The most basic task in this study of the first ten numbers (the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) is to find their origin and to relate them to that and to each other.

The Generation of the Digits
The Linear Generation
The Binary Generation
The Bipolar Generation
The Stupa and the Digits
The Tetraktys and the Stupa
The Bar Notation
Relative Weights of Digits and Numbers
The Four Elements and the Digits
Subtractive Complements
Review

Three different derivations or generations of the digits will be presented. These three generations provide the foundations for quantitative mathematics, digital computers, and qualitative mathematics.

Once these basic relationships have been explored, then many more will appear. These will in truth demonstrate a system latent in the digits. Such a system can then be applied to various types of qualities and classifications.

Consider zero first. It not only is the first and smallest of the positive digits, but it involves a concept so abstract that only two peoples have discovered zero - the Hindus and the Mayans. Numbers are counts of objects, but the number zero implies that there are no objects present to be counted. It is thus a negation of number and indicates a negation of the known type of existence. Zero says "no", "not", "The Void", "The Unknown". So it is a good symbol for God who is often described in the same terms. It represents Chaos, the Unknowable, the Undifferentiated. Thus it is comparable to the Mother from which all things are generated in the mythologies of many peoples - the Greeks, the Babylonians, and the early Hebrews.

The word Chaos is defined to mean all that we can not segment into logical pieces. It is assumed that, on some higher level or reference, chaos is indeed a cosmos.[top]

#### The Linear Generation

But obviously there is knowledge and there are differences, so from zero arise the other numbers, from Chaos arise the categories. However, these other numbers do not form a limited set, so they can not be listed completely in order to study them. Mathematicians get around this by starting with two things: first a starting point (=0) and then an operation which will produce all the other numbers by the simple repetition of the operation. This operation is called "addition-of-one". Adding one repeatedly to zero produces all the other numbers.

```0 -> 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 10 -> 11
```

In this diagram the arrows all represent the operation of addition-of-one. Since the generation forms a straight line it is call the Linear Generation of Numbers. This linear generation is the basic one because it defines the quantity involved. But it says nothing about the quality represented by a digit. And it is exactly this idea of quality so universally attributed to numbers that is the subject of mathesis.[top]

#### The Binary Generation

Another well-known generation is the ordinary binary notation for numbers. In this system there are two starting points (=0 and =1). Any number, no matter how large, can be written as a combination of zeros and ones. All that is needed in an agreement that putting a zero after a number doubles the original number. Putting a one after a number doubles the original value and then adds one. We start with zero and one in their ordinary meanings.

``` 2 = 10    since putting 0 behind the 1 doubles the 1
3 = 11    suffixing the 1 doubles the first 1 and then adds 1
4 = 100   by doubling 2
5 = 101   doubling 2 and adding 1
6 = 110   doubling 3
7 = 111
8 = 1000
9 = 1001
```

This binary notation uses only two different symbols, 0 and 1. Because of this it is used in computers where the 0 is an "off" vacuum tube and the 1 is an "on" tube. Leibnitz, the famous German mathematician, was enthralled with these simple formulas because they seemed to show each digit as a compound of "pure being" and "pure not-being". "He imagined that unity represented God and the zero the Void; that the Supreme Being drew all the numbers from the Void, just as unity and zero expressed all numbers in his system of numeration." [quoted from Dantzig]

But here the zero, the source of numbers, is both the source and an operational digit. The one indicates two entirely different operations - double and then add-one.[top]

#### The Bipolar Generation

Zero, as always, is the source and progenitor of the digits. But if there are two operations and a corresponding duality of symbols for them, then these will incorporate into mathesis the age-old ideas of equilibrium between dynamically opposing forces. This dynamic tension is the source of Newton's Second Law of Motion: "for each action there is an equal and opposite reaction". The philosophers of China enunciated same principle previous to history when they spoke of Yang and Yin.

Only when zero manifests in a duality is there equilibrium of contraries. Hence the digits 1 and 2 are the operative numbers.The 1 is still connected with the operation "addition of one." The 2 is connected with the operation of double or "multiply by two." These condiserations prodcuce the following generation.

```                             0   Source

1
Primary Digits
2

3 = 2 + 1                              4 = 2 x 2

6 = (2 + 1) x 2                        5 = (2 x 2) + 1

7 = (2 + 1) x 2 + 1                    8 = (2 x 2) x 2

9 = (2 x 2) x 2 + 1
```

The first two digits (= 1 and = 2) are called the primary digits. The remaining seven are termed the descendent digits since they are generated by the primary ones. The above arrangement indicates that the descendent digits are divided into two sets. This separation into two sets depends on the operations required to produce the digits.

Other combinations of the two primaries and the operations associated with each may be shown to produce a certain digit. But, it is easy to check that the above equations give the simplest possible generation for each digit. For example, 6 could have been generated by

```  6 = 2 x 2 + 1 + 1
```
but this generation uses both primaries twice and three operations.

Trivial it may seem, but from such simple ideas will develop a wealth of results. These preceding equations can be put into pictorial or diagrammatic form. In the diagram of the Linear Generation an arrow was used to indicate the single operation. Here there are two operations, so there will be two kinds of arrows. Addition-of-one is indicated by an upward arrow. Doubling or multiplication-by-two is an arrow across. Starting from a zero point and two primaries produces the following schema of the Bipolar Generation.

```                    7
^
|
|
1         3 ---> 6  5        9
^         ^         ^        ^
|         |         |        |
|         |         |        |
0  ---->  2  ---->  4  --->  8
```

As it stands above there is a square of {4, 5, 8, and 9} and a triangle of {3, 6, and 7}. The primary digits (=1 and =2) seem to share no common geometric configuration. To bring out these relationships, the Bipolar Generation can be slightly rearranged to this form.

```                      0

1

7
3   6

2

5   9
4   8
```

This arrangement of the digits is derived by moving the sets of digits from the Generation. The digits 0, 1, and 2 each form one set. Then 3, 6, and 7 form one more such logical set of numbers; and 4, 5, 8, and 9 form another such set.[top]

#### The Stupa and the Digits The above arrangement of the digits is the basis of the further development of mathesis. This arrangement is identical to the old Indian, Tibetan, Chinese, and Japanese figure called the stupa. The stupa is a diagram showing the qualitative elements of the cosmos and their relations. The top part is an ellipse which is emblematic of the Void, the source of all elements. The crescent is the element Air, the active force. The triangle represents Fire, the circle represents Water - the unstable, formative element that links Fire and Earth. The square is the solid and stable element of Earth.

The Bipolar Generation of the digits is one fact; the ideas associated with the stupa in the East form another fact. The justification for bringing these two facts together is that they provide a rich stream of further facts. Indeed, the only justification for any theory is that it is useful.

Some interesting relationships can be deduced from the numerical stupa. Three is the first number allocated to Fire, and the Fire element has a total of three digits. Four is the first Earth number, and the Earth square has four digits. Furthermore, three and six are the only triangular digits (in the Pythagorean sense). These are both placed on the Fire Triangle. Also, four and nine are the only square digits, and they are both on the Earth square.

The Bipolar Generation of digits is completed with the number nine. The next number is ten, which can be most easily generated by 10 = 5 x 2. This generation wouid put it among the Earth digits, whereas ten is a triangular number and so belongs on the Fire triangle. Numbers greater than nine are analyzed by the law of decimal octaves.

The stupa is a complex symbol and many meanings surround it. Both the oriental and the occidental imaginantions have seen many things in it. For a more complete treatment of its history and philosophy see the book by Goldsmith listed in the Biblioraphy. [top]

#### The Tetraktys and The Stupa

The Pythagorean tetraktys has some important points of connection with the theory developed here. These connections are legitimate because mathesis accepts the insights of other philosophers.

```                      o           Air
o   o         Water
o   o   o       first Fire digit
o  o   o   o     first Earth digit
```

This also shows the derivation of Fire and Earth, the descendent elements.

```          o
Fire  = o   o  = Water + Air   = 3 = 2 + 1

o   o
Earth = o   o  = Water + Water = 4 = 2 + 2
```

Empedocles started the four-element theory of the cosmos. It is certain he was either a Pythagorean or was well acquainted with their doctrines. He lived near the school in Crotona. Also, ten is the first re-occurrence of the number zero by the law of decimal octaves. The Greeks had no symbol for zero so they used ten.

Finally, an association of specific numbers to the various dots of the tetraktys develops a definite resemblance to the arrangement of digits on the stupa. This similarity is more easily show by changing the triangular array of the tetraktys into a right angle triangle.

```                      o
o o
o o o
o o o o
```

The digits from 0 to 9 are assigned to the dots in the above pattern in numerically increasing order.

```                          0

1    2

3    4    5

6    7    8    9
```

The Earth set {4, 5, 8, 9} stands out especially. The diagonally opposite pairs add to 13 as they do on the stupa. The primary digits 1 and 2 share no geometric configuration with the seven descendent digits.[top]

#### The Bar Notation

Since each of the descendent digits is an alloy of one or both of the primary digits, the descendants are alloys of Air and Water. But it is hard to visually grasp the formation of the descendent digits from either the formulas on page __ or the diagram on page __. By using the symbols of the old Chinese I Ching each digit can be diagramed so that its composition is evident.

```   0  will not be represented
1  will be represented by ------
2  will be represented by --  --
```

One is connected with the operation of add-one. Two is connected with the operation of multiply-by-two. The sequence of these elementary operations is indicated by stacking the single-bar and double-bar symbols in vertical sequence. The first operation is on the bottom; succeeding operations are placed in order upwards. Some examples follow.

```--  --   Four was derived by 4 = 2 x 2.  In the bar notation this
--  --   is shown by two double bars.

------   Three was derived by 3 = 2 + 1.  The bar form for three
--  --   shows this by a double bar on the bottom and a single
bar above it.

--  --   Six was derived by 6 = 3 x 2 = (2 + 1) x 2.  The bar
------   equivalent graphically displays this sequence of oper-
--  --   ations.
```

The following table presents each digit and its equivalent symbol in the bar notation. The number of elementary operations used to generate each digit is significant. This relationship is also shown on the table.

```         Digit     Bar Notation   Classification

0                      Source of the digits

1       ------     }
}   Primary digits
2       --  --     }

3       ------     }
--  --     }
}   Secondary digits
4       --  --     }
--  --     }

------     }
5       --  --     }
--  --     }
}
--  --     }
6       ------     }   Tertiary digits
--  --     }
}
--  --     }
8       --  --     }
--  --     }

------     }
--  --     }
7       ------     }
--  --     }
}   Quaternary digits
------     }
--  --     }
9       --  --     }
--  --     }
```

The digits are thus divided into four basic categories depending on the number of operations to produce the digits. One and two are primary digits. Three and four are secondary digits because these both require two operations. Five, six, and eight require three operations and so form the tertiary digits. Seven and nine are the most complex of the digits in structure. Three and seven are composed of an equal number of single and double bars, and so are balanced in composition. Seven is doubly balanced between the primordial Air and Water that create all the digits.[top]

#### Relative Weights of Digits and Numbers

The bar representation of the digits shows that they are composed of varying proportions of unity and duality. The relative weight gives a precise measure of these proportions. The greater proportion of the Watery element there is in a digit, the heavier it is. The rule for calculating the relative weights of the digits from the bar notation provides a simple method for find the numerical values for the relative weights.

```Rule:  To calculate the weight of a digit take the number of
all the bars and divided by the number of lines.

```

For example, in nine there are seven bars distributed in four lines, so 7 / 4 = 1.75, which is the relative weight of nine. The rest of the weights are given in the table following. The reason behind the rule is this. To one is given a weight of 1, and to two is given the weight of 2. These values are merely convenient choices; other values would show the relative weights just as well.

Some digits are equal in relative weight. These digits are not equal in meaning or quality. Although such digits have the same constituents, these elements are arranged into different structures to produce different numbers. Thus, 5 and 6, though they both have two double-bars and one single bar, are built with a different arrangement of the primary operations. In the case of 2, 4, and 8 the ratio fails because there is no Air alloyed into these digits. But again, they are not identical in meaning.

The following table presents the digits in order of their relative weights.

```               Digit         Weight
1              1
3, 7             1.5
6, 5             1 2/3
9              1.75
2, 4, 8             2
```

The lighter numbers are more Airy, masculine, active, and positive. The heavier numbers are more material, feminine, formative, and passive. This is similar to the Chinese idea of yin and yang. The Pythagoreans said: "All is number and all is harmony, because every number is a definite union of the odd and even. And what is union but harmony?" The bar notation for the digits shows the exact formulation of the digits from the odd (=1) and the even (=2) in a graphic picture.

The weight of larger numbers is calculated from the law of decimal octaves. For example, in a number such as 174, the basic element which gives the most important meaning. The 7 modifies the 4. The 1 in turn modifies the 7 and, indirectly, the 4. The calculation of the weight of 174 depends on the fact that there are three digits. Thus the 1 will be counted as worth one unit, the 7 will be counted 10 times; and the 4 - being the most important digit - will be counted 100 times. This will produce the correct weighted average weight. The weight computed from 100 times the weight of 4 plus 10 times the weight of 7 plus one times the weight of 1 averaged together.

``` 100 x 2  +  10 x 1.5  +  1 x 1  =  200 + 15 + 1
=  216
```

But 216 is not the relative weight because it has not been averaged. The average is taken by dividing 216 by 111 because the average is over 100 items plus ten items plus one item. So the weight of 174 is

```                  . .
216 / 111  =  1.945
```

The periods over the 9 and the 5 indicate that this is a repeating decimal fraction.

```    . .
1.945  =  1.945945945 ...
```

In a similar fashion the weight of any whole number at all can be readily calculated. For example, the weight of 29 is given by

```                              ..
(10 x 1.75 + 2) / 11  =  1.772
```

The weight of a number places it on the scale from the most active to the most passive. The bar notation leads directly to the means of calculating precise numerical measurements of the relative weights. The numerical weights show how far separated the one is from the rest of the digits. One has a relative weight of 1.0; three and seven have weights of 1.5; and all the rest of the digits are grouped together in the lower half of the interval from weight 1 to weight 2.[top]

#### The Four Elements and the Digits

The same bar notation can also be used to discover the alchemical elements which are related to each digit. For one and two the correspondence with the elements is quite definite because these two digits are the only ones associated with their elements of the stupa. The seven descendant digits have both a primary and a secondary element association. The Greek philosophers used a similar diagram to show the relationships among the elements that has since been called the Hermetic Square.

Elements diagonally opposite are complementary. Thus Air and Water are opposite or complementary. This arrangement echoes the stupa since going clockwise from Air produces the same sequence of elements as going down the stupa from Air.

The names must not be taken too literally. The only reason that they are used here is that they imply a division into four elements or parts. Even the Greeks knew quite well that these four elements were not really the basic building blocks of the universe if considered literally. To be more specific, we let Air represent the active element, Water the passive element, Fire - their combination - is mind or the mental element, and Earth stands for the physical or material element.

The three digits assigned to the Fire Triangle and the four digits assigned to the Earth Square suggest that these might have a secondary affinity to specific elements. The relative weights suggest that:
3 or 7 could be assigned to either Air or Fire
6 could only be given a secondary element of Water
5 could only be given a secondary element of Air
9 could only be given a secondary element of Fire
4 or 8 could be assigned to either Water or Earth.

The assignment of the four Earth digits is motivated by the requirement that a cyclic motion around the square match the linear movement down the stupa. This results in 8 having as both primary and secondary element Earth. So we are motivated to let 7 have both primary and secondary element Fire. A synoptic table in matrix form shows the numbers classified by both primary and secondary elements. The Fire and Earth quarters of the Hermetic square are subdivided into smaller Hermetic squares.">

Here 1, 5 and 7 are the active principles functioning in the mental (=7) and material (=5) worlds. The trio 2, 6, and 8 are the formative principles in the mental (=6) and material (=8) worlds. The 3 and 9 are the ordering principles in the mental and material worlds. The 4 is the principle of matter or inertia. [top]

Perhaps the most striking results from the study of elements are reached by looking for opposites or complements. A complement of one number is the number which implies the opposite meaning or quality. For instance, in the primary digits, 1 is the complement of 2 and vice versa. In the seven descendant digits the complements are found by diagonals of the Hermetic square. Thus the complement of Air is Earth, the complement of Fire is Water, and conversely.

The pairs of opposites on the first Hermetic matrix show that all the complementaries add to 3 or to 13, its equivalent.

```   1 + 2 =  3
5 + 8 = 13
4 + 9 = 13
6 + 7 = 13
```

Thirteen is the three of the first decimal octave. Thus thirteen and three indicate a neutral or transcendent mode of being. Although there are four pairs of complementary numbers, all these pairs are reconciled and harmonized by the same digit, just as all complementary colors combine to give gray. There is no additive complementary to three because there is no secondary element of Earth in the set that forms the primary Fire element. However, in a sense, the zero is a complementary to three. This is because three summarizes or contains all the non-zero digits. Hence three symbolizes all the digits as opposed to the Void from which they sprang.

These relationships appear more clearly in a diagrammatic presentation. Three diagrams follow; each demonstrates a facet of the qualitative structure of the digits. The first of these show the interlocking trinities formed by complementary pairs of digits. The other two show two types of complements in a three-dimensional format.   The middle form visually shows how one and two stand apart from the descendent digits and how the three is the central point of all the opposing pairs of numbers. Simon Magus, the first of the Gnostics, wrote of a similar qualitative structure.

Of the Universal Æons there are two shoots (= 1 and 2), without beginning or end, springing from one Root (= 0), which is the power invisible, inapprehensible silence. Of these shoots one (= 1) is manifested from above, which is the Great Power, the Universal Mind ordering all things, male; and the other (= 2) from below, the Great Thought, female, producing all things. Hence pairing with each other, they unite and manifest in the Middle Distance, incomprehensible Air (= 3), without beginning or end. In this is the Father Who sustains all things, and nourishes those things which have a beginning and an end.

Out of the incomprehensible Air emanate three pairs of Æons: Mind and its complementary Thought; Voice and its complementary, Name; Reason and its complementary, Reflection. These then were the seven Æons or Angels of the Gnostics.

The third figure shows the complementary relationships of the descendent numbers. This arrangement is the skeleton of the Ostwald color solid. [top]

#### Subtractive Complements

Those numbers whose sum is three or thirteen are additive complements. By definition, a subtractive pair of complements are two numbers whose difference is three. The following list gives all combinations of digits whose differences are three.

```               3 - 0  =  3
4 - 1  =  3
5 - 2  =  3
6 - 3  =  3
7 - 4  =  3
8 - 5  =  3
9 - 6  =  3
```

The last three pairs of complements can be most conveniently displayed in a hexagonal diagram.

```                              8

9         7

3

4         6

5
```

Putting the numbers in this way places the digits in circular order and puts the additive complements directly above each other. There does not seem to be any convenient way of drawing in the one and two. A separate figure is used to show the subtractive opposites for one and two, the primary digits.

```
1         5

3

2         4

```

Another set of both subtractive and additive complements is given by this rectangle. The two numbers on the left side are both complements of the two numbers on the right side.

```
4         7

6         9

```

The results from the study of subtractive complements are quite consistent with the previous results. Digits 8 and 5 are still complementary. Four is opposite 7, but 7 is very similar to nine, so this is still consistent. Six is opposite 9, but 9 corresponds to 7 as before.

In general, if two numbers add up to (or differ by) an even number, they are somewhat similar. If they add up to (or differ by) an odd number, they are opposite in meaning. But those combinations called complementary show the most directly opposite pairs of numbers.[top]

#### Review

This Chapter contains much of the meat of the book, so a moment should be spent in review. The digits have been derived from zero by using a duality of operations connected with a duality of primary digits. From this Generation the digits exhibit varying mixtures of the dual principles of positive and negative, yang and yin, masculine and feminine. This applies number to the many things that are in a state of flux due to alternations between members of dualities. No longer are numbers perfectly stable, unvarying ideas. They show different modes of action, different stages in the eternal flux. To a large extent this is the trend of modern thinking. Scientists do not look for stable objects or substances; they look for the unchanging laws which describe how everything changes. Greek geometry and Arabic algebra have been replace by the integro-differential equation and set theory.

Some numbers are the complements of others, and each of these pairs of complementary numbers neutralize to a single digit, just as all pairs of complementary colors combine to form gray. This relationship carries the idea of duality to the descendent digits and puts them into meaningful relationships.

Other such interconnections were developed: the correlation of each digit with the Hermetic square and its elements that shows the basic ideas of the digits. And finally, the important relationship shown by having the seven descendent digits divided into a light set of three in the Fire triangle and a heavier set of four on the Earth square. The whole next Chapter is an amplification on this one idea.

We need to keep our goals in mind. This report has as its purposes two things. First, to show that there is a qualitative system to be found in numbers. Second, to show how this system can be applied to qualitative problems. The first Chapter sketched the qualitative system developed from numbers by using the Bipolar Generation of Digits. The rest of the book amplifies this Chapter and applies its results to actual problems. Chapter Two presents three rather broad applications of the mathetical analysis. Chapters Three and Four present a very detailed analysis of just one corner of the whole problem which might be solved by mathetical means. [top]