Men all pay homage to what understanding understands, but no one understands enough to rely upon what understanding does not understand and thereby come to understand. -- Chuang Tzu [C55:5.288] Reality is not perfection except in the active equilibrium of the fusion of dynamic opposites. --Purce [P538:19] 3.1 The General Form of Simple Synthesis If we were to regard logic solely in terms of its mathematical foundation, then our discussion of its analytic forms in Chapter Two would suffice as a complete introduction to all the formal relations utilized by logic on its most abstract level. Bertrand Russell, the modern philosopher primarily responsible for popularizing the view that logic and mathematics are identical, would affirm that logic is concerned solely with "'analytic' propositions" [R517:204]--i.e. with propositions which are both a priori and non-empirical. This approach has been so widely accepted during the twentieth century because it promises to ground philosophical speculation once and for all in the solid rock of logical clarity, and likewise logical thinking in the solid rock of scientific, mathematical rigor. But in spite of the widespread acceptance of this modern presupposition concerning the nature of logic, we should not be satisfied with such a one-sided approach. For the application of the Geometry of Logic to systems in Part Two depends upon establishing the logical foundation of synthetic reality with equal rigor. Both modern and traditional logic do provide us with valuable tools for the clarification of thought; but each is valid only within the bounds of the perspective on which it is based. Modern logic, with its firm foundation in mathematics, is indeed, as Russell insists, entirely analytic. It must be, for it rules out beforehand the relevance of concrete experience. Traditional logic, too, is predominantly analytic. However, if we look beyond the narrow scientific limits of analytic thinking, and open up the field of logic to the realm of experience and its multifarious art forms, we will find a radically different way of constructing logical relations: for in addition to the analytic forms of relation discussed in Chapter Two, experience reveals certain synthetic forms of relation. In the immediate, non-reflective experience of everyday life, we are often confronted with situations in which there is no clear-cut distinction between "positivity" and "negativity". Numerous questions which arise empirically, or are asked from the synthetic perspective of immediate experience, cannot be answered with a simple "yes" or "no", but only with "yes and no". This fact cannot be forced into the neat compartments of a purely analytic system of logic; yet there is no reason to assume that it is therefore irrelevant to or incompatible with the structure of logical inquiry. On the contrary, such experiences represent the "unfolding" of "an essentially logical process" [W533:21]. Since our expressed interest is in developing an approach to logic which is intuitive as well as conceptual [see Introduction], the logical basis of such apparently illogical (i.e. non-analytic) experiences is well within the scope of the Geometry of Logic. Moreover, some logicians in the West, notably Hegel and the whole tradition of German Idealism, have expended a great deal of effort in constructing entire philosophical Systems on the basis of synthetic logic. Rabel's paraphrase of Kant's definition of the terms "analytic" and "synthetic" provides a helpful clue as to what is involved in this distinction: "Analytical judgements explain, synthetical judgements expand" [R422:106; cf. K105:10-4]. Analysis explains various elements (e.g. concepts) by dividing a unified whole into its constituent logical parts. Such analytical explanations sometimes distort the nuances inherent in the original whole, as when a twenty- four hour period is explained as being composed of "a night and a day"; but this is only to be expected, for the explanatory power of analysis depends on the ability to reduce what is complex to what is readily graspable (?!). Synthesis, by contrast, expands its elements by integrating two or more opposing terms in a "third term", a "higher unity" which is more than simply the sum of its parts. The third term transcends the simple opposition posed by the first two terms in a way which is completely foreign to analysis: it represents both the unity and the diversity, or indeed, the "unity-in-diversity" of its constituents. Whereas analysis at its simplest level is a one-step operation yielding two elements (i.e. a simple division of one into two), synthesis at its simplest level can be described tentatively as a two-step operation involving three elements. Using Hegel's well-known terminology, these two steps can be described as (1) that in which the "thesis" is confronted with an "antithesis", and (2) that in which the two opposing terms are united and transcended in the form of a "synthesis". Using arrows to denote the radically dynamic character of synthetic relations, we can depict these two steps as follows: Figure 3.1: Hegel's Logic of Synthesis (a) Step One (b) Step Two This initial way of mapping synthetic relations will be revised as we take a closer look at its full logical form, but for now it provides an adequate description of the general form followed by the simple synthetic operation which we shall refer to as "synthetic integration". 3.2 A Symbolic Notation for of Synthetic Logic We have already discussed [in 1.4] the crucial importance for any modern logical approach of choosing appropriate symbols to represent the key terms and relations at the outset of any given inquiry. Since the Geometry of Logic abstracts from all specific types of material terms and relations, we will once again require only one set of symbols, which can be used to represent the general character of all synthetic forms of relation. In our elaboration of the analytic forms of relation the two symbols "+" and "-" were sufficient to define all the levels of analysis in its most abstract structure. These same symbols are can also be used to describe the first of the two steps in a synthetic operation mentioned in 3.1, in which one element (+) is confronted by its opposite (-) [cf. Figures 3.2a and 3.1a]. But the second step poses a problem, for we clearly require a new symbol to represent the crucial third term of a synthetic relation [cf. Figures 3.2b and 3.1b]. Figure 3.2: Symbols of Synthesis (a) Step One (b) Step Two Because of the naturally obscure nature of synthetic relations (at least in comparison to the clear-cut nature of analytic relations), and because no symbol for "synthesis" (in this non-analytic sense) has been generally accepted in the past, it will be helpful to discuss several ways of symbolizing this form of relation. Discussing the pros and cons of several options will help to clarify the nature of the operation of synthetic integration. The most obvious way of symbolizing the integration of "+" and "-" in a single synthetic term is to combine the two to form the symbol " ". This option has the advantage of explicitly stating the logical roots of the synthesis in such a way as to reveal why our response to empirical situations evincing synthetic integration is often "yes and no". But it is also misleading for this very reason. The mere appending of one symbol (+) to another (-) would seem to imply that the third term in a synthetic relation is no more than an artificial holding-together of two entities which remain discrete. (A more adequate response to synthetic reality might be "yes-no"; yet even this, as we shall see, is expressing a synthetic situation in analytic terms.) Moreover, the term " " is used in mathematics to refer to a choice between "+" and "-": "1" means "+1 or -1" (though the force of "or" is often intended to be inclusive rather than exclusive). The symbol " " therefore fails to carry with it the implication of transcendence, which is vital to the process of synthetic integration. Using this symbol would be like mixing a bucket of black paint with a bucket of white paint and referring to the end product as "black and white", instead of simply "grey" paint. The symbol "x" avoids all the drawbacks which plague " ". As the mathematical sign for multiplication, it has a natural place alongside "+" and "-", yet it refers (rather appropriately) to a radically different kind of operation. Thus if these three terms are used to refer to logical forms of relation rather than to mathematical operations, one would expect them to have parallel, but clearly distinguishable functions. Such would be the case if we adopted "+", "-" and "x" as symbols for the three constituents of the synthetic form of relation. For just as multiplication is related to, yet transcends, the operations of addition and subtraction, so also synthetic integration is related to, yet transcends, the analytic division between two opposing terms. The letter "x" is also used as the archetype of all mathematical variables. Using it in our present context would not imply a simple choice between one of the two root terms "+" and "-", or both together, but would open up the possibility of assigning some intermediate value to the third term. The extent to which a given synthetic relation leans more toward the "+" or more toward the "-" in its third term could then be determined only on the basis of a concrete experience involving the particular synthetic operation under consideration. The suggestiveness of its two mathematical functions, therefore, makes "x" a strong candidate for a symbolic representation of the third term of a synthetic operation. Its only drawback is that it could be mistakenly taken to imply that a given synthetic relation is itself "variable", when in fact, it is every bit as determinate as are its "+" and "-" roots. A third option is to attempt to preserve the useful aspects of both " " and "x" but avoid their drawbacks by using a familiar set of symbols whose ordinary use is rather more distant from that of "+" and "-". The symbols "0" and "1" fit nicely into this category. In 1.2 we noted the common use of these symbols to represent "nothing" and "everything", or "the null class" and "the universe class" in a given universe of discourse. The distinction between "0" and "1" is itself based on a first level analytic division; for in relation to "0", "1" is "+", and in relation to "1", "0" is "-". But if "0" and "1" are regarded as referring not to numbers or classes, but to synthetic forms of relation, then the function of "+" and "-" will be to mediate between the extreme functions of "0" and "1". For the "0" would serve as a hidden term, or presupposition, of all synthetic operations, and the "1" would represent the "third term" which unites or integrates "+" and "-" by transcending them both in the form of some sort of conclusion. ("1" is particularly appropriate to fulfil this integrating, or unifying, role because in its mathematical meaning it represents the integer which is defined as a numerical unit.) Thus the full pattern of synthetic integration is a three step process, which can be mapped as follows: Figure 3.3: The "0" and "1" of Synthetic Integration Here, of course, the symbol "1" would not represent "the universe class", but rather a "unit of synthesis"; likewise "0" would not represent "the null class", but rather a "pre-synthetic state".3 This symbolism is particularly appropriate when used to represent a chain of two or more synthetic relations. For the "1" of a given synthetic operation can act as a "0" for a subsequent synthesis, thus giving rise to the compound symbol "0". A full treatment of such "complex" synthetic relations will be given in Chapter Four. At this point, however, we should note that, by combining "0" and "1" in various ways, we can actually achieve much the same "variable" result as is suggested by the use of "x", but without implying that the third term of the synthetic relation is an unknown. The flexibility of these symbols is evident when one is divided by the other in various combinations: the standard algebraic symbol for an "indefinite number" is "0/0" and for "infinity" is "1/0" [B506:6.10-1]; likewise "1/1" represents unity and "0/1" emptiness, or Nothing. These four ways of dividing the numbers "0" (+) and "1" (-) are related in terms of a perfect second level analytic division: Figure 3.4: The Analytic Combination of 0 and 1 Indefinite!(0/0) # # # Unity (1/1) Infinite (1/0) # # # Nothing (0/1) The arithmetical meanings for these combinations of 0 and 1 can be transferred directly to the logic of synthesis simply by noting that a synthetic operation ideally passes through each of these stages as it progresses from an indefinite "pre-synthetic" state (0), through the extremes of infinite positivity (+) and empty negativity (-), to a single unified solution (1). Moreover, the reality that synthetic relations do not always operate according to such a neat plan can be expressed by representing their outcome in terms of fractions (i.e. real numbers between 0 and 1). In this way the logic of synthesis can be regarded as the formal basis of the logic of probability, which is the application of logical laws to real situations, with a view towards determining the appropriate fraction to represent the chances of completing a given synthetic operation. Although the symbols "0" and "1" are obviously superior to " " and "x" as representations for the third term of a synthetic operation, it will be helpful to consider one final option by attending to the geometrical shape of the symbol rather than to its mathematical meaning. That the mathematical symbols "-" and "+" are appropriate terms for representing analytic relations is confirmed by the fact that they have the same geometrical shape (a short horizontal line and a symmetrical cross) as the standard maps for the first two levels of analytic division [cf. Figures 2.2 and 2.3]. In choosing a symbol for synthetic integration we should therefore keep in mind the importance of its geometrical value, as well as the implications, if any, of its ordinary mathematical use. On these grounds the simple combination of "+" and "-" in " " seems even less appealing, as does the symbol "x", which is essentially a "+" turned sideways. (Note the "x" formed by the diagonals of the square, which was established as a standard map for the inversion of second level analytic division in 2.2.) Neither of these is sufficiently distinct from "+" and "-" to represent the radical difference between analytic division and synthetic integration. At first sight the symbols "0" and "1" fare no better. Since no synthesis stands alone in actual experience, but is interrelated with other synthetic operations, the third term not only concludes the given synthesis, but sets the scene for another; so the symbols "0" and "1" in Figure 3.3 will occur together in every real synthetic operation. This means that, even though we originally introduced synthesis as a twofold operation, and even though it appears to involve four elements when we view it analytically, as in Figures 3.3 and 3.4, it is properly regarded as a threefold operation. The most appropriate geometrical figure for mapping the constituents of a synthetic relation, then, is the figure with three vertices and three sides: the triangle. The simplest and most regular triangle, the equilateral, is the most likely candidate for an intuitive representation of these conceptual relations. Mapping synthesis onto the triangle (using "0" and "1" in the above sense for the time being) enables us to construct the circuit given in Figure 3.5, in which the primary direction of synthetic influence is indicated by arrowheads.@ This figure symbolizes the way in which the roots of a synthesis are opposed to each other, as it were, "horizontally", yet "rise" to a common, "higher unity" in the third term of the synthetic relation. This same operation can be mapped onto the well-established analytic figure of a square by artificially separating the "0" into "0" and "1", and joining two right triangles by a 11 common hypotenuse [see Figure 3.6].@ Using the right triangle in this Figure 3.5: The Triangle of Simple Synthetic Relations Figure 3.6: An Analytic Map of Simple Synthetic Relations way is even more appropriate than using an equilateral triangle, since the difference between the relation of "+" to "-" and the relation of these to "0" and "1" is symbolized by the difference in length between the hypotenuse and the shorter sides of each triangle. (Alternatively, if the two triangles in the area of the square are labelled with "+" and "-", then their common hypotenuse would represent the synthesis of the two extremes.) The triangle is clearly more suggestive as a symbol for the threefold operation of synthesis than is the rather ambiguous "x" or the somewhat awkward "0". Aside from its rather obscure, technical use in the construction of formal languages, as a symbol representing sets of "well-formed formula", the triangle does not have the potentially misleading connotations, resulting from familiarity of use, which are possessed by all the alternative symbols considered above. The triangle has been defined in terms of the overall process by which "+" and "-" produce "+", "x", or "0". However, we have not actually been searching in this section for a symbol which can represent the entire threefold operation of synthetic integration, but only the third term of the synthesis. Whenever the synthetic operation as a whole is concerned, it will therefore be best to use the symbol " ". The symbol "0" can then 1<1 be reserved for instances in which the notation requires us to break the synthesis down into its constituent elements. When the third term alone is referred to without any intention of distinguishing between its "0" and "1" aspects, the symbol "x" will sometimes be substituted as its equivalent. Taking advantage of the benefits of both "01" and "x" will help us avoid wherever possible the rather unfortunate mixture of metaphors involved in using both mathematical signs and numbers in the notation of a single expression. We can test our symbolic notation by applying it to a straightforward example of synthetic operation, drawn directly from the experience of every human being. In 1.4 the male (+) and female (-) distinction was mentioned as an example of a first level analytic division, at least as it applies to the biological function of the sexes in procreation [see note 1.20]. The whole topic was treated from an entirely analytic perspective, with a view towards establishing the nature of such an exclusive (A = -A) relation. Hence the logical status of the child 1/ was not even considered. When this same distinction is treated synthetically, by contrast, the child takes on an integral role in the overall situation. For it completes a process which remains partial as long as the "+" and "-" are kept in opposition: synthesis, whatever it is, certainly involves more than merely two polar opposites being held together side-by-side. Procreation is in a sense an imperfect example of synthetic integration, since it does not result in a third sex, but in a child who is one or the other. As such, the symbol " " would be appropriate to represent this synthetic role, at least insofar as it is read as "+ or -". For the offspring will inevitably be fit into one category or the other. But before it is actually born, the symbol "x" would be even better, since (barring the intervention of modern medical techniques) the child's sex is unknown to its parents. If we look beyond the (analytic) sex label, however, we find that every child is in fact a perfect synthesis of its parents, for the chromosomes which determine the potential size, shape and character of the child are drawn from both sources, and are mixed always in quite a unique way. The " " fails to represent such a mixture, whereas the "x" implies the potential for a wide spectrum of results. But the "0" is even better, for it implies 1&1 that the resulting "third term" is a separate individual (1), though defined in terms of its overall synthetic context ( ), and that some of its characteristics may well have been present in its parents only covertly (as "0"). Thus we can say that a child draws "everything" (1) from its parents, yet maintains a transcendence from them which somehow goes both beyond and before their influence, so that we can just as accurately say that the child (as the instigator of new syntheses) draws "nothing" (0) from its parents. This example serves as a "preview" of the way our new notation-- particularly "x" and "0"--can be used to clarify the various constituents of a synthetic relation. We must now take a step backwards, and ask what laws form the basis of this operation of synthetic integration. Only in this way will we then be adequately prepared to examine in 3.4 other examples of various types of synthetic integration and fully to appreciate what is involved in the "complex" forms of relation (discussed in Chapter Four), or in the "compound" forms of relation (discussed in Part Two), which combine both analytic and synthetic operations. 3.3 The Logical Laws of Synthetic Integration We began our investigation of the Geometry of Logic by discussing the fundamental logical laws of identity (A = A) and non-contradiction (A = -A). We then witnessed in Chapter Two how these two laws function 1/ as the necessary presuppositions of analytic division. As such, they should be regarded as "set apart" from other, secondary laws of formal logic. Yet, because of their exclusive interest in propositional logic, and their consequent reluctance to abstract all material elements from their formal logic, most modern logicians now bunch these laws together with a number of other "fundamental" formal laws, such as the logical equivalents of the algebraic laws of the commutative, associative, and distributive relations between elements (i.e. "a+b = b+a", "(a+b)+c = a+(b+c)", and "a(b+c) = ab+ac", respectively). The irrelevance of these other formal laws to the operation of analytic division in its most abstract manifestation (as described in Chapter Two) clearly indicates their secondary nature: they come into play only when some measure of content is added, even if it be nothing more than the content implied in the consideration of "propositions in general". The question we are now faced with, however, is: What logical laws form the basis of synthetic integration? Content in some form must be taken into consideration in order to answer this question; for, whereas analytic division is the operation which determines the logical structure of thought in general, synthetic integration is the operation which determines the logical structure of what we might call (with Kant) sensible intuition in general. These two elements can be regarded as the formal and material elements in all experience. Hence the logic of synthesis, which at its simplest and most general is the logic of sensible intuition, is the polar opposite of the logic of analysis. If the latter is defined in terms of "analytic a priori" (+-) laws, we can therefore define the former in terms of "synthetic a priori" (-+). The traditional "laws of thought" must give way in this context to an equal and opposite set of "laws of intuition". Our task in this section will be to determine just what these alternative laws are and how they can best be stated. This will prepare us to discuss in 3.4 some specific examples of how synthetic integration is applied in actual situations. The main problem in dealing with the logic of synthetic integration is that, by its very nature it cannot be expressed accurately in conceptual terms. Expression in words (logoi) is naturally suited to describing analytic division, but must be strained in order to describe synthetic integration (ho logos). In fact, the only way a logic of synthesis can be developed is to borrow terms and laws from the logic of analytic division, redefining them in terms of their polar opposites. The precise meaning of the logical system which results will be difficult, if not impossible, to conceive in abstraction, since conception is primarily analytic; but it will nevertheless provide a useful and much needed explanation of the intuitive form according to which all human experience is structured. The law which governs the second step in a synthetic operation is the opposite of the law of non-contradiction, so we shall refer to it as the "law of contradiction". This new law states that "A = -A". For in the first step the "+" empties itself completely of its positive nature and loses itself in the negativity of its opposite. "A" becomes equal to "-A", yet without losing its discrete nature as "A". This law of contradiction is the logical basis of the essential tenets of many world religions, from the Hindu doctrine of the self-emptying of man to become Brahma, to the Christian doctrine of the self-emptying of God to become man in Christ. Such doctrines are perverted when they are interpreted in such a way that the contradiction is resolved or explained away: for the power of religious discourse is inextricably bound up with its dependence on the laws of synthetic integration. To interpret them analytically is to render them powerless. As we shall see, the law of contradiction, "A = -A", "+" becoming "-", is at the heart of much of our everyday experience, even in the secularized modern world. For in real experiences we are constantly met with things which are what they are, yet are their opposites as well--with questions that can be answered not by "yes" or "no", but by "yes and no". The law which governs the first and third steps in a synthetic operation is the opposite of the law of identity, which we shall refer to as the "law of non- identity". This new law states that "A = A". For 1>/ in the third step the "+" transcends the entire +/- context, and becomes, as it were, "born again" into a new form. (A similar process occurs in the hidden first step.) "A" no longer can be equated with "A", for it is now radically transformed. "A" is the same entity, yet it is no longer "A": in union with "-A" it is now integrated, it is now "1". We experience the effects of the law of non-identity whenever we feel something (either sensibly or emotionally). For to feel is to be confronted with the non- identity of oneself with oneself, or of an object with itself, to be jarred into an awareness of the non-conceptual basis of the real world, to realize that A = A. 1+/ Some of the differences between these two laws of synthetic integration and the two traditional laws of analytic division can be brought to light by comparing the properties of their respective geometrical representations, the triangle and the square. The triangle is the only rectilinear figure for which equality of angles guarantees equality (i.e. proportionality) of shape. This is exemplified in Figure 3.7: Figure 3.7: The Relation Between Angles and Shape (a) In the Triangle (b) In the Quadrilateral No matter what length is given to the sides of various triangles, if the three angles are the same in each case, then the shape of each will be equivalent. If "+" represents the relation "sameness of angles" and "-" represents the relation "sameness of shape", then the equation "+ = -" (reading "=" as "guarantees") holds for triangles. For quadrilaterals and all other rectilinear figures, by contrast, sameness of angles does not guarantee sameness of shape--thus "+ = - ". The consistency of geometrical figures with the laws of synthesis and analysis, respectively, is due to the fact that geometry is essentially synthetic, whereas mathematics is essentially analytic: the former deals (albeit with the full use of analytic tools) with the form of space, the realm of experience, whereas the latter deals with the realm of thought, in complete abstraction from experience. As mentioned above, these two laws of synthetic logic have been described using the terms of analytic logic. This has two primary consequences, one beneficial and one detrimental. The beneficial implication is that the four fundamental laws of logic, two synthetic and two analytic, can be mapped in terms of a perfect second level analytic relation by contrasting "A = ..." (+) with "A = ..." (-) on the first level and "...A" (+) with "...-A" (-) on the second: Figure 3.8: The Four Laws of Formal Logic (a) Mapped onto the Cross (b) Mapped onto the Square These maps clarify why the two traditional laws have the appearance of all-sufficiency: because they occupy the "++" and "--" positions (the first and the last in Figure 3.8a), it is easy to ignore the polar opposites of each and assume the relation to be a first level analytic relation between "+" and "-". They do have a logical priority for this reason, but this should not deceive us into emphasizing their importance to the complete exclusion of their second level partners. Figure 3.8b is particularly helpful in this regard by positioning the laws of analysis on one diagonal and those of synthesis on the other, thus implying a more balanced status. The detrimental consequence of such an analytic frame of reference stems from the fact that we in the West have been trained from an early age to assume (unconsciously) that all of life (thought, intuition, and all the experiences which they compose) can be adequately interpreted only in terms of the principles of identity and non-contradiction. As a result we find it difficult, if not impossible, even to countenance the idea that there may be a different way of thinking properly about aspects of the world which are not bound by the formal limits of our analytic thought patterns. The analytic frame of reference merely heightens the stark contrast between the two methods. Eastern cultures tend to be more open to the possibility and value of such "non-logical logic". ("To know non- knowledge is the highest good", says Lao Tzu [L532:71].) Indeed, their openness to synthetic integration and their interest in its formal basis may help to explain why, as any reader of ancient Chinese or Indian philosophy will readily affirm, the philosophers of the East often seem from our analytic perspective to be inconsistent and unclear in their mode of expression. For just as the intuitive experience of the Western philosopher too often lacks integrity because of his inability to force it into the analytic mould, which he has mastered and in which he excels, so also the Eastern philosopher too often lacks consistency because of the inherent difficulty in expressing certain thoughts according to the mould provided by the synthetic integration which characterizes his approach to life. True integrity and logical consistency are possible only when both of these extremes join hands and learn from each other, so that synthetic integration and analytic division are seen as complementary modes of philosophical inquiry, each with their own legitimate logical foundation. (In Part Two we will investigate the ways in which these two types of logic work together to produce compound forms of relation.) From the logical perspective, analysis precedes synthesis, for the "++" and "--" relations are themselves a first level analytic relation, which only includes "+-" and "-+" when it is extended to the second level [see Figure 2.3a]. For this reason a logical treatise should discuss analytic division before discussing synthetic integration. Without doing so we would have had no means of describing the latter operation. From the empirical perspective, however, synthesis precedes analysis: conceptualization depends on the prior existence of a body of material which is given to it and upon which it can impose its forms. Prior to analysis, the positive and negative roots of synthetic integration remain "unknown" (i.e. unspecifiable): all we know is that we experience the contradictory self-opposition of the synthetic process. Once synthesis is complete, though, we can turn around and analyze the constituents of the synthesis. Thus analysis can be mapped onto the standard synthetic map simply by separating the "0" and "1" in "0" and reversing the arrows [cf. Figure 3.6]: Figure 3.9: Analysis, Viewed Synthetically We reverse the flow of synthesis in this way whenever we adopt the laws of identity and non-contradiction in order to classify the contents of our experience into similar groups. The step from "1" to "+" and "-" is analytic division as described in Part One; the step from there to "0" verifies the result by insuring that the cross between "+' and "-" is always the null class (hence Boole's equation for the law of non-contradiction: "x(1-x) = 0" [see 1.2]). Such analytic "exclusion" must not be confused with the synthetic "inclusion" presented in experience itself, for as we have seen, the latter operates on the basis of its own quite distinct logical laws. 3.4 Perspectives in Synthetic Applications The reversal of direction which characterizes the change from analysis to synthesis and from synthesis to analysis represents a fundamental difference in the weight which is given to one's perspective in these two operations. Perspectives arise out of reflection upon one's experience, so the original non- reflective experience of synthetic integration is in a sense perspectiveless. All the perspectives we can adopt in reflecting on our experience are present in experience, but only in a potential state, meshed together in synthetic unity. Synthetic integration establishes the perspectival basis on which alone analytic division can be used to discern different perspectives. Synthesis proceeds from "0" to "1"; analysis begins with the "1" and divides it into the "many". If we attend to the perspectival shifts involved in synthetic integration, shifts which can be identified only from the standpoint of analysis, then the laws of synthetic logic turn out to be less objectionable than they appear to be at first. For the contradiction can often (though not always) be made more palatable by taking note of the perspectival difference between the terms on either side of the "=". Two examples will help to illustrate this point. First, the whole of modern sub-atomic physics is based on a fundamental contradiction which has been accepted, but made palatable by means of an awareness of the perspectival relationship between the two sides of the contradiction. The contradiction is best known for its application to the physical theories of light, which can be viewed either as a particle, or as a wave, but not both. The general principle upon which the theory of light is based is that the method by which the observer views sub-atomic "objects" (or "observables") determines, or even changes, its very state. Thus the physicist can measure either the position or the velocity of an observable, but it is impossible to measure both at the same time. Once the synthetic experience of a single object manifesting two contradictory qualities is analyzed in terms of its perspectival basis, it becomes quite plausible to hold both views simultaneously. The second example is simply a reminder of what we have already said about the triangles in Figure 3.7a. If we are asked whether these triangles are equal, we find ourselves answering "yes and no": yes, they are equal from the perspective of their angles and their shape; but no, they are not equal from the perspective of their size. In geometry the "=" sign is often used to relate two terms which are equal in one sense, though unequal in others. In co-ordinate geometry, for example, the equation "AB = Y" can be used to describe a line segment (AB) which is parallel to the Y-axis [W516:165-6], even if it is not equal in other respects (e.g. does not cross the X-axis at the same point). It is crucial to understand that such attempts to explain the perspectival basis of the contradictions of synthetic integration are primarily valuable as aids to seeing the compatibility of the analytic and the synthetic. "A = -A" and "A = A" still hold in synthetic integration, even in those cases for which we can "explain away" their contradictory appearance by analyzing, e.g., the influence of time in the changes which have taken place. We learn to cope with the rhythm of "contradiction-resolution" to such an extent that we become unaware of our constant participation in it--that is, until a contradiction crosses our path which refuses to submit to the neat categories of perspectival analysis. In such cases we are left with two alternatives. First, we can refuse to admit its contradictory nature, and hope we will not suffer any adverse consequences as a result of our inability to understand it. Or second, we can seek to adopt a new, higher perspective, from which the contradiction can be taken up into ourselves and therefore "understood" in a non-discursive sense. Only by adopting the latter approach is the reciprocal relation between analysis and synthesis fully realized. The remainder of this section will therefore be devoted to a discussion of several examples of the many diverse ways synthetic integration manifests itself as a means of encouraging a heightened perspective. The simplest and most general description of the content of synthetic integration is as the process by which "this" becomes united with "that", so that both together occupy common ground in a new dimension of reality. This union of "this" and "that" marks the starting-point of philosophers as diverse as the ancient Chinese Taoist, Chuang Tzu, and modern Western Idealists, such as Hegel [see e.g. H112:???]. Those who turn around and regard these same terms analytically end up with a considerably more sterile result: "This remains identical [i.e. remains a discrete object] and is thereby distinguishable from that" [S515:404]. But without the synthetic recognition of a common universe in which "this" and "that" both take part, and in which they may be combined, such an analytic distinction between them would never be possible. Philosophers who base their entire Systems on synthetic logic proceed not from the abstract to the concrete, but from the concrete synthesis between "this" and "that" to more abstract syntheses such as "self" and "not-self" [e.g. Fichte, F : ], eventually leading to some ultimately abstract Synthesis--e.g. between "Being" and "Time" in the form of "Dasein" [Heidegger, H : ], or (for Hegel) between God and the World in the form of Absolute Spirit (i.e. of Religion and Art in the form of Philosophy). Synthetic logic manifests itself in one form or another in virtually every corner of philosophy. For synthetic operations are involved in the construction of many arguments. We have already mentioned [in 1.3] that the simple syllogism is a good example of synthetic reasoning. But even arguments which are not intentionally syllogistic often manifest the essential characteristics of synthetic integration by positing a starting-point, connecting it with a condition with which it would not ordinarily be connected, and drawing a conclusion which transcends either condition on its own. Kant's "Analogies of Experience" [K105:218-65] are a typical case in point. He describes three principles as governing the relations between empirical objects: a permanent substance persisting through all change (+), the law of cause and effect governing all change (-), and the integration of these in the postulation of the interrelatedness of all empirical objects (x). Only by presupposing that substance allows itself to undergo change (i.e. only when A = -A) are we able to conceive of the objects of our experience as constituting an unbroken network of integrated relationships. The contradiction that the permanent changes is easily resolved by viewing the situation analytically in terms of the distinction between a substance (which is permanent) and its attributes (which change). But this analytic perspective in no way nullifies the essential contradiction entailed in adopting a synthetic perspective on the same subject. Even the explicitly "analytic philosophy" of the twentieth century is based on a concept which is primarily synthetic (though their failure to recognize this fact often results in a distorted appraisal of the role of philosophical inquiry). Analytic philosophy is based on an analysis of meaning, and the meaning of a word, if one follows Wittgenstein's lead, is to be found by examining its use. The analytical tools of modern logic are therefore used to pry the meaning from the use of philosophical propositions, as well as from those of interest which are found in ordinary language. What seems to be ignored too often is the synthetic foundation of all meaning. For, as Stebbing rightly says in S515:499: "Means, or symbolizes, is a triadic relation; its three terms are: (i) the sound or mark, (ii) the person using the sound or mark, (iii) that which is referred to." The synthetic foundation of meaning can be represented by mapping this description directly onto the triangle of simple synthetic integration: Figure 3.10: Meaning as a Simple Synthetic Relation the symbol used 1 to refer the object the subject of reference who refers a referent) Stebbing adds: "When...we say that the word has meaning, then the word must not be identified with the sound (or mark); it will be the sound (or mark) as used to refer to something." "Meaning", therefore, refers not just to the third term of the relation, but to the entire operation of synthetic integration. The meaning of a word is not to be found in the meaning of the word(hence A = A), but in the use to which it is put in the real act of reference (where A = -A). This conformity to synthetic logic does not preclude the possibility of treating the meaning of a word, once it is given, from a purely analytic perspective. Indeed, analytic philosophers take this as their starting-point when they analyze the constituents given in such syntheses. Such analysis may clarify meaning, but on its own it can never create meaning; nor can it explain how meaning is created. Hence, although an exclusive concentration on analysis certainly allows the philosopher to "make sense" (i.e. to produce coherent explanations), it prevents him from making sense (i.e. from originating new syntheses). The philosophical question at the heart of the science of cybernetics (viz.: Can computers think?) can be readily answered once the fundamental difference between the logic of synthesis and the logic of analysis is understood. A computer is a logic machine, which can be programmed to do analytic operations on command. To this extent a computer can think. But real thinking--i.e. thinking of the sort performed by human beings--is ordinarily more than bare analysis. For it includes both analysis and synthesis in combination. No machine will ever be designed which can think synthetically. Programming a computer with software based on the laws of non-identity and contradiction would be like switching the machine on while the hardware is immersed in water: the whole structure of the machine would short-circuit. The most a computer can do is to ape synthetic operations by mapping them onto analytic models, as exemplified in Figure 3.6. Just as man's ability to conceptualize depends on intuitive content upon which its generalizing activity can operate, so also the computer is powerless until it is programmed and fed specific information and instructions. It never gives itself its own original intuition; for this it depends on the intervention of some human agent. Human reasoning is unique precisely because it is able to combine synthetic and analytic logic in such a way as to produce compound forms of relation. With these forms, which shall be discussed thoroughly in Part Two, the Geometry of Logic reaches its climax. Moral Philosophy, too, is incomplete without a recognition of the synthetic integration of good and bad in every ethical action. For in moral experience the good comes out of the bad and the bad out of the good. The contradiction implicit in the synthetic basis of morality must be taken into account by any adequate philosophy. The same is true in the philosophy of religion, or in theology. The doctrine of the Trinity is an obvious example. According to the traditional Christian doctrine of the Trinity, either the Father is regarded as the ultimate source from which both Son and Spirit proceed [see Figure 3.11a] or the Son is regarded as on a level with the Father, so that the Spirit proceeds from both the Father and the Son [see Figure 3.11b]. Figure 3.11: The Traditional Conceptions of the Trinity (a) Priority of the Father (b) Priority of Father and Son Together In either case the threefold relation has obvious synthetic characteristics, though in both cases something is missing if the relation is to be a perfect example of synthetic integration. Along these lines Jung suggests that the Trinity is in fact a Quaternity in which one element of the relation has traditionally been ignored [J473: ]. The fourth "Person", he suggests, is Satan, who can be regarded as the dark side of God [J524:81]. This puts Satan in a position which seems strange indeed to most Christian theologians, yet Jung's psychological interpretation of this doctrine can be defended as being perfectly consistent with the logical structure of synthetic integration. Any attempt to describe the various forms in which God manifests himself will be a description of the synthetic integration which begins with the transcendent God (0) and ends with the Christ (1), the Chief representative of man before God. The spiritual struggle through which this incarnation of God must pass is naturally conceived as a struggle between the "Holy" Spirit (+), the Spirit of Light, and the "Unholy" Spirit (-), the Spirit of Darkness. This suggests the following analytical map of the synthetic relations inherent in the Godhead: Figure 3.12: The Godhead as a Synthetic Relation The Christ-man Dark Spirit (-) Holy Spirit (+) The Father-God (0) Jung's inclusion of Satan in the Godhead is actually quite consistent with the traditional conception of Satan as originally occupying an important position in the Heavenly City. The story of the incarnation of God in the Christ-man then becomes the account of how God conquers evil by taking its power (the Dark Spirit) back up into himself, where it belongs, and where it existed before the Fall. Shortly before realizing this purpose in the act of his death, Christ asserts his Oneness with the Father (see John 14:9-10 and 17:11). This unity of "1" and "0", which results when "+" submits to "-" without being totally overcome by it, perfectly exemplifies the logical form of simple synthetic integration. (Note that when Christ ascends, when the "1" is withdrawn, the disciples are given the "Comforter" (+) to help them in the task of taking up the "-" into themselves and realizing the "1" in themselves, as they await the return of the ultimate "1" which will mark the end of time.) Eastern philosophy and religion, as we have already seen [see notes 3.7,11-14], are much more explicit in their dependence on synthetic logic than Western philosophy and religion. God, or Ultimate Reality, is always regarded as "a supreme non-duality" which "is inexpressible and non-communicable in itself, i.e. ... it is totally transcendent [0] (or totally immanent [1], which amounts to the same [x]...)" [B528:200]. But this ultimate synthetic reality "can be expressed only in terms of a diametrical polarity", such as "man and woman" or "god and goddess" (i.e. "+ and -"). The distinction between an emphasis on the "1" and an emphasis on the "0" in one's religious quest is clearly exemplified by the distinction between the Hindu, who seeks "the unfoldment of the world" through the union of opposites in "1", and the Buddhist, who "seeks its [the world's] regression into the 'unborn, unformed' [the "0"] which is at the basis of all unfoldment" [214]. Indian Buddhists use the concept of "sahaja" to try to explain the ineffable starting-point of synthetic integration. The Buddhist, Bhusukapada, says: "Where there is no self [+], how can there be any non-self [-]? ...nothing goes or comes, there is neither existence [+] nor non-existence [-] in sahaja" [29]. This clearly resembles many Chinese descriptions of the "Tao": "The Way cannot be thought of as being [+], nor can it be thought of as nonbeing [-]" [C525:25.293], for this duality is bound up in an eternal cycle between "+" and "-", whose ultimate resolution is apparent only from the higher (x) perspective of the Tao. From this higher perspective [see W533:74] L532:40 says: "All things under heaven [all "+" and "-" objects] come about in existence [in "1"]. Existence [1] comes about in non-existence [0]. "The Tao is the overall " " on which this insight is based. The logic of synthetic integration is evident in many areas which are quite removed from the philosophical and theological applications described above. Not only do we experience time as the synthesis of past (+) and future (-) in the form of the present (x), but we also experience space and time together in the synthetic form of space-time, as modern physics now tells us. These intuitive syntheses are easily ignored by our conscious mind as it goes about the business of analysis. But when we are asleep, synthetic logic tends to exercise dominance over analytic logic. Jung and other psychologists have proved that dreams have a logic of their own which has clear implications for the dreamer's waking experience. In fact, the logic of dreams is identical to the logic of immediate, non-reflective conscious experience. Dreams often include elements which seem quite natural and acceptable as we dream them, but which are obviously contradictory when we recall them while awake. The obvious interpretation of this fact is that in dreams the laws of synthetic integration, A = -A and A = A, operate without being constrained by the laws of analytic division, the laws we use to reflect upon our dreams once we are awake. We can conclude this overview of some of the common applications of synthetic logic by comparing the general difference between the work of the scientist and the work of the artist. When the scientist observes the world, he does so in order to gather (synthetic) data for theoretical reflection. But his theoretical perspective on the empirical world operates primarily in terms of the laws of analytic logic (especially in the case of the "reductionist"); for the scientist's task is to discern similarities (A = A) and differences (A = -A) which regularly occur in the world. The synthetic basis of his observation is irrelevant to his concern, unless he wishes to adopt a more phenomenological approach-- e.g. by viewing his work as a form of art, or by philosophizing on the overall nature of the scientific method. When the artist observes the world, by contrast, he does so in order to represent what he sees in a new form. So he must first have some (analytic) understanding of the difference between the forms which are involved. But his empirical perspective on the world operates primarily in terms of the laws of synthetic logic; for the artist sees that in experience "A" is not simply "A", but goes beyond itself into the domain of "-A", and he attempts to depict this process in his artwork. "A" now represents and is represented by that which it is not. As a result, the observer of a work of art should be primarily concerned not so much to determine what it represents through rational analysis, as to perceive the work, to experience it in such a way that the synthesis it represents becomes a part of his own experience. Stebbing classifies logic as essentially a science, although she adds that the ability to think clearly may be regarded as an art [S515:473]. But the radically formal logic with which we are concerned is best regarded as neither an art nor a science, since it serves as the basis of both. For we have seen that, just as science is rooted in synthetic observation and is fulfilled in an analysis of the results, so also art is rooted in analytic observation and is fulfilled in the synthesis of the results. Thus the principles established by the Geometry of Logic clarify the reciprocal relationship not only between Eastern and Western philosophical perspectives, but also between the perspectives adopted in the practice of art and science.NOTES TO CHAPTER THREE
1. See e.g. R517:194, quoted above in Introduction. The details of Russell's approach are heavily dependent on the ideas of Frege and Peano. His achievement was not to initiate the modern approach to logic, but rather to develop a systematic defence of the unity of logic and mathematics [in R507], and to use this as a tool for revolutionizing philosophical inquiry, at least in England [see R517 and his numerous other philosophical publications]. 2. The distinction between "static" and "dynamic" interpretations of logical symbols will be discussed fully in 4.4. 3. The function of "0" in Figure 3.3 provides a logical basis of explanation for why the concept of "Nothing" is treated almost as a fetish by philosophers such as Heidegger [see e.g. H144d], who are concerned to expound the metaphysical foundations of the world as experienced--i.e. on its "existential" structure. Such endeavors can be rendered rather more easily comprehensible if they are regarded as attempts to take full account of synthetic forms of relation. 4. This fractional notation would be an appropriate way of representing the type of synthesis implied in Figure 1.6b. The ideal equivalent to this figure would be one in which the "B" circle which intersects the "A" circle is replaced by "-A". Synthetic integration could then be expressed (rather too simplistically) with the progression of geometrical figures given at the top of the following page. Another interesting way of representing this fractional notation is by the use of successively smaller tangent circles, arranged on a coordinate graph as shown at the bottom of the following page. The large circles represent all the whole numbers. The smaller circles, as specified between "0" and "1", represent all real fractions, so that any definite degree of probability in the third term of a synthetic operation could be represented by one such circle. Irrational numbers such as n or /2, and imaginary numbers such as /-1, are not represented 1- by any circle, no matter how minute, since they cannot be represented by a simple real fraction. 5. See e.g. S515:472-3. Stebbing lists the laws of identity and non-contradiction first on her list of fundamental logical laws, but adds eight others, implying that each deserves the same formal status. But with the exception of her third law, the "law of the excluded middle" [see note 3.8], these other laws are all principles of deductive or inductive argumentation. Only the first three apply to the relations between thoughts in general. It is for this reason that they are traditionally referred to as the "laws of thought" (even by Boole, the founder of the "Algebra of Logic" [see B506:]). 6. In the context of second level analytic relations, where "+-" is set in opposition to "++", and "-+" to "--", this law manifests itself as a "law of polarity", since one term always stays the same. In synthetic relations, however, there is no such continuity between opposing forces, so the relation remains an out-and-out contradiction. 7. B528:19 lists "the four great Upanisadic dicta, 'this atma is brahma', 'I and brahma', 'thou art that', and 'the conscious self is brahma'." Similar contradictions abound in Chinese philosophy, as when Chuang Tzu describes the "Tao" in C525:77: "The Way has its reality and its signs but is without action or form. You can hand it down but you cannot receive it; you can get it but you cannot see it. It is its own source, its own root.... It exists beyond the highest point, and yet you cannot call it lofty; it exists beneath the limit of the six directions, and yet you cannot call it deep. It was born before Heaven and earth, and yet you cannot say it has been there for long; it is earlier than the earliest time, and yet you cannot call it old. "The poetry of Lao Tzu is rich with such contradictions. In L532:41, for instance, he says: The clear DAO appears to be dark. The DAO of progress appears as retreat. The smooth DAO appears to be rough. The highest Life appears as a valley. The highest purity appears as shame. The broad Life appears to be stealthy. The true essence appears to be changeable. The great quadrant has no corners. The great instrument is completed late. The great tone has an inaudible sound. The great image has no form. 8. Consider again the synthetic character of a child: "A child is real and not yet real, it is in history and not yet historical. Its nature is visible and invisible, it is here and not yet here" [T523:95]. Such inclusiveness entails, of course, that the third great traditional law of logic, the "law of the excluded middle", does not pertain to the logical relations involved in synthetic integration. This law is subordinate to the laws of identity and non-contradiction because it introduces a new term, "B", so that "A = B or -B". But the synthetic law of contradiction, "A = -A", implies that, for some values of "B" at least, "A = B and -B" (i.e. "0 = + and -"). 9. Hume's willingness to compartmentalize his life into the pleasant experiences of dining, playing backgammon, and making merry with his friends on the one hand, and the business of constructing a philosophical system on the other [H58:1.4.7], is a typical example of this lack of integrity--i.e. of the lack of unity between the life of one's philosophy and the philosophy of one's life. At least Hume recognized the insufficiency of the analytic as a mould for the synthetic. 10. As with all generalizations, this comparison between East and West should not be regarded as universally applicable, but only as representing the trends and patterns which tend to dominate the two approaches to logic and life. In reality, of course, both analytic and synthetic methods are evident in virtually all philosophers; their difference lies in the degree to which they suppress one method and emphasize the other. Moreover, just as the West has had thinkers such as Hegel and Heidegger for whom synthetic forms of logic were more attractive than analytic, so also the East has had philosophers, such as the "Mohists" in ancient China, whose analysis of "disputation" was concerned with judgments which "follow by strict necessity from the definitions of names" [G522:37]. But in both cases these examples are exceptions to the general rule. (As Graham says, "the Mohists were men of a kind very unusual in traditional China though familiar in Western civilization..." [59].) The fundamental difference between Eastern and Western tendencies is epitomized by certain actual differences in how objects are referred to. For example: "whereas in English a stick has two ends, in Chinese it has two beginnings" [310]. 11. Chuang Tzu, whose views on the "perspectiveless" are treated more fully in note 3.12, believes that the fact "that all things are the same from one point of view [the synthetic, "A = -A"] and different from another [the analytic, "A = - A"]" is proof of "the pointlessness 1/ of all disputation" [G522:340]. But as we shall see in Part Two, such an extreme emphasis on synthesis is not necessary: analysis and synthesis can be seen as compatible operations. 12. A good example of the way in which synthetic contradiction can be explained away through adopting an analytic point of view is given by the following quotation of eight rules of the ancient Chinese Taoists, each followed in brackets by the translator's "reconstructions" [C525:31.]: 1. Eyes do not see. [The mind does.] 2. An orphan colt has never had a mother. [When it had a mother {it was not an orphan colt.] 3. If a rod one foot in length is cut short every day by one-{half of its length, it will still have something left after {ten thousand generations have passed. [By definition. There {will always be a half left to cut in half again.] 4. The egg has feathers. [In potentiality.] 5. Things die as they are born. [Through their transformation {from one state to another. Thus the caterpillar perishes to {become the chrysalis, and the chrysalis to become the butter{fly.] 6. The shadow of a flying bird has never moved. [Each succeeding {shadow is a different one, appearing in sequence.] 7. That which has no thickness has no bulk yet covers a thousand {square miles. [The plane, presumably.] 8. Killing robbers is not killing people. ["Executing" is not {"murdering" and "felons" are not "people".] While some of these reconstructions are perhaps helpful, they all conceal the original motivation for putting forward these propositions as logical principles: viz. to reflect the synthetic laws of contradiction and non-identity which govern our non-reflective experience of the world, and in so doing to encourage a heightened awareness of one's perspectival limitations. 13. The presence of so much contradictory language in religious discourse is surely due to the writer's attempt to challenge his reader into taking up such a higher level of awareness. B528:18 describes the reaction of Hindu and Buddhist tantrics to such contradictions: they attempt to reconcile "discursively contrary notions by raising them to a level of discourse where these contradictions are thought to have no validity." 14. In C527:34-5 Chuang Tzu says: "'that' comes out of 'this' and 'this' depends on 'that'--which is to say that 'this' and 'that' give birth to each other.... [The sage] too recognizes a 'this,' but a 'this' which is also a 'that,' a 'that' which is also 'this.'" Chuang Tzu's works must be read in full in order to appreciate the extent to which he uses language such as this, which is based on the logic of synthetic integration. At one point [38], for example, he says: "There is nothing in the world bigger than the tip of an autumn hair, and Mount T'ai is little. No one has lived longer than a dead child, and P'eng-tsu [the Chinese Methuselah] died young. Heaven and earth were born at the same time I was, and the ten thousand things are one with me." (The Mohist's analytical reply to Chuang Tzu's merging of "this" and "that" is given in C529:B68,82 [see R522:36-40,440-1,456-7]. But even the Mohist admits that analytic logic does not hold true in all cases, for "at the moment of death a thing is both 'horse' and 'non-horse', a man is both alive and dead" [G522:58; cf. C529:A50,88].) Chuang Tzu is careful to explain, however, that such contradictions are not intended to be understood at all, but to break through the barrier of the understanding to reveal a deeper level of reality: "If the Way is made clear, it is not the Way" [C527:39-40]; "You can talk about it, you can think about it; but the more you talk about it, the farther away you get from it" [C525:25.293]. Thus, as Graham puts it, "according to Chuang-tzu it is precisely when we distinguish alternatives, the right and the wrong, the beneficial and the harmful, self and other, [+ and -,] that we cut ourselves off from the world we objectify, and lose the capacity of the angler, the carpenter and the swimmer to heed his total situation with undivided attention and respond with the immediacy of a shadow to a shape and an echo to a sound" [G522:21]. Unfortunately, Graham's conclusion, that the Mohist "has progressed a long way ahead of Chuang-tzu" [40] reveals his failure to grasp the distinction between synthesis and analysis as two valid ways of approaching the world with logical principles. Indeed, the word "ahead" here simply means "closer to the Western bias"! 15. Strawson draws an important distinction along these lines "between questions of meaning and questions of fact" [S535:11]. He also notes that "character-ascribing expressions have complementaries [i.e. -A's for every A], and individual-identifying expressions do not" [7]. This is because the latter refers to a synthetic operation and the former to an analytic operation. 16. According to W533:73, some of the early Church Fathers were rather more inclined towards such a "high" view of Satan. 17. The same insight can be found in Western mystical traditions in various ways. In the Cabbalistic cosmology, for example, "The universe...becomes manifest through the materialization of four progressively denser worlds: Azeluth (Material), Briah (Creative), Yetzirah (Formative), Assiah (Material)" [P538:22; s.a. 122]. The exact correspondence between these stages and the "0, +, -, 1" pattern of synthetic integration is not accidental.