A system consists of elements standing in certain relations. --Stebbing [S515:174] 5.1 Logical Systemization and its Compound Form of Relation In Part One we limited our attention to analytic and synthetic forms of relation as they operate in abstraction from each other. We are now equipped with a set of symbolic geometrical tools with which we can examine the ways these logical forms of relation can be combined. Analysis, with its basis in the logical laws of identity and non-contradiction, might seem at first to be so diametrically opposed to synthesis, with its basis in the logical laws of non- identity and contradiction [see 3.3], that the two could never work together consistently in a single operation. Yet in Chapter Four we saw that, even when these operations are viewed in abstraction from each other, they are thoroughly reciprocal. Analytic and synthetic forms of relation define the different perspectives from which the same objects can be viewed, so the construction of a coherent system is bound to depend on a thoroughgoing interrelation between these two types of logical operation. The way forward, as was suggested at the close of Part One, is not to choose between analysis and synthesis, but to combine them in a single unified system of logical forms of relation. In Chapter Four we discussed the ways in which simple synthetic integration can multiply itself to produce "complex" forms of synthetic relation. This might involve, for example, viewing each of the three steps in a synthetic operation as itself based on simple synthetic integration, thus resulting in a ninefold network of "second level" relations [4.2]. But synthetic integration on its own often lacks the systematic structure of such formal patterns. In other words, when a person adopts a synthetic perspective to the exclusion of its analytic counterparts, his experience will almost inevitably be unorganized and recalcitrant. Conversely, if an analytic perspective is allowed to eclipse the synthetic forms upon which it depends for its very existence, the result is a dry, dislocated view of reality, which is in danger of quenching the life force itself. Neither of these extreme perspectives provides a sufficiently firm foundation for the construction of a "system" in its true sense. For a systematic treatment of any material must strike a balance between the analytic and the synthetic forms which lie at its base. Our task in Part Two will be to strike such a balance by limiting our use of synthetic integration to its simple threefold form, and by seeking to determine the extent to which such synthetic relations can be combined in a progression of relations which fit together into patterns established by analytic division. To combine in this way the passive receptivity of intuitive experience with the active spontaneity of conceptual thought is the essential requirement of any attempt to build a truly coherent system. Hence we can refer to this new operation as "compound systemization". For the logician it is not so much actual systems that matter, as the formal patterns (i.e. the forms of relation) which tend to structure real systems when considered in general. It is important, therefore, to explain what is meant by an analytically ordered "progression" of simple synthetic relations. In contrast to a complex synthetic relation, which multiplies simple synthetic relations, a "compound" (i.e. analytic and synthetic) relation is based on a stream of simple synthetic relations which are added together in such a way that the third term of each synthesis (i.e. the "1") is the starting-point (i.e.the "0") for a new synthetic operation. This process, which is central to all coherent argumentation, can be depicted in the following way: Figure 5.1: Compound Synthetic Relations Such a string of synthetic relations could go on indefinitely, and need not follow an analytic pattern any more than a line of dominoes toppling each other one by one. A good example is Hegel's seemingly endless string of syntheses which surprises us by suddenly ending in the "Absolute", the ultimate "1" (though he may have intended to depict the Absolute not as the end of this synthetic progression so much as the "higher unity" which encompasses an infinite, or at least indefinite, stream of synthetic relations). But if such relations are grouped together in such a way as to form an analytic pattern, then and only then do they constitute a proper logical system. Only through the cooperative interaction involved in compound systemization can logic become "activated", for such interaction enables the dynamic, synthetic forms of synthetic relation to breath life into the static, analytic forms of relation. The use of the syllogism in argumentation provides a helpful example of how compound systemization operates. For the form of relation which holds between the elements of a simple (i.e. threefold) syllogism is synthetic, whereas the form of relation which holds between the (four) main types of simple syllogism [see Figure 2.12] is analytic. As a result, a set of syllogisms can be used to exemplify the most fundamental of all compound relations, the twelvefold relation. When each of the four types of syllogism appears in its simple, three-step, form in a single line of argument [see 6.4], the twelve elements in the resulting argument evince an elegance which is nothing short of logical beauty. (Indeed, Leibniz described "the form of syllogisms" as "one of the most beautiful which the human mind has made... It is a kind of universal mathematics whose importance is not sufficiently known" [a.q.i. S515:81].) But before discussing the many nuances of this twelvefold pattern of compound systemization [see Chapter Six], it will be helpful to review the results established so far by summarizing the symbolic use which has been, and is to be, made of certain numbers. 5.2 "2" and "3" as Standard Roots for All Numbers The integers "0" and "1" have played a foundational role in the Geometry of Logic. "0" has been used primarily as the symbol for the mysterious, empty starting-point of synthetic integration. But it also entered our discussion of analytic division, where the original operation of twofold division was depicted as a distinction between everything (1) and nothing (0). An even more important part has been played in analytic division by the number "1": it stands for that original unity from which all analytic diversity is derived. In the context of synthetic integration, however, "1" stands not for a prior unity, but for the unity towards which all experiences point, the unity of polar opposites in a higher synthesis. These numerical symbols will continue their foundational function throughout our discussion of compound systemization in Part Two. But the symbolic function of the next two numbers, "2" and "3", make them more appropriate than this original pair to fulfil the task of defining the symbolic meaning of all subsequent numbers. The integers "2" and "3" have been just as essential to the Geometry of Logic as the integers "0" and "1". "2", along with its more complex derivatives (viz. "4", "8", "16", etc.) functions primarily within the context of analytic division [see Chapter Two]. Likewise "3", along with its regular complex derivatives (viz. "9", "27", "81", etc.) functions primarily within the context of synthetic integration [see Chapter Three]. All other positive integers can be formed by combining "2" and "3" in various ways. (For example, we have just made the provisional suggestion that "12" functions primarily within the context of compound systemization [see Chapter Six].) This is demonstrated in Table 5.1, where numbers which are composites of "2" and "3" (i.e. "2+3" or "2n.3", where n-1) are classified as "impure", while those which require only one base number (i.e. either "2" or "3") are classified as "pure". (Pure numbers can often be defined also in terms of impure composites; but Table 5.1 does not specify such derivations, because the pure derivation is always far more significant symbolically.) The most important numbers on this list are the first twelve, from which the patterns for all others can be constructed. Of these we have already thoroughly discussed "0", "1", "2", "3", and "4", as well as "8" and "9"; so it will be necessary to concentrate in more detail on the symbolic relations implied in the others, and particularly in the trio of numbers "5", "6", and "7". 5.3 The Symbolism of "5", "6", and "7" The simplest compound forms of relation, according to Table 5.1, are those in which the numbers "5" and "6" are regarded as the sum and product, respectively, of "2" and "3". These two examples can be used to illustrate an important logical distinction between two types of impure definition for positive integers: a number whose impure definition includes a sum will tend to represent synthetic integration more than analytic division; one which does not contain a sum will tend to represent the latter more than the former; and a number which can be defined with or without a sum will tend to be equally relevant to both aspects of logical relations. Table 5.1: The Whole Numbers in Terms of the Addition and Multiplication of "2" and "3" Third 2+ Power 2+2 2+32+22+3 2x3 3+32x3 Impure Second2+3+3=2+3 Deri- Power 2+3(2x3)3+2 2x3 vations First 2x3= Power 2 3 2+3 3+3 Whole Numbers 2 3 4 5 6 7 8 9 10 11 12 13 16 24 27 32 36 6472 81 128 243 256 512 Second Power 2 3 (4) (8) (9) (16) Third Power 2 3 (4) (8) Fourth 2= 3= Power (2) (3) (4) Fifth 2= 3= Pure Power 2x2 3x3 Deri- vations Sixth 2= Power (2)= (2) Seventh 2=2x Power (2) Eighth 2= Power ((2)) Ninth 2= Power (2) The organization of the natural world itself reflects, or at least parallels, the logical status of the numbers "5" and "6". For an investigation reveals that "Hexagonal patterns...are more common in inorganic than in organic nature, which favours pentagonal patterns" [D491:79]. Thus, for example, the organic structure of a pair of human hands may be used to explain the wide acceptance of the base-ten number system (i.e. "5.2" (fingers.hands), or "2+8" (thumbs + fingers)), even though the base-twelve system (i.e. "4.3", or "3+4+5") would be more suitable from the (inorganic) perspective of logic [see Chapter Six]. A good example of the ubiquitous twelvefold (6.2) tendencies of inorganic nature is the snowflake, which is always shaped in some perfectly symmetrical hexagonal form: "Each snowflake is restricted to one pattern, repeated and reflected twelve times" [79]. The quite obvious geometrical representations for five- and sixfold patterns are the pentagon and the hexagon, or alternatively, the pentagram and the hexagram (i.e. the five- and six-pointed stars). A brief discussion of each is enough to demonstrate the synthetic tendencies of fivefold patterns and the analytic tendencies of sixfold patterns. A regular pentagon can be divided into periodic, triangular patterns in several different ways. In Figure 5.2a a line is drawn from each vertex to the center, thus dividing the pentagon into five equal triangles. If each of these lines is extended so as to bisect the opposite edge (see dashed lines in Figure 5.2a), then the pentagon is divided into ten equal triangles. By contrast, connecting each vertex to all the others with lines will inscribe a pentagram inside the pentagon [see Figure 5.2b]. The center of the resulting pentagram is, interestingly enough, another pentagon, in an upside-down position. Finally, by combining these two ways of dividing the pentagon, the "complete" division of this figure into its thirty "elementary" triangles is attained [see Figure 5.2c].@ This three-step differentiation of the pentagon is a perfect example of synthetic integration: an undifferentiated pentagon (0) is divided first in such a way as to occupy the center (+) and then in such a way as to leave the center empty (-); the two are then synthesized in order to produce the completely integrated whole (1). Figure 5.2: The Triangular Elements of a Pentagon (a)Five Triangles (b) The Inscribed Pentagon (c) The Thirty Triangular Elements This division of the regular pentagon into its elementary triangles gives rise to two types of triangles, both of which are relevant to the Geometry of Logic. The five triangles which make up the "points of the star" in Figure 5.2b are all isosceles triangles: the ratio between the size of the short side and that of each of the two long sides approximates the "golden" (2/3, 3/5, 5/8, etc.) relationship, which is characteristic of many synthetic proportions [see 4.3, especially Figure 4.13]. And in Figure 5.2c the ten triangles which make up the central pentagon are all Pythagorean triangles [D491:6,9]. The Pythagorean triangle is also known as the "3-4-5" triangle, since these are the simplest whole numbers which solve the algebraic formula for all right triangles (i.e. the "Pythagorean Theorem": "a+b = c", which is solved by "3+4 = 5"). This provides another interesting way of relating the symbolic value of the number "5" to that of its synthetic and analytic forerunners. "5", of course, is the sum of "2" and "3". On their own the symbolic value of the two latter numbers is primarily logical. But when added they primarily symbolize natural or organic structures [see D491:1-13]. If the operation of squaring in the Pythagorean Theorem is regarded as an analogy of the logical process of "thinking about" something, then the version of this formula appropriate to the Geometry of Logic would be: Thinking about our own experience (5) is equivalent (=) to thinking about its simple synthetic aspect (3) and its second level analytic aspect (4) as in combination (+). If the process of thinking (squaring) is then abstracted, we end up not with the number "5", but with either "12" (= 3.4) or "7" (= 3+4 or 12-5), depending on whether the mode of combination is taken to be addition or multiplication. Both these cases will be treated in more detail shortly. A regular hexagon can be divided into triangles according to a rather simpler periodic pattern than that required by the pentagon. The hexagram itself is made up of two equilateral triangles bisecting each other on all three sides. However, the construction of its complete differentiation into elementary triangles proceeds by the same synthetic steps as did that of the pentagon: Figure 5.3a constructs the twelve lines proceeding through the center; Figure 5.3b constructs the six lines which form the hexagram; and Figure 5.3c synthesizes both, thus dividing the hexagram into thirty-six half-equilateral (i.e. 30-60-90) triangles.@ Like the pentagram, it has at its center the same figure into which it has been inscribed, only shifted by 30\ (i.e. by half the length of one side). Now because the equilateral triangle was used in Part One to represent simple synthetic integration, the triangles inside the differentiated hexagon contain some of the basic patterns used to symbolize various sorts of synthetic relations. The diamond shape is an obvious example [see Figure 5.4a]: any set of four triangles in Figure 5.3c which are connected by their right angles form this shape, which was used in 3.3 to depict the expanded version of simple synthetic relations. Furthermore, the configuration of triangles given in Figure 4.6b to represent second level complex integration is also inherent within the triangular elements of a hexagram [see Figure 5.4b]--indeed, it occupies exactly one-third of the total area.@ The affinity of the thirty-six-fold division of the hexagon with synthesis is represented in Table 5.1 by defining "36" as "3+3"-- i.e. as the synthesis of two synthetic terms (27+9). But the most important point raised by the division of the hexagon into its thirty-six triangular elements is not the presence of these synthetic figures; rather it is the systematic way in which these elements are organized. Although the triangular elements of the pentagon were also arranged in an orderly fashion, the periodic pattern was not Figure 5.3: The Triangular Elements of a Hexagon (a) Twelve Triangles (b) The Inscribed Hexagon (c) The Thirty-Six Triangular Elements Figure 5.4: Synthetic Elements in the Differentiated Hexagon (a) The Synthetic Diamond (b)The Hexahedron's Net nearly as simple as the one we have here. Not only is there just one type of triangle (the half-equilateral), but every edge fulfills the same function in each of the triangles which it helps to compose (e.g. always the hypotenuse, or always the short base, etc.). Moreover, the thirty-six triangles fall into three sets of twelve: twelve triangles lie outside the hexagram; twelve lie inside the hexagram but outside the inner hexagon; and twelve lie inside the inner hexagon. As we shall see in 7.1, this pattern of three sets of twelve is one of the most profound of all patterns in compound systemization. In Table 5.1 this systematic character is represented by the definition of "36" (6) as the product of analysis (2) and synthesis (3)--just as "6" is the product of "2" and "3". These speculations really only scratch the surface of what could be said about the numbers "5" and "6" and their application to the Geometry of Logic. Before moving on it is interesting to note that the icosahedron--the regular solid constructed out of twenty isosceles triangles--can actually be used to represent the perspectival shift that can occur between "5" and "6" (as between addition and multiplication, or synthesis and analysis). For the cross section of the icosahedron is either a pentagon or a hexagon, depending on where the plane passes through the solid [see D491:80]. The pentagon and hexagon, together with their corresponding "stars", have played an important role in art and architecture down through the ages, especially when these have had religious connotations (e.g. the six-pointed "Star of David" [s.a. B528:257]). But it is unnecessary to digress at this point into a detailed study of such applications of "5" and "6" [see D491 for such a study], for our primary purpose in discussing them here is to prepare ourselves for the discussion of similar compound relations which are more relevant to the task of systemization, and for which we will therefore supply plenty of examples. The next number, "7", is rather less complex, insofar as it is defined in only one way, as "4+3" (= 2+3). It has already been mentioned in 4.4 as the standard numerical symbol for the simple combination of analysis and synthesis. On its own, however, this definition of "7" is not of much use to the logic of compound systemization. This can be seen by noting the complicated results of dividing a regular seven-sided polygon (or "heptagon") into its elements, as we did with the pentagon and hexagon. In the case of the heptagon (0) such division involves first connecting each vertex, through the center, with the side opposite it (+), then connecting each vertex to all other vertices (-), thus constructing a seven-pointed star (or heptagram), and finally synthesizing the two in a single figure (1). The result is in one sense far too irregular to serve as a practicable map for the Geometry of Logic, for the figure ends up with a total of ninety-eight segments, grouped into seven different types (one of which is not even a triangle), each of which occurs fourteen times [see Figure 5.5].@ In spite of this irregularity, however, there is also a sense in which the periodic pattern given in the differentiated heptagon has profound implications for the Geometry of Logic. Indeed, a closer look at Figure 5.5b reveals that, for the first time, not only is the original figure inscribed in the star (i.e. in the heptagram), but the star itself is also inscribed within this inscribed heptagon-- complete with its own inscribed heptagon at its center! Altogether there are three heptagons and three heptagrams explicitly constructed within the differentiated heptagon; these can be clearly depicted by shading the appropriate lines in Figure 5.5c, as shown in Figure 5.6.@ If these diagrams are regarded as maps of a dynamic process, then these six figures would be regarded as pointing the way inwards to the center (to a hidden fourth heptagon), or perhaps outwards beyond the perimeter (to a transcendent fourth heptagram). Either way, the differentiation of the heptagon reveals a dynamic sevenfold symbolism which actually supersedes the systematic perfection of the hexagon with its more Figure 5.5: The Elements of the Heptagon (a) Fourteen Triangles (b) The Inscribed Heptagon (c) The Ninety-Eight Elements: Figure 5.6: Analysis of the Differentiated Heptagon (a) The(i) outer: (ii) middle:(iii) inner: 1Three Heptagons (b) The(i) outer: (ii) middle: (iii) inner: Three Heptagrams =2 static sixfold symbolism. It symbolizes the formal processes which are reflected, for example, in the Old Testament story of Creation, in which the day of rest is just as important in completing the act of Creation as are the other six days, which end in a merely apparent completion; for in the same way, the natural power of the cyclone is most awesome at its motionless center. Hence we can expect sevenfold distinctions to represent a higher level of systematic integration: a System of systems. The mathematical equation which best describes this way of organizing the elements in a heptagon is not so much "4+3 = 7" (though, as we have seen, this does come into it) as "6+1 = 7". For it is in the polarity between the center (1) and all the peripheral figures (6) that the heptagon displays its profound symbolic meaning--a meaning which will be fully discussed in Chapter Seven. Incidentally, we could have used a similar method to describe the numbers "5" and "6" as well: since "5 = 4+1" and "6 = 5+1", fivefold distinctions could be mapped onto a square (4) with the center (1) specified, and sixfold distinctions could be mapped onto a pentagon (5) with the center (1) specified [see 4.3]. But in these cases, as with "2", "3", and "4" as well, the definition given without the use of "1" (i.e. with only "2" and "3", as in Table 5.1) is more fundamental to the Geometry of Logic. 5.4 Summary and Concluding Remarks on "7" and "12" The integers between "7" and "12" need not be discussed in any detail here. "8" and "9" have already been discussed thoroughly in terms of third level analytic division and second level synthetic integration. "10" has been discussed in its role as the point at which our number system begins to repeat [see 2.3 and note 5.2], hence making it the numerical equivalent of what "7" is in the Geometry of Logic". It can also be regarded as twice five, in which case much of what was said with regard to the pentagon and fivefold distinctions would apply here as well. And in some traditions a rhombus is inserted into the center of a hexagon to produce a map of the "6+4 = 10" relation [see e.g. B528:275]. "11" is the sum of "5" and "6" (= "2+3" or "2+3") and as such is the least systematically relevant of the first twelve positive integers. And "12" will be our main concern in Chapter Six. But before concluding this section with a summary of the numerical symbolism established in this chapter, several interesting points should be raised about the relation between the numbers "7" and "12". The numbers "7" and "12" will both play key roles in the distinctions made throughout Part Two. These two numbers stand in the same relation to each other as addition stands to multiplication, since "2+3 = 7" and "22.3 = 12". (Compare the similar correspondence between "10" and "16" ("2+2" and "23.2") and between "13" and "36" ("2+3" and "22.3") in Table 5.1.) The fact that the sum of the sides of the simple (3-4-5) Pythagorean triangle is "12", whereas the sum of the two bases is "7" could perhaps be used to make a similar point. And if the "5" which is the hypotenuse is used to determine the number of edges on each face of a regular solid, that solid (i.e. the one constructed out of pentagons) will have twelve faces (the dodecahedron). Without bothering to interpret these rather suggestive geometrical similarities, it will suffice to mention that another way of making the same distinction is to compare the square and the triangle in simple conjunction, as discussed in 4.4, with the more thoroughgoing interrelationship between the vertices of these figures which will be discussed in 6.3. The implication of all these distinctions, however, is the same: sevenfold distinctions will be more likely to represent systematic relations between groups of systems, whereas twelvefold distinctions will be more likely to represent formal relations between the elements of a single system. On this basis we shall discuss the latter in Chapter Six and the former in Chapter Seven. We can now summarize the primary logical functions of each of the integers between "0" and "12" in the following way: 0 = the starting-point of synthetic integration; 1 = the endpoint of synthetic integration; the starting-point of analytic division. 2 = the polarity between 0 and 1 as the first "level" of analytic division. 3 = the number of steps in the process of simple synthetic integration, and the number of explicit elements; 4 = the number of elements in second level analytic division; 5 = the combination of "2" and "3" as found in natural organisms (addition); 6 = the combination of "2" and "3" as found in inorganic systems (multiplication); 7 = the symbol of logical systemization on a higher level, as a dynamic process involving self-renewal and transformation (= 3+4); 8 = the number of elements in third level analytic division; 9 = the number of steps (and elements) in the operation of second level synthetic integration; 10 = the perfection of the number system: the point at which it begins to repeat itself; 11 = the symbol of an imperfect or incomplete system (5+6); 12 = the perfection of logical systemization on the level of a single system (3.4). The symbolic value of these numbers (and others related directly to them) as they interrelate in the structure of various types of compound logical systemization will be our concern throughout the remainder of Part Two.NOTES TO CHAPTER FIVE
1. Concerning the analytic and synthetic numerical progression, C495:68 notes that "the two progressions 1,2,4,8 and 1,3,9,27 stand at the head of Adrastus" list of geometrical progressions of the primary and most perfect kind." Cornford adds that Adrastus stopped at the cube (i.e. "2=8" and "3=27") because "the cube symbolizes body in three dimensions." But given the modern tendency to regard space and time together as a four-dimensional unit, the same analogy would now suggest that the progressions should be carried out one step further (i.e. to "16" and "81"), as we have done. 2. Cornford raises the interesting point that this series of numbers "adds up to the perfect number, 10" and "contains the numbers forming the ratios of the perfect consonances: 2:1 (octave), 4:3 (fourth), 3:2 (fifth)" [C495:69; cf. D491:8-13]. 3. In the system of Cabalistic cosmology the Cosmos is regarded as a series of four interlinking worlds, each of which consists of ten elements plus a hidden eleventh, which represents the "hidden treasure" of Knowledge ("Daat") [P538:22-3,108-9,122-3]. However, "11" is not actually the best number to represent this ultimate element in the system; since this element stands for both the beginning (birth) and the end (death) of each tenfold world system, it would be more accurate to use our synthetic notation and refer to its function as the "0" and "1" of the system. (Differentiating between its roles as the "0" and the "1" reveals a total of twelve elements in each "world".) 4. An interesting example of the sum "5+7 = 12" is the difference between the musical scales of the East and West. The pentatonic scale has five notes, whereas the Western scale has seven (the repetition of the first being called the "octave"). This is the same relation which holds between the black and white keys on the piano (i.e. between A#, C#, D#, F#, G# on the one hand, and A,B,C,D,E,F,G on the other).