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.
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Return to Mathesis Home
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
Additive Complements
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]
Additive Complements
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]
