CHAOS
(OR: THE RECIPROCITY OF ANALYSIS AND
SYNTHESIS IN COMPLEX AND DYNAMIC
FORMS OF RELATION)
Once Chuang Chou dreamt he was a butterfly, a butterfly flitting and fluttering
around, happy with himself and doing as he pleased. He didn't know he was
Chuang Chou. Suddenly he woke up and there he was, solid and unmistakable
Chuang Chou. But he didn't know if he was Chuang Chou who had dreamt he
was a butterfly, or a butterfly dreaming he was Chuang Chou. Between Chuang
Chou and a butterfly there must be some distinction! This is called the
Transformation of Things. --Chuang Tzu [C525:45]
4.1 Complex Synthetic Integration in General
In 2.1 we distinguished between different levels of analytic division by
noting the increasingly complex structure of each successively higher level. The
same distinction between "simple" and "complex" can now also be applied to the
various levels of synthetic integration. Just as the complex analytic forms of
relation are all based on the simple root "2", so also the complex synthetic forms
of relation are all based on the simple root "3". (Compound forms of relation, by
contrast, are all based on a combination of the roots "2" and "3"; we shall
discuss them in Part Two). The main difference between analytic and synthetic
complex forms of relation is that the latter are not always as regular and clear-cut
as the former. Because synthetic relations are not bound by the analytic "laws of
thought", they can be organized according to a wide variety of patterns. In this
chapter we will consider some of the less complicated examples of complex
synthetic integration in its regular [4.2] and irregular [4.3] manifestations, and
then conclude with a discussion of its dynamic character [4.4]. As we proceed,
the fundamental reciprocity between analytic and synthetic operations should
become more and more clear.
Complex synthetic integration is the logical title given to any set of simple
synthetic operations in which each operation is related to one or more of the
others by sharing one or more of its three terms. Any number of syntheses can
combine together in this way, regardless of whether or not their total number
enables them to be fit into some analytic mould. (Compound relations are those
complex synthetic relations in which the conclusion of one synthesis (1) is the
starting-point (0) only for the next synthesis, and in which the number of
synthetic operations corresponds to a level of analytic division [see 5.1].) In our
everyday experiences, so many syntheses are constantly uniting together to form
a single web, or network, of relationships that it would be virtually impossible to
reduce these to a logical formula, much less to represent them accurately with
some complicated geometrical model. Nevertheless, our treatment of a few
examples which are rather less intricate will be sufficient to clarify the rules
followed by all complex synthetic operations.
Every rectilinear geometrical figure which is either two- or three-
dimensional can be constructed out of triangular segments. Since no other
geometrical figure can claim such infinite applicability, the triangle is perfectly
suited to symbolize the infinite variety we find in experience. And since the
underlying fourth term in a synthetic relation, the "0", remains hidden (x) until
such relations are subjected to analysis, complex synthetic relations can be
symbolized by the use of multiple triangles.
The most appropriate type of triangle to use in such modes of complex
synthesis is the equilateral. This is not because of its regularity, for many
synthetic relations are irregular. Rather it is because of a remarkable logical
property shared by all triangles. If the angles of a triangle of any size or shape
are trisected, and the points at which each line intersects with another are made
the vertices of a new triangle, the inscribed figure will always be equilateral.
Only is size will vary. The three examples in Figure 4.1a-c should make this
evident.@ This geometrical operation of triangular trisection is a symbol of the
logical operation of analyzing a synthetic integration. The three new angles at
each vertex represent the three steps, and the four lines represent the four discrete
terms which are distinguished in the analysis. Thus each vertex can be labelled
with the four terms of an analyzed synthetic relation, as in Figure 4.1d. Mapping
a complex synthesis onto a geometrical figure is always a way of analyzing its
properties. Since all triangles, when analyzed (i.e. trisected), are inscribed by
equilateral triangles of various sizes, a geometrical representation of complex
synthesis can accurately symbolize all possible synthetic variations simply by
varying the size and position of the equilateral triangles. The original irregularly
shaped triangles would then represent complex synthesis as such (i.e. apart from
our analysis of it in geometrical terms), which cannot be intelligibly diagrammed.
Following this suggestion, we can symbolize most real experiences by a
huge mass of triangles assembled together on a flat surface, some sections of
which display obvious order, and others, little or no order [see Figure 4.2]. The
portion of the diagram which contains the triangles symbolizing the focus of
one's attention is in this case depicted as a regular polygon, divided into its
constituent triangles, which converge at a single point in the center. Around this
regular figure is a less organized mass of triangular syntheses. This diagram
represents what Kant called the "manifold of intuition" [see e.g. K105:103].
Within this manifold, which is not yet "organized" by our conceptual powers,
each synthesis has its own "1", but the complex synthesis as a whole also has a
"1" at the point where the most simple syntheses converge (which in the case of
Figure 4.2 is the center of the hexagon). This area of focus within the manifold
becomes data for conceptualization, and hence, for analysis.
Figure 4.1: Trisection of the Angles of a Triangle
(a) An Acute Triangle (b) An Equilateral Triangle
(c) An Obtuse Triangle(d) A Trisected Vertex
Figure 4.2: A Partially Organized Complex Synthesis
It should be noted that in complex diagrams constructed out of triangles,
a single line or point often belongs to two or more different triangles. Although
this makes it difficult, if not impossible, to label such diagrams with the symbols
"+", "-", and "x", it nevertheless accurately reflects the fact that in experience a
single item (i.e. a single intuition within the manifold) often serves more than
one function. For example, my intuition of this piece of paper is related on the
one hand to my intuition of this pen and the ink which is coming out of it onto
the paper, and on the other hand to the table which is providing the support for
both it and numerous other similar scraps. The difficulty of labelling a diagram
such as we have in Figure 4.2 is a direct result, therefore, of its purely synthetic
character--i.e. of its dependence on the synthetic laws of non-identity and
contradiction.
Figure 4.2 should not be taken to imply that complex synthetic
integration is a haphazard mishmash of elements connected together without any
rules. On the contrary, the rules it follows have been set out in 3.3. The point is
that when these rules are followed they do not necessarily give rise to the kind of
system which looks neat and orderly, or "perfect", to our analytic minds, though
they may and often do form interesting patterns of one sort or another. This
flexibility results from the fact that in any synthetic process the outcome is
determined primarily by the material with which we are dealing. Unlike analytic
processes, wherein we impose some pattern or order upon our material, in a
synthetic process, we watch more or less passively as patterns emerge, often
surprising us with unexpected results. In analysis the thinker controls the object;
in synthesis he allows himself to be controlled by the unfolding of the object's
transcendent order. But in neither case is it correct to assume that detached
"facts" can be merely "read off" from a supposedly objective world. For the
nature of "facts" always changes in accordance with the observer's changes in
perspective.
The question we now face is: To what extent does complex synthetic
integration provide us with patterns which are orderly enough to be mapped onto
geometrical figures? In answering this question I will begin in 4.2 by examining
a few regular patterns (which could be represented as being at the center of a
manifold), and give examples of experiences or natural objects in which such
patterns can be discerned. The reciprocity between the analytic and synthetic
perspectives will be most evident if we confine our attention for the most part to
figures which have already been discussed in some context. In 4.3 we can then
discuss the role of irregular geometrical figures as it relates to the approximation
often required by complex synthetic integration. Finally we will conclude Part
One in 4.4 with a general discussion of the reciprocal nature of analysis and
synthesis as it is evinced in their respective static and dynamic functions.
4.2 Regular Geometrical Models for
Second-Level Synthetic Integration
The most regular type of complex synthetic integration is that in which
the simple triadic function is reflected in on itself, much as in the progression of
the levels of analytic division. Thus the "second level" of synthetic integration is
based on a ninefold form of relation (3.3), the "third level" on a twenty-seven-
fold form of relation (3.3.3), and the fourth level on an eighty-one-fold form of
relation (3.3.3.3). The synthetic expressions pertaining to each level are listed
analytically in Table 4.1, much as we did in Table 2.1 with the analytic forms of
relation.
The nine second level synthetic relations can, interestingly enough, be
mapped onto the standard figure used for third level analytic division, by
combining the double cross [see Figure 2.17b] with the
Table 4.1: Levels 0 - 3 of Synthetic Integration
double square [see Figure 2.19], thus forming a grid with nine squares [see
Figure 2.27]. Instead of labelling the eight endpoints, or the sixteen vertices, or
the twenty-four sections of line, the nine spaces can be labelled with the synthetic
notation introduced in 3.2 [see Figure 4.3]. The new rule for mapping is that the
synthetic "x" is always placed between the analytic "+" and "-" when mapped
onto an analytic figure such as this.
This grid map is invaluable as a tool for demonstrating the reciprocity
between second level synthetic integration and various levels of analytic division.
Notice for example that the expressions with an "x" in them form a cross in the
middle of the figure. If the "x" is ignored in each of these expressions, the
remaining terms form the same pattern as the second terms in a second level
analytic relation, as mapped onto the standard cross [see Figure 4.3a]. Similarly,
the four corners of the grid (i.e. the four expressions without an "x" in them) are
arranged in exactly the same pattern as in the inversion of the standard cross, the
second level analytic square! This is not a coincidence. On the contrary it is
rooted in the logical reciprocity of
Figure 4.3: An Analytic Map of Second Level Synthetic Relations
(a) With Second Level Analytic (b) With Third Level Analytic (c) With Fourth
Level Analytic Expressions in ()
analysis and synthesis: second level synthesis combines second level analysis
and its inversion by specifying the point at which they intersect, the "xx" which
is left unspecified at the center of the analytic cross. It does so by representing
one-dimensional lines as two-dimensional spaces. When the "x" in each
expression is taken into consideration, it turns out to be a reflection of the
combination of the opposite third terms in each of the third level analytic
expressions which label the endpoints of the lines on either side of each synthetic
space [see Figure 4.3b]. Thus the "x" in "x+" reflects not only the synthesis of
the "-" and "+" in the first terms of "-+" and "++", but also the "-" and "+" in the
third terms of "++-" and "+++". Finally, the third and fourth terms of the fourth
level analytic expressions, which label the four corners of each synthetic space in
Figure 4.3c, can be combined to produce the synthetic labels for each space. In
the spaces at the corners of the grid all four fourth level analytic expressions have
the same third and fourth terms, so that (for example) "++-+", "+--+", "---+",
and "-+-+" combine to define the "-+" space. In the other four perimeter spaces,
one set of fourth level terms (i.e. either the third or fourth in each expression) is
the same; but the other is different, so it defines an "x", as when "+-++", "+++-
", "-++-", and "--++" combine to define the "+x" space. The expressions
labelling the corners of the central space have no terms in common, so they
define the central space as "xx". (Incidentally, the full, fourth level analytic,
definition of these four spaces using synthetic notation would include two x's
before each of the nine synthetic expressions. These can be dropped without
ambiguity, however, since they do nothing to differentiate the status of the nine
expressions in relation to each other).
The reason this model is so useful in showing the reciprocity between
analytic and synthetic forms of relation is that it uses synthetic notation on a
figure which is defined primarily in analytic terms, since its primary geometrical
properties correlate with the principles of analytic division. But the very
analyticity of this model, which makes it helpful in the above respects, also
makes it an inadequate model for synthetic relations as such, since it fails to
represent the various threefold relations with triangular shapes.
The simplest way of constructing a synthetic figure to represent second
level synthetic integration is to trisect an equilateral triangle (i.e. divide it into
three equal triangles). One problem with this method is that the resulting figure
has only six lines and only four points of intersection between them, so the nine
expressions cannot be mapped unambiguously onto such a figure. Each of the
three lines must be labelled twice (once for each triangle they border), as in
Figure 4.4a; or else each of the outer vertices must be labelled twice and the
center three times (once for every triangle they define), as in Figure 4.4b. As we
shall see towards the end of this section, such ambiguity can sometimes be used
to symbolize ambiguous, or double aspect, relations between real objects. But in
many other cases such ambiguity does not correspond to real relations.
One way of alleviating the ambiguity in Figure 4.4 is to separate each of
the three triangles while retaining their general triangular relation to each other
[see lines in Figure 4.5]. The result is a figure with nine lines and nine vertices,
so that the expressions can be mapped unambiguously onto either set of
elements. Note that the three synthetic terms in the center of Figure 4.5a form a
new synthesis, and hence are represented as labelling the vertices of a fourth
triangle.@ In Figure 4.5b the same relation is represented as the outer edges of
the large triangle. This "higher" (or "deeper") synthesis plays an important part
in systems of logic, such as Hegel's, which are primarily synthetic. Rather than
discuss its implications here, however, we shall wait until Chapter Seven, where
the systematic unity implicit is such synthesis can be discussed in a rather more
balanced context. Moreover, as we shall see below, the same relation is often
nothing more than a second level representation of the first level synthetic
relation.
Figure 4.4: The Trisected Triangle as a Map of Second Level Synthesis
(a) Labelling the Lines(b) Labelling the Vertices
Figure 4.5: The Triple Triangle as a Map of Second Level Synthesis
(a) Labelling the Vertices (b) Labelling the Lines
Turning now from two to three dimensions, we find several solid
geometrical figures which can act as suitable representations for second level
synthesis. One of the best examples is the regular hexahedron, since its nine
edges are grouped into three sets of three. The rules for mapping the nine
synthetic expressions onto this figure correspond to the rules for mapping
analytic expressions: on the first (horizontal) level, right is "+", left is "-", and
middle is "x"; on the second (vertical) level, top is "+", bottom is "-", and middle
is "x". The "middle" in both cases can be represented as a discrete line (or
vertex) by mapping it onto the third dimension. In Figure 4.6 the second level
expressions (in curved brackets) are mapped onto the vertices according to these
rules. The proper label for each line is then determined by synthesizing the
values of its two endpoints, using the following rules for calculating the new
values: a term plus itself yields itself; "x" plus any other term yields the other
term; "+" plus "-" yields "x".@ It is important to note that the six triangular faces
of this hexahedron (e.g. those made up of the lines "-+", "x+", and "+x") do not
represent simple synthetic relations when mapped in this way. Instead the first
two steps in each simple synthetic operation are represented by two continuous
vertical edges (viz. "-+" and "--", or "++" and "+-", or "x+" and "x-"), and their
synthesis is represented by a horizontal edge which appears to be between the
other two edges (viz. "-x", "+x", and "xx", respectively). (Note that the third
dimensional synthetic expressions (i.e. "x+", "x-", and "xx") are not joined by a
common vertex, as are the other two.) By contrast, the "hidden face", which cuts
the figure into two tetrahedrons, is now represented by the lines of a seventh
triangle (viz. the lines "+x", "-x", and "xx"). The importance of this "6+1 = 7"
pattern of faces will be brought out in 5.3.
Figure 4.6: The Hexahedron as a Map of Second Level Synthesis
(a) The Solid(b) The Net
Figure 4.7: The Half-Cube as a Map of Second Level Synthesis
(a) Labelling the Vertices(b) Labelling the Lines
The two-dimensional "net" for this figure [4.6b] consists of two large
(first level) equilateral triangles whose bases are bisected by each other's apexes,
so that the four remaining sides also bisect each other. Including the line
connecting the points of intersection between these four sides, six smaller
(second level) equilateral triangles are formed. This "unfolded" form has thirteen
lines instead of nine, because the four lines which are artificially separated (viz.
x+, x-, +x, and -x) have to be written twice (once for each face they border).
The "hidden face" is unfortunately destroyed in this process of analyzing the
solid. But the advantage of depicting the hexahedron in this analytic form is that
all the "mixed" expressions (i.e. those containing an "x" together with a "+" or "-
") are relegated to the perimeter, so that the five "pure" terms (i.e. xx, ++, --, +-,
and -+) become the focus of attention. The "xx" line is the perfect geometrical
representation of the synthesis of syntheses since it actually connects the
midpoints of the two converging triangles. Moreover, the internal "diamond"
shape is labelled with precisely the same expressions--even in the same order--as
those given in Figure 2.9 for the inverted form of second level analytic division.
Here again is evidence that analytic and synthetic forms of relation, like the
formal laws on which they are based, are thoroughly reciprocal.
The hexahedron is just one example of a figure with the characteristics
appropriate to represent complex synthetic integration. Other figures do so with
equal clarity. One example of such an alternative is the figure which is created by
bisecting six of the twelve edges of a cube with a plane passing through it at a
45\ angle to the top face. The resulting figure has a large hexagonal face, three
triangular faces, and three pentagonal faces [see Figure 4.7].@ The edges
connecting each triangle to the back corner constitute the fourth synthetic relation
which crops up in any good representation of second level synthesis. Once
again, this extra relation is actually nothing but the second level representation of
the first level synthesis.
The application of the second level of synthetic integration to concrete
examples is rather more difficult than was the application of the higher levels of
analytic division. For synthesis is not as clear-cut as analysis--at least not when
taken on its own terms, without the structure provided by analysis. Most of the
examples of simple synthesis given in 3.4, for instance, do not lend themselves
readily to being broken down into nine components, related in three sets of three.
Nevertheless, it will be helpful to discuss examples of how each of the five
figures introduced in this section can be used as maps for logical relations.
The double cross as used in Figure 4.3 (i.e. attending to the nine
squares, rather than the eight endpoints) is essentially the same figure as that
used in the child's game "noughts and crosses" (or "tic-tac-toe"). The "0" and
"X" markers in this game are both symbols we have used to describe the
operation of synthetic integration. Unfortunately, the game is only played by two
players, so its logic does not correspond perfectly to second level synthetic
integration. As a result of the simple logic on which it is based, the game is
usually able to draw the interest only of children. Both of these difficulties can be
avoided by devising alternative ways of playing the game which depend on the
slightly more sophisticated logic of second level synthetic integration. I shall
discuss two such variations.
Both variations of tic-tac-toe are played with three players, using "+", "-
", and "x" as markers, and changing the rules to compensate for the fact that
making a line of three similar symbols would now be virtually impossible under
normal circumstances (i.e. assuming intelligent players who desire to win). One
new rule could be that each square can be filled in with two symbols, either of
which can be used to make a line of three squares using the same symbol. To
make the game more fair (since "+" and "-" have a definite advantage over "x" as
it now
Figure 4.8: Synthetic Variations on Tic-Tac-Toe
(a) A "Win", Using Two Symbols for Each Square:
(1) +'s turn-'s turn x's turn(2) +'s turn-'s turn x's turn xx x-x-
(3) +'s turn-'s turn x's turn(4) +'s turn-'s turn x's turn x-
(b) A "Draw", Using Two Symbols for Each Square:
(1) +'s turn-'s turn x's turn(2) +'s turn-'s turn x's turn - - - - -- x -
(3) +'s turn-'s turn x's turn(4) +'s turn-'s turn x's turn - x -- x -- x --+x --(5)
+'s turn-'s turn x's turn(6) +'s turn-'s turn x's turn -+x--x-+x--x-+x--x-(c) A
"Loss", Due to Three Different Consecutive Symbols:
(1) +'s turn-'s turn x's turn(2) +'s turn-'s turn x's turn + + x + x + x + x +
(3) +'s turn-'s turn x's turn
stands), the rules could also require one of the three consecutive squares to be
filled with two of the winner's symbols [see Figures 4.8a and b]. A rather
different alternative rule would be for a player to lose if he places his symbol in
such a way as to make a line of three different symbols (as in a simple synthetic
relation!) [see Figure 4.8c]. The latter version of the game, with its nine plays,
reflects the structure of second level synthesis more accurately than the former.
But the former is much more challenging to play, especially since an intelligent
and competitive "+" and "-" can always force "x" to lose in the latter version! An
interesting characteristic of both versions is that one player (usually the "-"
player) is often put in a rather awkward position (that is, if he is playing with
friends). The "-" player in the game played in Figure 4.8a, for example, could
have allowed the "+" player to win instead of the "x" player simply by altering
his final move. Likewise, the "-" player could have chosen to lose in the game
played in Figure 4.8c by putting his third symbol in the other open square. This
necessity for arbitrary choice, along with the temptation to engage in conspiracy
against one player (which spoils most games), is perhaps why these versions of
the game will probably never become popular. But their radically synthetic
structure at the same time makes them interesting models of the synthetic
experiences we often have in real life.
A more profound example of complex synthesis has to do with the three
modes in which any chemical substance can exist--viz. as gas, liquid, or solid.
These natural relations can be mapped directly onto Figure 4.4. The geometrical
ambiguity in labelling the three interior lines now works to our advantage, since
it accurately represents the logical ambiguity which arises at the point of
transformation between any two modes. For the transformation can go in either
direction, yet occurs at precisely the same point under the same conditions. We
can map these relations onto the trisected triangle in the following way. The large
triangle in Figure 4.9a represents a given chemical compound (e.g. a molecule),
which has the same essential structure no matter which mode it is manifesting at
any given point of time. The simple synthetic relation, symbolized by the three
interior spaces, or by their outer sides, is between the stability of the solid (-),
the extreme activity of gas (+), and the middle road between the two, the liquid
(x). The three conditions which combine together to determine the mode of
existence of any substance are the atmospheric pressure exerted upon it (+), the
volume it fills in order to resist this pressure (-), and the temperature (x), which
largely determines the details of the other two. These three conditions can be
regarded as coordinate axes defining the state of a given substance under given
conditions. If temperature is measured on the vertical axis, the horizontal axis
could measure either pressure or volume, with the one not chosen either
remaining constant or else being measured on a third axis. (We will assume
constant pressure.) The path followed through the triangle differs under different
pressures, as well as between different substances. The second level synthetic
relations represented by the internal lines indicate that each of the three modes of
existence can be transformed into either of the other two, given proper
conditions.
The familiar chemical compound "HO" can serve as an example of how
the general transformations mapped onto Figure 4.9a operate in a particular case.
At room temperature under normal atmospheric pressure, HO (the large triangle
in Figure 4.9b) exists in the form of water. Raising the temperature under
constant pressure causes an increase in the distance between the electrons of a
molecule and a decrease in order (i.e. entropy), so that the volume increases,
especially when it reaches the boiling point (B) and begins to vaporize into
steam. Lowering the temperature causes a decrease in the distance between
electrons and an increase in order, so that the volume decreases, until it reaches
the freezing point (F) and solidifies into ice [see Figure
Figure 4.9: A Map of the Synthetic Structure of Chemical Substances
(a) The Transformations of (b) As Exemplified by HO Substances in General
Figure 4.10: Second Level Synthetic Family Relations
4.9b]. At this point the volume increases again because of the peculiar way in
which HO restructures itself in its solid form. For HO, like most substances, the
normal transformations are between solid and liquid, or between gas and liquid,
but not between solid and gas. This is represented accurately by our logical
notation, since the path from "+" or "-" to "x" is far more "normal" than that
from "+" to "-", the two contradictories. Yet our diagram implies that the latter
transformation, though abnormal, should be possible. Indeed, this turns out to
be the case. For if the pressure is altered appropriately, it is possible for any
substance to move directly from solid to gas or vice versa. This process is called
"sublimation". Moreover, our diagram also implies that there should be some
point, at a single combination of a specific temperature, volume, and pressure,
where every substance exists simultaneously in all three modes (so that "+ = - =
x"). Once again, though it seems impossible to our analytically oriented minds,
this striking implication of complex synthetic logic, that a single substance can
exist in three modes simultaneously, turns out to be confirmed by scientific
experimentation and proof. The very name given to this unique point also
confirms the felicity of such geometrical models; for physicists name this state
the "triple point", which is precisely what it is in the model provided by the
Geometry of Logic!
Neither substances nor their chemical modes of existence are actually
shaped like a trisected triangle. In other words, the maps used in Figure 4.9 do
not merely copy the size or shape of some observed object, the way a road map
reproduces the relative length and bearing of roads. Rather, the triangle is an
appropriate model because it exhibits a logical structure which is identical to the
logical structure exhibited by the modes of transformation of chemical
substances. Both sets of relations are rooted in the second level synthetic form of
relation. This example, therefore, like all others we have been and shall be
discussing, testifies both to the conformity of the natural world to logical patterns
(a quite remarkable affinity!) and to the importance of the Geometry of Logic, as
a systematic means of highlighting the logical structure of real relations by
mapping them onto geometrical figures which exhibit the same logical relations.
The archetypal relation between "father, mother, and child" [see 3.1] can
be used again here as another example of a complex synthetic relation, structured
this time along the lines of the tripled triangle [see Figure 4.5]. For the father and
mother are both a child from the perspective of some previous simple synthetic
relation, and the child has the potential of fulfilling the role of a parent in some
future synthesis. The "hidden" synthesis in the resulting ninefold relation is thus
the first level relation under consideration, as is accurately depicted by the central
triangle in Figure 4.10.@ The "x-" of the child's potential spouse and the "x+"
of the grandchild, it should be kept in mind, describe how these persons are
related to the child's parents, not to the child as parent. The arrows, by contrast,
denote the direction of relation within each simple synthetic relation (i.e. within
each triangle). From the perspective of the new synthesis originated by the child,
the logical expressions would therefore be mapped rather differently. For
example the child would not be "xx", but "+x" if he is male or "-x" if she is
female, with the spouse fulfilling the other category and their child becoming the
new "xx". The participation of every man, woman, and child in this ninefold
relation is perhaps one reason why the number "9" is regarded in some traditions
as "the number of Perfect Man" [P538:74].
An advantage of the tripled triangle as a geometrical model for the logical
structure of real relations is its versatility. As used in Figure 4.10 it represents a
slight variation on standard second level synthesis, since the overall relation
flows continuously in one direction--i.e. from grandparents to parents to child to
grandchild. The same figure can also be used to represent ninefold relations in
which each synthesis moves from the outside in [see Figure 4.5a] or from the
inside out [see Figure 4.5b]: in such cases all the elements mapped around the
extreme inner (or outer) triangle are either derived from or else produce the two
elements mapped onto each of the three main triangles. The former movement
represents standard second level synthetic integration, whereas the latter
represents three sets of first level analytic divisions, related by a simple
synthesis. The nature of the combination of analysis and synthesis, such as is
implied in the latter, will be discussed in Part Two. The former, however, is the
same type of complex synthetic integration as that which is symbolized by the
hexahedron, the "hidden face" in Figure 4.6 representing the same form of
relation as the inner triangle in Figure 4.5a. Thus both can be used as models for
the same examples (though the two-dimensional figure is usually preferable
because of its clarity and simplicity).
The structure of the HO molecule discussed above can be explained by
mapping its elements onto Figure 4.6. Its H-O-H pattern indicates that the
oxygen atom holds together (synthesizes) two otherwise repellant hydrogen
atoms; and this threefold pattern can be ordered in such a way as to produce
water, ice, or steam [see Figure 4.9b], depending on the distance between the
electrons and atoms forming the bond. Thus the three modes of existence for HO
can be mapped onto either Figure 4.5a or Figure 4.6, as shown in Figure
4.11.@ Figure 4.11a is a good representation of this molecule in its ordinary
state, assuming only one of the main triangles applies at any time. Figure 4.11b,
however, is a better representation of the molecule at the triple point where,
paradoxically, it has all these characteristics, while yet preserving its basic H-O-
H structure. The latter figure is also a good representation of the real structure of
HO, since it correlates each pair of "H" lines with the "O" line furthest from it.
(The position of the "+x" and "-x" expressions is interchanged in Figure 4.11b
[cf. Figure 4.6a]; this change can be explained as a change in the rules for
mapping, such
Figure 4.11: The Ninefold Relations in an HO Molecule
(a) Mapped onto the Tripled Triangle(b) Mapped onto the Hexahedron
(c) Conventional Map of HO(d) The Eight Electrons in Each State
that each line is now labelled with the same expression as the vertex opposite it.)
The conventional map for the HO molecule is a tetrahedron, with the oxygen
atom in the center, the hydrogen atoms at two vertices (thus about 109\ apart),
and the eight electrons grouped in pairs between the center and each vertex [see
Figure 4.11c]. The hexahedron is simply two tetrahedrons joined by a common
face. Our map also represents oxygen as in the center (the center face, that is),
and the hydrogen atoms as maintaining maximum distance from the oxygen atom
in each state. And the eight electrons can be mapped onto each set of three lines
by positioning one electron at each vertex defined by each line [see Figure
4.11d]. Raising the perspective from points to lines distorts the angles of relation
between the various elements (the H's are now \ apart); but all the other essential
relations are preserved, and we have gained the considerable advantage of
mapping the structure of HO in all its states simultaneously onto a single
diagram! Although other chemical elements would have to be mapped rather
differently, most would be relevant in one way or another to the Geometry of
Logic. Indeed, an interesting study could be made of the logical patterns
exhibited by such geometrical maps of real molecular substances.
Yet another variation of the tripled triangle turns out to exhibit the same
relations as those mapped onto the "half-cube" in Figure 4.7. The three terms of
the inner triangle can be regarded as basic elements which can be combined with
each other in various ways to produce new, synthetic elements. In such cases the
position of the three main triangles must be rearranged so that two vertices of
each correspond to two on the inner triangle. An example of this variation is
provided by the logical structure of colors. If "color" is taken to refer to
pigmentation, then the three "primary colors" are red, blue, and yellow. If it
refers to spectral bands, the three primary colors are red, green, and bluish
violet. Mixing all the colored pigments produces brown; mixing all the spectral
bands produces white. Moreover, colorless pigment is white, whereas colorless
light is black. Despite such differences in detail, these two types of color have an
analogous structure in several respects. For instance, in both cases color is
described by its hue, its brightness (or lightness), and its saturation. These three
aspects of color could be related to the three primary colors much as HO was
related to its three modes of existence in Figure 4.9b. But instead we will look
more closely at the mixture of the colors themselves, choosing pigments as the
object of our mapping exercise.
The primary pigment colors can be mapped either onto the three vertices
of the inner triangle in Figure 4.5a [see Figure 4.12a], or onto the three
triangular faces of the half-cube [see Figure 4.12b]. The three secondary
pigments (purple, green, and orange) are then represented either by the outward-
pointing vertices of the three main triangles (which themselves are arranged in a
triangular pattern), or by the three pentagonal back faces of the half-cube. The
main difference between these two synthetic maps is in the way they depict
brown. In (a) three shades of brown are represented as being derived from the
synthesis of two secondary colors. (The same shades can be mapped onto the
lines forming the back corner of the half-cube in (b).) If such synthetic
operations were carried out far enough, all the shades of brown would begin to
take on the same (rather unappealing) shade. Significantly, this same shade can
be derived in a single step, simply by combining equal portions of all three
primary (or secondary) colors. This simple combination of red, blue, and yellow
is represented in (a) by the central point of the inner triangle, and in (b) by the
hexagonal front face of the half-cube (which is bordered by three primary and
three secondary pigments). The vertex opposite the hexagonal face in (b)
represents the fact that the same mediocre shade of brown can be derived from
the mixture of the three initial shades of brown (the
Figure 4.12: Second Level Synthesis of Colored Pigments
(a) Mapped onto a Series of Tripled Triangles
(b) Mapped onto the Half-Cube
three lines of the vertex). These two maps are each valuable in their own way,
the former for its potential for minute detail (other colors could be mapped as
subsidiary triangles, such as "turquoise" as the synthesis of blue and green), and
the latter for its limited completeness; but both provide perfect examples of the
fundamental laws of all synthetic relations, that "A = -A" (i.e. "center =
circumference"; "hexagonal face = opposite vertex") and "A = A" (e.g. "red =
red" when it is synthetically related to "blue" in the form of "purple"). There is
no need to go into such detail in examining the properties of higher levels of
synthetic integration. Indeed, the complexity of even the third level, with its
twenty-seven expressions [see Table 4.1], would be difficult to represent in
terms of Euclidean geometrical figures. One possibility, perhaps, would be to
arrange three hexahedrons in a form similar to how the triangles are arranged in
the tripled triangle. The result would be accurate, inasmuch as we would have
three systems of ninefold relations related to each other according to a simple
synthesis; but the complexity of such a figurative representation would far
outweigh its practical value, to the extent that it would be difficult to see any
value at all in it outside the purely formal embellishment of the Geometry of
Logic. Instead, we must take a brief look at one type of synthetic operation
which is complex, even though it is analogous to simple synthetic operations in
certain ways. We will then be prepared to examine a selection of irregular
geometrical models for synthetic operations of a less symmetrical sort.
Nothing has been said so far about synthetic operations which combine
more than two elements of equal status into a single whole. For instance, the
recurrence of sets of five elements, especially in qualitative relations such as
color (the three primary colors plus black and white) or tone (e.g. the pentatonic
scale), calls for some explanation. In many cases, a fivefold relation is actually a
perfect (fourfold) second level analytic relation, with the fifth element being the
synthesis of the other four. The human hand, with four fingers working together
and a single thumb set in opposition to them is an example of this "4+1 = 5"
pattern of relations. Whenever a relation includes both a set of elements related
analytically (whether it be 2, 4, 8, etc.) and a term synthesizing them all, the
latter term can be mapped onto the appropriate analytic diagram by placing it at
the center. The double cross was used in Figure 4.3b precisely in this way: the
eight perimeter squares were supplemented by the central square, which serves
as the synthesis (xx) of all the others. Our discussion of the "levels" of synthetic
integration assumed that complex synthesis is limited to synthesis of a simple
synthetic relation (i.e. multiples of 3). But we can now add that the specification
of the center for any level of analytic relation can be regarded as a simple
synthesis: thus "2+1 = 3", "4+1 = 5", "8+1 = 9", "16+1 = 17", etc., are all
arithmetical representations of potential synthetic relations. The addition of "1" to
these (or any other) numbers stands for the synthetic integration, or unification,
of the elements of the original relation.
Elements which are so similar that no distinction between "+" and "-"
relations within the group can be made (e.g. a bowl with five apples in it) do not
fit very neatly into any of the categories we shall discuss. Nevertheless, their
equal status can be depicted by mapping them onto an appropriate polygon, such
as (in the case of fivefold relations) a pentagon divided into five triangles, or a
five-pointed star (i.e. a pentagram). In the pentagon the point in the center could
represent (where relevant) the synthesis of all its elements, while in the
pentagram the inscribed pentagon would fulfil this role [see 5.3]. In many cases
such a "synthesis" will actually be nothing but the concept which unites these
things in a single class; hence the different shape given to the synthetic element in
these maps (viz. a point or a pentagon, versus a pentagon or a pentagram) is
appropriate.
Defining a notation with which to formalize these alternative types of
regular synthetic forms of relation would require the introduction of extra terms
to supplement the use of "+", "-", and "x". Such a task may prove valuable at
some later stage in the development of the Geometry of Logic, but it is beyond
the scope of the present work. Instead we shall now turn our attention to the role
of approximation and of various irregular geometrical figures in representing
complex synthetic integration.
4.3 Irregularity and Approximation in Synthetic Logic
A key difference between the patterns which arise out of analytic division
and synthetic integration is that the former are concerned with exact
classifications, whereas the latter often make use of approximation. This
difference stems from the different focus of control mentioned in 4.1: analytic
forms are conceptual and hence under the control of the person who uses them to
impose patterns on his world; synthetic forms are intuitive and hence have their
root in the complexities of nature. Approximation is legitimate (indeed,
necessary) in certain mathematical contexts, for all "irrational" numbers,
including such key notions as ??? and /2, can be stated only approximately. In
mathematics it is often necessary to reduce the margin of error by specifying
such approximations as precisely as is practicable. But for the purposes of the
Geometry of Logic a simple rounding off to the nearest whole number is often
sufficient, since the geometrical analogy is not intended to be exact when applied
to synthetic logic.
A good example of how approximation is essential to the process of
discerning the logical patterns given by the synthetic elements of experience is
the famous proportion referred to as the "golden section". The golden section is
the point at which a given quantity can be divided such that the lesser is to the
greater part as the greater is to the original undivided quantity:
1A : B::B : A+B
Approximations to this proportion emerge with surprising frequency in
most forms of art, architecture, and even the formations of Nature--indeed this
pattern (often combined with the spiral [see Ch. Seven]) seems to define the very
principle of growth in many organisms. Its numerical value ((/5-1)/2) is
approximately "0.618 : 1.0", or alternatively, "1.0 : 1.618". These numbers can
be approximated with greater and greater precision by proceeding along the
"Fibonacci series", which is the series of numbers "in which each number is the
sum of the two previous ones: 1,2,3,5,8,13,21,34,55,89,144,233,377 etc.".
Yet the further we proceed along this series, the more difficult it is to utilize such
numbers for determining patterns in the Geometry of Logic.
A simple way of representing the golden section geometrically is to show
how a "golden rectangle" can be constructed with the help of a Pythagorean
triangle [see D491:3,41,47]. (Incidentally, this combination of a fourfold figure,
which symbolizes analysis, with a threefold figure, which symbolizes synthesis,
is another example of the reciprocity of analytic and synthetic forms of relation.)
The hypotenuse of the Pythagorean triangle can be divided into its golden section
by dropping a line from the vertex opposite it (i.e. the right angle) so that it is
perpendicular to the hypotenuse. If this hypotenuse is used as the base for a
rectangle, and if the opposite side of the rectangle is tangent to the triangle's right
angle, then the perpendicular line will divide the larger rectangle into two smaller
"golden" rectangles [see Figure 4.13a]. The vertical rectangle (on the left) is
golden because its sides stand in a "0.618 : 1" ratio, while the horizontal
rectangle (on the right) is golden because its sides stand in a "1 : 1.618" ratio. If
the base of the main rectangle in Figure 4.13a is used as the diameter of a circle
(another synthetic symbol [see 4.4]), the circle" arc will be tangent to the
triangle's right angle.
Figure 4.13: Construction of the Golden Rectangle
(a) Using the Pythagorean Triangle (b) Using the Circle
Figure 4.14: "Trees" in Graph Theory
A square can then be cut out of the center of the original rectangle, so that
equal golden rectangles appear on either side [see Figure 4.13b]. This circle
makes it possible to construct a new Pythagorean triangle at the point where it re-
enters the rectangle, and so also to construct a new horizontal golden rectangle
on the left.
The golden section applies primarily to natural and artistic proportions,
and only secondarily to logical relations. Nevertheless, it is interesting to see
how the rectangle, triangle, and circle even here are used in ways wholly
consistent with their use in the more regular branches of the Geometry of Logic.
Consider, for instance, the interesting fact that the area of the circle (r) in Figure
4.13b approximates "4" ((1.118) = 3.927), and its circumference (d)
approximates "7" (2.236 = 7.025). The connection between the circle and the
numbers "4" and "7" will be of utmost significance in 4.4, and throughout Part
Two.
A rather different example of an irregular application of complex
synthetic forms of relation is the role they play in the branch of mathematics
known as "graph theory". Graph theorists are concerned with the properties and
functions of such irregularly shaped geometrical structures as "paths", "trees",
and "circuits". A path is a sequence of adjoining edges and vertices in a given
graph (a graph being any finite set of vertices and edges connected according to a
rule) [B520:9]. A tree is "a connected graph which contains no circuits" [37],
and a circuit is a path which encloses an area, leading back to the point of
departure. All the possible trees with five or less vertices are listed in Figure
4.14. These figures have an obvious affinity with the synthetic forms of relation.
For example, any point on such trees can be designated as its "root", and used to
connect the tree with other trees (a process called "unification" [39]!). Trivalent
trees--i.e. ones in which all vertices have three lines leading from them--play an
important part in graph theory [103,195-7] and have a form particularly similar
to that of synthetic integration in its complex, but regular, forms. Other
reminders of synthetic integration within graph theory, such as the use of the
word "operandator" to define the synthetic process by which a symbol "partakes
of the nature of an operand and an operator" [40], cannot be discussed here
without straying too far from our main line of inquiry.
The reciprocity between analysis and synthesis is actually the logical
source of many irregularities or discontinuities exhibited by natural objects which
are ordinarily regarded as continuous. This can be readily demonstrated by
means of a geometrical figure which was invented by Seeman specifically as a
tool for representing the mathematical properties of such unexpected irregularities
(called "catastrophes"). The figure is a curved plane with three more or less
straight edges and one edge folded in an "S" shape; it is plotted onto a three-
dimensional coordinate graph, two dimensions of which are used to give a linear
representation of the area covered by the fold [see Figure 4.15a].@ The lines
composing the back, right, and left edges of the plane each move continuously in
one direction, connecting pairs of extreme points. There are no surprises,
because (as we shall see) these paths follow the laws of analytic logic.
Movement across the plane from the middle of the back edge towards the front is
"divergent", since two paths leading in very similar directions end up at
completely different destinations. This, clearly, represents an analytic division of
unity (one level on the plane) into duality (two levels on the plane). The line
comprising the front edge, by contrast, changes direction twice, so that in order
to go from one corner to the other without changing directions, a "catastrophic
jump" is necessary: proceeding from the right to the left requires a downward
jump from the upper to the lower level at the point labelled "A" in Figure 4.15a;
and proceeding in the opposite direction requires an upward jump at point "B".
Figure 4.15: Seeman's Discontinuity Plane as a Map of Synthetic Irregularity
(a) Labelled with Seeman's Terminology
(b) Labelled with Logical Notation
This geometrical model can be used to represent sets of relations which
are characterized by a maximum (+), a minimum (-), and a point of equilibrium
(x). Examples can be found in virtually every corner of science, ranging from
psychology and sociology to physics and chemistry. The way an animal's (or a
person's) mood effects his behavior is an example of divergence: the balance of
fear and rage may be equivalent in two dogs, yet if their moods differ slightly,
they may react in opposite ways (i.e. one might fight, the other might flee). And
the chemical transformation called "sublimation" [see 4.2] is an example of a
catastrophic jump. Ordinarily, chemical transformations occur only between
solid and liquid or liquid and gas, but under high pressure they surprise us by
transforming directly from solid to gas (or vice versa). The detailed application
of this map can best be accomplished by relating it to a very mundane example
which can be easily tested.
Compressing any hard, flat object, such as a board or a card, at two ends
will produce a continuous bowing effect in the middle. If pressure is applied to
the middle, the bow will gradually change its shape. But at some point the bow
will suddenly "jump" back into its original shape, bending now in the opposite
direction. This can be tested easily enough by folding a sheet of paper in half
twice and holding it between two fingers, while applying pressure to the middle
with the other hand. With only a small amount of compression (mapped onto the
"x" axis in Figure 4.15a) there will be no jump at all. This state corresponds to
the back edge of the plane. As the pressure at the ends increases, more pressure
must be exerted on the middle (mapped onto the "y" axis) in order to move the
bow. This soon gives rise to the desired jump, which increases in intensity (i.e.
travels further on the "z" axis) as the pressure on the ends increases. When the
plane is projected onto a two-dimensional surface below, the fold (the "z"
dimension) is depicted as two curved lines converging at a point (equivalent to
where the fold starts). This provides a clear distinction between the area where
continuous behavior can be safely predicted and the area where discontinuous
behavior is likely to occur.
The reason Seeman's discontinuity plane is applicable to such a wide
variety of real relations is that the geometrical relations between its elements is
directly related to the logical forms of relation defined by analysis and synthesis.
The projection of the fold in the curved plane onto the line in the flat plane, for
example, divides the latter into two areas, one of which operates according to the
laws of analysis ("A = A" and "A = -A"), and the other, those of synthesis ("A =
A" and "A = -A"). The logical basis of this difference can be / brought out by
labelling the four corners of the flat plane with the expressions appropriate to the
square (i.e. to the inversion of the standard second level analytic form of
relation), and the angles of the projected fold with the expressions appropriate to
the triangular shape it resembles (i.e. to the simple synthetic form of relation).
Transferring these labels onto the three-dimensional plane raises the synthetic
aspect of the graph to the second level. Thus the analytic expressions remain at
the same four corners, while the expressions expression unique to second level
synthesis are mapped onto the four edges and the edge of the fold. (Note the
correspondence between the map in this form and that given in Figure 4.3. The
top row of spaces on the latter figure corresponds to the right edge on the
former, etc.) Supplying two "+x" labels for the S-shaped front edge, one for
each of its two main levels, signifies the fact that in synthetic operations "A =
A". This law of synthetic logic is the law which governs the process which
Seeman calls catastrophe. The other law of synthetic logic, "A = -A", governs
the process of convergence, in which the two sides of the fold ("x+" and "x-")
are equated at the center ("xx"). Convergence is the flow from the front edge of
the plane, with its two levels, to the single level of the back edge. It gives rise to
the very possibility of continuity. Divergence, as mentioned above, flows in the
opposite direction. Its splitting of the one level into two is based on the analytic
law, "A = -A" ("-x" = "+x"). Finally, continuity is based on the other analytic
law, that "A = A", without which neither analysis nor synthesis could be
described (nor could catastrophic and convergent operations be measured
without presupposing continuity and discontinuity). By using the Geometry of
Logic to define the logical relations inherent in geometrical maps such as
Seeman's, the task of applying them unequivocally to concrete examples is
greatly facilitated.
An excellent set of examples of complex synthetic relations which are
orderly, yet infinitely variable, so as to defy any strict categorization in terms of
"levels", is provided by the branch of mathematics sometimes called the "fractal
geometry of nature" [M536]. A "fractal" is a complex empirical object or
geometrical figure which appears to have an ordinary "intuitive" dimension (i.e.
"0", "1", "2", or "3"), but turns out to have a real dimension which exceeds the
apparent intuitive dimension by some fraction [M536:15,37]. "Nature exhibits
not simply a higher degree but an altogether different level of complexity", as
compared to the complexity possible in a Euclidean universe [M536:1]. Indeed,
in nature, according to Jean Perrin, "curves that have no tangents are the rule,
and regular curves, such as circles, are interesting but quite special" [a.q.i.
M536:7]. In support of this view Mandelbrot adds that "a line that at first sight
would seem to be satisfactory [i.e. as a tangent] appears on close scrutiny to be
perpendicular or oblique." The fractal geometry of nature thus establishes that
irregular objects have two different but equally legitimate "dimensions" [A = -A]:
one which applies to its mathematical structure and the other to its intuitive
structure.
A typical example of a fractal is the length of any coastline. The task of
measuring a coastline seems straightforward, until one realizes that the smaller
the standard of measure, the larger the final measurement will be [M536:33]: a
rod one mile long would miss quite a lot of the inlets and turns which
characterize most coasts; one which is one foot long would measure these
accurately, but would miss the many spaces between pebbles and other small
irregularities which a smaller rod would measure; under a microscope still more
irregularities would show up; and the process could be continued indefinitely.
Thus the length of a coastline--or of any infinitely irregular object--would seem
to be "as long as you want to make it" [28]. The problem is that the conventional
means of comparing different lengths now seems rather arbitrary, especially
since different types of irregularity actually give rise to different rates of increase
in length. In order to distinguish between these differences, the object's "fractal
dimension" must be calculated. This number, which differs for different
coastlines [see M536:33], represents the most accurate mathematical description
of the real dimension of a given object.
The study of fractals raises a number of interesting points as regards
synthetic logic. The first is that for an irregular object with a Euclidean
dimension of "1", the first approximation of the fractal dimension is "3/2"
[M536:30]. This is a mathematical reflection of the fact that our experience of an
object first appears as "unity", but our full understanding of it can be described
as an analytic division (2) of a synthetic reality (3). The second point has to do
with the synthetic logic which guides the interpretation of fractal geometry. For a
frequent occurrence in the study of fractals is that "some quantity that is
commonly expected to be positive and finite turns out either to be infinite or to
vanish [i.e. it tends toward the synthetic "1" or "0"]. At first blush, such
misbehavior looks most bizarre and even terrifying, but a careful reexamination
shows it to be quite acceptable..., as long as one is willing to use new methods
of thought" [19]. This illustrates the principle that "A = -A": the finite is either
infinite or non-existent. Similarly, the correlate law, "A = A" is illustrated by the
fractal geometry of coastlines: "A point chosen at random almost surely flips
between being inland [A] and in the ocean [-A], without end" [62]. This amazing
flipping phenomenon, as well as the contradictory fact that an object can have an
infinite number of equally valid lengths, can be made more palatable by taking
into consideration the principle of perspective. For it is only when the
perspective (e.g. the length of the rod) is left unspecified that the synthetic
ambiguity of such objects is troublesome.
A final example of the synthetic form of fractal geometry is the "Koch
curve", which defines "the limit of decreasing approximations". The "triadic"
version of the Koch curve is especially relevant. It is formed by repeatedly
trisecting the sides of an equilateral triangle by attaching progressively smaller
equilateral triangles to the middle of each side. Figure 4.16 [adapted from
M536:42-3] shows the first four steps in this ideally infinite operation. Although
this particular way of trisecting the triangle with triangles does not correspond
exactly to the standard levels of synthetic integration, the potential for using the
resulting figures as logical models for real synthetic relations is obvious. These
geometrical representations of regular irregularity are called "Koch Islands" by
Mandelbrot, because their form is analogous to that of a real island. Their fractal
dimension, incidentally, is "log 4/log 3 = 1.2618" [M536:36,39]. Another
interesting Koch curve is given in Figure 4.17: it resembles a row of trees, but
its fractal dimension is actually just under "2" [M536:57]. Other Koch curves
have been reproduced in Figure 2.31 and on the title pages of Parts One and Two
of this book [cf. M536:36,39].
One final example of irregular, yet reciprocal, logical relations is the
complex network of family relationships in which every human being
participates, and which is sometimes reconstructed and recorded in the form of a
"family tree" [cf. Figure 4.17]. As long as we proceed backward in time (i.e.
counterclockwise) from any given individual, we have a perfect analytic division
at every step: for every person has exactly two (biological) parents, four
grandparents, eight great-
Figure 4.16: The Triadic Koch Island
(a) Step One`(b) Step Two
(c) Step Three(d) Step Four
Figure 4.17: A Koch Curve Resembling a Row of Natural Trees
grandparents, etc. Proceeding forward as well as backward converts this into a
complex synthetic relation [see Figure 4.10]. The difficulty--and the challenge
that motivates many a genealogist--is to include not only parents and
grandparents, but also brothers and sisters, aunts and uncles, and all the children
of each couple. Every family tree, therefore, exemplifies the logic of synthesis
and its thoroughgoing reciprocity with the logic of analysis. For each "leaf" on
the tree is the synthesis (1) of some pair of opposites (i.e. parents); each also is
either a "+" or a "-" in at least one sense (viz. male/female) and possibly more
(e.g. uncle/aunt, brother/sister, etc.); and some continue the process by joining
with an opposite to initiate a new series of synthetic relations. But the variety in
the structures of the nuclear families in such a family tree reflects the tendency of
all synthetic structures in the natural world to manifest themselves in irregular
forms which cannot be consistently reduced to simple geometrical figures.
4.4 The Static and Dynamic Aspects
of Analysis and Synthesis
The reciprocal relation between analysis and synthesis which has been
evident throughout our discussion of the various complex synthetic forms of
relation is perhaps most pronounced when it comes to the complementary logical
"directions" which characterize these two operations. Logicians agree that every
logical relation between elements "has a sense, i.e. the direction in which it goes.
The analytic and synthetic forms of relation also have such a direction, only not
with respect to the set of elements which can satisfy the relation, but rather
between the various expressions in a given form of relation. By distinguishing
between the "static" and "dynamic" nature of the various forms of relation, the
difference in direction appropriate to analysis and synthesis should become more
clear. As a result, we shall discover important differences in the rules for
mapping these two fundamental types of logical operation.
The elements which participate in an analytic relation all enjoy an equal
status. That is, no element is subordinate to any other: the extreme positivity of a
"++" element does not (in itself) make it any more important than, for example,
the corresponding "--" element. The relations between the elements of any
analytic relation can therefore be labelled "static". This simply means that, when
a set of elements is viewed in terms of their analytic relations, each element is
assumed to preserve its original identity (A = A) through any and all
transformations.
By contrast, the elements which participate in a synthetic relation are
naturally unequal. That is, the elements work together in a process in which each
element can take on quite different relations to other elements. The relations
between the elements of any synthetic relation can therefore be labelled
"dynamic". This simply means that, when a set of elements is viewed in terms of
their synthetic relations, no element can be assumed to preserve its original
identity (A = A), but is likely to be transformed into some other element as the
process develops.
In mapping static and dynamic forms of relation onto geometrical figures
the rules specified so far have differed in one key respect. In both cases we have
included arrowheads on the line segments of a geometrical map whenever it has
been necessary to depict a logical progression between the elements of the figure.
(The use of the arrow as a symbol for relational direction is quite common
among logicians [see e.g. S515:201; H485:175-81,186], and does not require
any justification here.) But these arrows refer to different sorts of progression in
synthetic than in analytic maps. In synthetic maps an arrow is used to represent a
dynamic change in the relational value of an element into that which it is not--e.g.
from having a value of "0" to one of "+", or from "+" to "-", etc. [see 3.2]. In
analytic maps, by contrast, an arrow indicates a difference in the relational value
of two opposed expressions: the arrow always begins with the value which
stems from the lowest level of division (e.g. the "++" and "--" expressions in
second level analytic division stem from the "+" and "-" of the first level,
whereas the "+-" and "-+" expressions do not), since it is from such points that
the opposite poles are "activated". The analytic arrow therefore reflects a static
hierarchy of levels.
The actual shape of the triangle and the square makes them appropriate
geometrical symbols for dynamic and static relations, even apart from their use
as standard maps of synthetic and analytic forms of relation. The triangle, onto
various combinations of which most synthetic relations can be mapped, is itself a
kind of two-dimensional arrowhead, in the sense that it is the plane figure with
the "sharpest" vertices. And the square, with its perfect symmetry, can stand on
any of its sides and still connote a blunt "firmness" which excels that of any
other rectilinear figure. Interestingly, as more vertices are added to a regular
rectilinear figure, the "sharpness" of its corners is progressively reduced, so that
it approximates more and more closely to a circle, a figure which will play an
important role in symbolizing the compound forms of relation discussed in Part
Two.
As a single unified experience describable in three steps, simple synthetic
integration is based on a 1:3 ratio, just as is the triangle ("one circumference :
three sides"; or "one area : three boundaries"). But when we analyze this
relation--when we, as it were, "freeze" the motion and examine its constituent
parts--we find in addition to the three main terms being related ( "+", "-", and
"1") a submerged fourth term ("0"), which serves as the starting-point for the
operation in its dynamic manifestation. The fourth term does not actually show
up in the dynamic manifestation of the synthetic relation because it is united with
its (or some other) conclusion: as "x", "0" and "1" are indistinguishable (A = -
A). Once the hidden presence of this fourth term is recognized, however,
synthetic integration can be seen to be based on a 3:4 ratio (between steps
relating and terms related) [see note 4.1]. But of course, this ratio is just another
name for the relation between a simple synthetic operation and its analytic
counterpart.
The suitability of the triangle and the square as symbols for the two main
forms of logical relation can be intuitively confirmed by taking note of the
symbolic construction of certain ancient (and modern) Chinese characters. The
written form of the Chinese language originally developed as a way of
representing ideas in pictorial, or indeed geometrical, forms. Among the most
common elements in the construction of Chinese characters are the line, the
triangle, the cross, and the square. A detailed study of the logical basis of the use
of such symbols in Chinese characters could therefore be based on the principles
provided by the Geometry of Logic. Although the reason for the shape of many
characters cannot be discerned, a striking example of one which can is the
character for "relation" (or in logical contexts "form of relation") which is " "
(pronounced "ho"). This word literally means "be together" [G522:553], and
was used by ancient Chinese logicians (the Mohists) to refer to the connection
between a name (as symbolized in analysis by a square) and the object to which
it refers (as symbolized in synthesis by the triangle) [30]: "The mating of name
and object is " [C529:A80]. Mohist epistemology is itself structured on the 3:4
ratio which is suggested by " "; for it is based on a threefold account of how one
knows ("By hearsay, by explanation, by personal experience") and a fourfold
analysis of what one knows ("The name, the object, how to relate [them], how
to act"). Here again, the dynamic character of the threefold distinction stands in
sharp contrast to the static character of the fourfold distinction.
Another interesting example of a 3:4 ratio is given in the approximate
relationship between the circumference of a circle and the square constructed on
the circle's diameter. A circle with a diameter of "1" has a circumference of " " (=
3.14159...), which can be approximated to "3". If a square is constructed in
such a way that its horizontal sides are bisected by this diameter, while its
vertical sides are parallel to it [see Figure 4.18], then the square will have a
circumference of exactly "4".@ Like the triangle, the circle in this figure stands
in a dynamic relationship to the static figure of the square. Again this was
recognized on an intuitive level by the ancient Chinese philosophers, for whom
"circle and square were stock examples of free and hampered movement"
[G522:328]. A circle connotes dynamic fluidity because it can be easily rotated
(cf. the wheel), whereas the square connotes static rigidity because it will not
naturally rotate on a flat surface.
The similarity between the symbolic functions of the circle and the
triangle in the Geometry of Logic is not accidental, but is rooted in a purely
geometrical similarity between the two. For "the circle holds the same
fundamental position among curvilinear figures, as does the triangle among
rectilinear figures. Any two circles are similar figures; they differ only in scale"
[W516:133]. This formal similarity is depicted in Figure 4.19.@ One important
mathematical use of this circle-triangle relation is in defining a "sine" function.
The sine function compares the difference between the perpendicular (AB) of the
diameter and the arc which joins the point of intersection between diameter and
circle (C) with the point of intersection between perpendicular and circle (A), by
relating both of these to the radius of the circle (OA). The resulting equation,
AB/OA = sine AC/OA, defines the periodic function known as the "sine wave"
[see e.g. W516:141], a function which has proved to be very useful in
describing numerous natural phenomena. Mapping the sine wave for "y = sine
x" onto a co-ordinate graph yields a valuable example of how a certain "A" can
equal both "B" and "-B" [see Figure 4.20; cf. note 3.8].@ For in this function,
"Y = 0" yields both "x = " and "x = -"; likewise, "Y = 1" yields both "x = /2"
and "x = -/2"; etc. The periodic nature of such synthetic functions will concern
us more at various points in Part Two.
Figure 4.18: The 3:4 Ratio of Circle and Square
Figure 4.19: The Geometrical Similarity Between Circles and Triangles
Figure 4.20: The Sine Wave
In conclusion, the important reciprocal relationship between the dynamic
experience of synthetic integration and the static conceptualization of analytic
division can be effectively symbolized using the model of a child's top. When a
top is spinning it is fulfilling its purpose, even though its constitution (i.e. its
shape, colors, etc.) is not easy to see. Indeed, a multi-colored top may appear
when spinning to be a single (blurred) color. As we have seen, the same is true
of synthetic integration in general, as well as of the art of reasoning in particular:
it reaches its goal (1), but only by contradicting the laws of analytic logic.
However, it is sometimes necessary to stop the top, examine its properties,
perhaps even alter its shape on the basis of one's examination. For a faulty top
will not spin true no matter how fervently one practices. In the same way, the
unity given by synthetic integration must be subjected to analysis before it can be
understood and so also before its validity or consistency can be accurately
judged. Only in this static state does the "science" of logic hold its dominion, for
it is only when reasoning is at rest that the laws of identity and non-contradiction
are clearly discernable. Just as the skilled top spinner is able to describe his top at
rest as well as to set it in motion, so also the ultimate goal of the Geometry of
Logic must be to combine analytic and synthetic forms of relation together in a
single system, so that they can be regarded as working for, rather than against,
each other. The four chapters of Part One have provided a sufficient (albeit static)
foundation upon which we can base our discussion of the dynamics of logical
systems in the three chapters of Part Two. As we now change gears we must
keep in mind at all times that, just as a faulty top may wabble and a stopped top
will topple over, so also analysis and synthesis must always work hand in hand
in the construction of logical systems.
NOTES TO CHAPTER FOUR
1. In P494:53c-55c Plato uses triangles alone to construct the objective structure
of the world, beginning with the "four elements" as represented by the four
primary solids. In so doing he provides a geometrical confirmation of the
synthetic (threefold) basis of all analytic (fourfold) explanation. Interestingly,
this 3:4 ratio shows up not only in the distinction between the three-sided
triangles and the four solids, but also in the types of triangle which Plato uses:
for the first three solids can all be constructed with various combinations of half
an equilateral triangle, whereas the fourth solid (the earth, or "--", element in the
second level analytic relation) has to be constructed out of triangles which are
half of the square. The significance of the 3:4 ratio will be discussed more fully
in 4.4.
2. The traditional (analytic) way of diagramming the relations in the center of the
synthetic manifold given in Figure 4.2 would be to represent them as circles (i.e.
classes) within a common square (i.e. universe of discourse), and intersecting in
such a way that they all share some amount of common area.
3. Such a grid can also be used to illustrate the fact that the sum of any number
of successive positive odd integers (beginning with "1") is always a square
number. For the "gnomon" [see note 4.22] which is added in each case to form
the larger square is always two squares larger than its predecessor. Thus three
squares must be added to a unit square to make a larger (four unit) square; five
must be added to the fourfold square to make a ninefold square; seven must be
added to the latter to make a sixteenfold square, etc. This demonstrates the direct
geometrical relationship between the progression 1,3,5,7,9... and the
progression 1,4,9,16,25...
4. Each of the latter faces is in the shape of a square divided into eight equal right
triangles, with one of the corner triangles missing. Thus the synthetic use of this
primary analytic symbol (the cube) requires us to divide "8" not as "2", but as
"7+1" [see below].
5. Occasionally these higher levels of synthetic forms of relation do have some
measure of practical applicability. An understanding of fourth level (eighty-one-
fold) synthetic relations could, for example, be of some value in understanding
the structure of the Tao Te Ching (The Book of Meaning and Life) [L532]. This
ancient Chinese classic contains about 5000 (= 5.10) words and is divided into
81 (= 3) sections. Such orderly divisions are unlikely to have been chosen
arbitrarily. But as is typical with poetic utterances, the book is concerned so
exclusively with its synthetic subject-matter that no indication is given as to how
it has been structured or even as to whether or not its structure is intended to be
significant.
6. See D491:passim. Doczi shows how all "the manifold diversities of nature",
despite their many differences, share "the same simple, dinergic and harmonious
limitations", by demonstrating the formal similarities between plants, insects,
animals of all shapes and sizes, and even man himself, as well as all manner of
inorganic natural objects and human art forms.
7. D491:5. Doczi continues: "Any number in this series divided by the following
one approximates 0.618... and any number divided by the previous one
approximates 1.618..., these being the characteristic proportional rates between
minor and major parts of the golden section."
8. The triangle inscribed in a golden rectangle in this way is actually only
approximately Pythagorean. A perfect Pythagorean triangle with hypotenuse
2.236 would have sides measuring 1.097 and 1.950; so the margin of error here
is 0.0786 and 0.0479, which works out to about a 4% error.
9. The operation of dividing a golden rectangle into a square and a smaller
golden rectangle can be repeated on the smaller rectangle, and on each
successively smaller rectangle, to infinity. If a circular curve, such as that which
passes through both of the vertical golden rectangles in Figure 4.2b, is passed
through each successively smaller square, then a spiral can be constructed, as
follows [cf. D491:cover diagram]:
10. B520 is an introduction to this science which is admirable not only for its
exhaustive treatment of the history and development of its various topics, but
also for its lucidity (much of it is quite intelligible to the non-mathematician). The
problem which set Graph Theory in motion as a theoretical science was the once
famous "problem of the seven bridges of K|nigsberg". The challenge was to
walk in a circuit around the city, crossing each bridge once, but only once. The
river (see below) flows roughly in the form of a square, with branches at three of
the four corners. The bridges were arranged as follows [cf. B520:2-3]:
It does not take long to discover that such a circuit cannot be completed.
In 1736 Euler proved why it is impossible, yet in 1875 it was announced that the
problem was solved! (An eighth bridge had been built [12].)
The significance of this problem for the Geometry of Logic lies in the
actual layout of the river and bridges. For they clearly form the pattern of a
square with a triangle pointing to it:
This pattern is a remarkably accurate representation of the relation
between analytic and synthetic operations. Bridge "7" serves both as the
synthesis of the two opposing bridges, "5" and "6", and as the synthesis of the
four opposing bridges "1", "2", "3", and "4". Of course, these logical relations
were not used by Euler in proving the impossibility of a circuit in this particular
case; nevertheless, the story illustrates the complexity of synthetic relations, as
well as their tendency to form ordered relations, even in empirical situations
where we least expect such patterns to emerge.
11. I first learned of this model in a lecture given by Professor Seeman at
Wolfson College, Oxford (11 February 1986), entitled "Continuity,
Discontinuity, and Catastrophe". Most of the terminology and examples used
below were presented in this lecture. But the model's basis in the logical forms
of relation was not mentioned.
12. The word "catastrophe" is derived from the Greek "Katastrephein", which
means "to turn down".
13. The similarity between this S-shaped edge and the S-shape which divides the
t'ai chi symbol [see Figure 1.7] is not accidental. For both are symbols of the
synthetic laws of transformation--i.e. of the logical fact that synthetic operations
cut across the divisions of analysis, yet hold them together in the very process.
The fact that the word "synthesis" itself begins with the letter "s" is appropriate,
though accidental.
14. Mandelbrot uses the word "intuitive" to describe the validity and value of
fractals [e.g. M536:16]; but he must be using this word in some sense other than
that of the Kantian sensible intuition, for he never even tries to explain how a
fractal dimension as such can be visualized. Objects and geometrical figures with
fractal dimensions can, of course, be visualized, and the beauty of their complex
structure can be fully appreciated; but none of Mandelbrot's figures--
sophisticated graphics notwithstanding--succeed in providing any more than an
idea (not a sensation) of what a fractal dimension is like. Only if "intuition" is
taken to mean something more like "insight", or "an elucidating hint as to the true
nature of the situation", can such figures properly be called "intuitive".
15. Mandelbrot gives a good illustration of the importance of establishing the
perspective when he points out the "different effective dimensions implicit in a
ball of thread" [M536:17-18]: "To an observer placed far away, the ball appears
as a zero-dimensional figure: a point.... As seen from a distance of 10 cm
resolution, the ball of thread is a three-dimensional figure. At 10mm, it is a mess
of one-dimensional threads. At 0.1 mm, each thread becomes a column and the
whole becomes a three-dimensional figure again. At 0.01 mm, each column
dissolves into fibers, and the ball again becomes one-dimensional, and so one,
with the dimension crossing over repeatedly from one value to another. When
the ball is represented by a finite number of atom-like pinpoints, it becomes zero-
dimensional again."
16. M536:43. Defined in this way, "a Koch curve is not significantly more
complicated than a circle!" [41]. This is because a circle can be defined as a
construction out of "an infinite number of infinitely short strokes" [41].
17. Cesro was so enamored with the "inner affinity" of the triadic Koch curve
that he regarded its "similarity between the whole and its parts...as truly
marvelous. Had it been given life, it would not be possible to do away with it
without destroying it altogether for it would rise again and again from the depths
of its triangles, as life does in the Universe" [a.q.i. M536:43].
18. S515:167. The direction of a relation is defined by determining whether it
sometimes, always, or never fits into each of the three categories of symmetry,
reflexivity, and transitivity. For example, the relation "gave birth to" has a single
direction from its subject (a) to its object (b) because it is never symmetrical (b
cannot have given birth to a), never reflexive (a cannot have given birth to
herself), and never transitive (if b gives birth to c, then a cannot have given birth
to c). Or again, the relation "is the same size as" is omnidirectional, since it is
always symmetrical, always reflexive, and always transitive.
19. Jung recognizes and puts much stress on the significance of the frequent
recurrence of these 1:3 and 3:4 ratios, which he believes belong to "a definitely
psychological realm" [J524:100]. He gives no hint, however, that they are
grounded in the logical forms of relation.
20. Other suggestive characters could be mentioned--such as " " ("mu") for
"mother"--but could not be discussed in this context without straying too far
from our main topic.
21. C529:A80. These four "objects of knowledge" are surprisingly similar to the
four stages of knowledge which Kant outlines in K105, though the order is
slightly different. The latter will be discussed in some detail in 6.4, where it will
also be combined with a threefold distinction; but it will there be used to
exemplify not the simple addition of synthetic and analytic forms (3+4 = 7), but
their compound multiplication (3.4 = 12).
22. Adapted from W516:134. Of course, any regular geometrical figures (such
as a square, a cube, or a sphere) also differ from other figures of the same sort
only by their different scale. The uniqueness of the triangle and the circle is that
they are the simplest regular figures to satisfy this condition.
It may also be helpful to point out here that a figure which is added to a
smaller figure to produce a larger one of the same form was referred to by the
ancient Greeks as the "gnomon"--i.e. "a thing enabling something to be known"
[H488:1.370], for "from it the whole is known" [371]. A gnomon is any figure
which, when added to another figure, produces a figure whose form is the same
as the original figure, but whose content (or scale) is different. Thus the gnomon
does for geometrical figures what "generalization" or "formalization" does for
logical relations: it enables the same form to describe numerous different
contents.
23. A similar analogy is used in a rather different way in B528:32-4 to
distinguish the difference between an understanding of the path which leads to
mystical enlightenment and an actual experience of such a goal.