and as a boundary between an integer and a fractional part of that integer. This leads to a very useful convention for representing any real number with an arbitrary level of accuracy. Thus a YYU number will have an integer part to the left of a base point, and a fractional part to the right of the base point. The same convention as we currently use in the decimal point representation of the real numbers.

Because we can add increasingly smaller and smaller fractions to the right, for example, .00000001, or progressively more and more accurate fractions to the right, for example .3856231, we can represent any number to an arbitrary level of resolution. However, this now introduces a concept of indeterminacy into our understanding of number. This exists in the real numbers where many real numbers, such a square root of 2 or ∏ cannot be exactly portrayed in terms of Natural Numbers and Fractions, as discussed earlier. However, they can be represented to whatever level of accuracy that we require for pragmatic or practical application by using increasingly precise fractional parts.

YHU is the final YVU word that is proposed for Ododu. This refers to a general matrix notation that will include one by two and two by one spinor matrices and two by two matrices that are essential for expressing the most comprehensive and fundamental nature of number. This will be the quaternion. It will be argued in the Mathematics section that the quaternion is the numerical representation and meaning of the four Primary Vowels and their Relational structure in Ododu, and that this is also the best representation of the basic space-time structure of the universe itself.

Using these concepts it is now straightforward to construct the true Ododu numbers in terms of consonants. Using J as any one of the sixteen consonants, the words for the numbers in the YJV format are presented in the Grammar Dictionary (link) for YJU, YJI, YJE, YJA, YJO, and YJQ. The number words in the YJY format, and their extensions to words containing more than one consonant, are presented below. This uses the base point notation with zero to generate Ododu number systems in base four and base sixteen that behave in the same way as the base ten number systems that are commonly and almost universally used today.

Base four and base sixteen systems are used because these naturally follow from our General Principle Of Relativity and how it interprets the basic relational structure of Ododu.

First the Integers, words with a YJY format.

Base Ten Base Four Base Sixteen

1 1 P

2 2 R

3 3 B

4 10 C

5 11 L

6 12 T

7 13 K

8 20 S

9 21 G

10 22 F

11 23 X

12 30 N

13 31 M

14 32 W

15 33 Z

16 100 P0

17 101 PP

18 102 PR

19 103 PB

20 110 PC

1 1 P

2 2 R

3 3 B

4 10 C

5 11 L

6 12 T

7 13 K

8 20 S

9 21 G

10 22 F

11 23 X

12 30 N

13 31 M

14 32 W

15 33 Z

16 100 P0

17 101 PP

18 102 PR

19 103 PB

20 110 PC

The Rational Numbers as expressed with words with a YJY format, are fractions from one sixteenth to 15 sixteenths

Some of the Rational Numbers with more than one consonant, and that may use 0 as a placeholder, are presented below as fractions from 1/256 to 36/256

The YJH words are the initial numbers that are considered to be hypercomplex. These are numbers that include quaternions and other numbers generated by the Clifford Algebras and they can be best represented using matrix notation with complex numbers. The simplest possible such numbers, using a matrix notation and containing only one consonant, are presented below, and comprise the one by two and two by one spinor matrices. They will include a zero part, where 0 is used as a place holder, and a numerical part, which may include any one of the numbers 1, -1, i, and –i, where i is the square root of minus one. They are for now arbitrarily assigned and presented as

These spinors can be used to generate the basis elements of a quaternion and hence represent the most foundational components of the concept of number as it is derived in Ododu. The hypothesis is that to best understand the nature of ourselves and the universe using mathematics and numbers we have to use a quaternion format. The quaternion is a form that derives its characteristics from the fundamental relational nature of the universe itself. It is a best available mathematical and numerical representation or metaphor for the universe, and for ourselves as conscious entities. The fact that it does not generate deterministic solutions to our equations that we use to generate descriptive predictions of the universe and ourselves is simply a basic fact of nature. A linear reductionist determinism does not fit the world as it is.

To understand this more fully it is useful to review the nature of quaternions as follows. This is anticipating some of the derivations that will be presented in the Mathematics Section but it is relevant here to understand the concept of the quaternion as a number, albeit a hypercomplex number.

In existing mathematics and number theory, quaternion is a number with a form of;

a1 + bv + cj + dk

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

v * v = - 1, j * j = - 1, and k * k = - 1

(* represents multiplication)

but v, j, and k are not equal to each other.

The other requirements for the basis elements of a quaternion are;

To understand this more fully it is useful to review the nature of quaternions as follows. This is anticipating some of the derivations that will be presented in the Mathematics Section but it is relevant here to understand the concept of the quaternion as a number, albeit a hypercomplex number.

In existing mathematics and number theory, quaternion is a number with a form of;

a1 + bv + cj + dk

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

v * v = - 1, j * j = - 1, and k * k = - 1

(* represents multiplication)

but v, j, and k are not equal to each other.

The other requirements for the basis elements of a quaternion are;

1 * 1 = 1

1 * v = v * 1 = v

1 * j = j * 1 = j

1 * k = k * 1 = k

v * j = - j * v = k

j * k = - k * j = v

k * v = - v * k = j

1 * v = v * 1 = v

1 * j = j * 1 = j

1 * k = k * 1 = k

v * j = - j * v = k

j * k = - k * j = v

k * v = - v * k = j

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

i * i = -1

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

i * i = -1

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

We also define minus one in two by two matrix notation as;

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

When this is applied to the basis elements of v, j, and k;

We can generate the basis elements for a quaternion, 1, v, j, and k, by multiplying certain pairs of spinors and then adding the products. This is the reason that the spinors were assigned to the consonant YJH words as they were. Thus YDH * YPH + YRH * YBH gives the unit basis element, 1, (where * and + mean matrix multiplication and addition) as follows.

This is continued for the other quaternion basis elements v, j, and k, as

It will be argued in the Mathematics section that the quaternion is the numerical representation and meaning of the four Primary Vowels and their Relational structure in Ododu, and that this is also the best representation of the basic space-time structure of the universe itself.