Theories of Correspondences -- and potential equivalences between them in correlative thinking

Judge, Anthony

Alternative view of segmented documents via Kairos

23rd September 2007 | Draft

Theories of Correspondences

and potential equivalences between them in correlative thinking

-- / --

Annex to Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks (2007)

Introduction
Disassociated theories of correspondences: fractured cognitive connectivity
Relating different ways of knowing
Pointers to complementarity
Variations on any "theory of correspondences"
Correlative thinking
Similarity and analogical reasoning
"Pathological" correspondences?
Number mysticism -- pathology or remedy?
Varieties of connectivity
References

Introduction

The displacement of one theory of correspondences by another epitomizes the relationship between two contrasting cognitive styles or mindsets. The displacement has been a feature of the Age of Enlightenment, and the Scientific Revolution, in which reason is promoted as the primary basis of authority. The displaced theory was a characteristic of the Renaissance and remains fundamental to many cultural perspectives in which aesthetic qualities are especially valued. Curiously, whereas the implications of the latter theory were, and continue to be, widely comprehended, the authoritative theory is known only to specialists. Neither theory figures as such in online encyclopedias despite the historical role of the displaced theory and the scientific merits of its substitute.

It could be inferred that the algebraic theory of correspondences of science has displaced the symbolist variant, much as some religions are notable for constructing their places of worship on the precise location of the deprecated "pagan" forms performing an "analogous" function in earlier times. Historically the displacement process has also been associated with physical violence done to the proponents of the older theory, notably in the form of witch hunts. More problematic is the psychocultural violence done to society in which a variety of cognitive styles is arguably desirable.

The exploration below is intended to clarify the contrasting implications of such cognitive styles for credible representations of connectivity -- especially in conditions where unsubstantiated conjectures within the dominant mindset are given serious consideration, despite being labelled "moonshine" and being comprehensible (if at all) to only the very few. This is intended to highlight the cognitive challenges implied by the Monster Moonshine Conjecture (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007) and the extent to which its explorations surreptitiously call upon processes characteristic of the deprecated theory of correspondences.

The consideration given here to a variety of approaches to cognitive connectivity follows from much earlier work of the Integrative Knowledge and Transdisciplinarity Project which resulted from 1976 in some 600 profiles (plus bibliographic resources) on integrative and unitary concepts -- incorporated into successive editions of the Encyclopedia of World Problems and Human Potential.

Disassociated theories of correspondences: fractured cognitive connectivity

The challenge of comprehending the connectivity of "moonshine" is well-illustrated by use of any web search for "theory of correspondences". Two contrasting sets of references emerge from some 9,400 hits. In the surreal real world of today, they are notably distinguished in that the authors of one set would find the content of the other to be quite meaningless, if not dangerously so:

"algebraic" theory of correspondences: This is summarized in a standard work (E. Klein and A. C. Thompson, Theory of Correspondences, 1984) but has a long history in relation to group theory (cf Joseph Edmund Wright, Correspondences and the Theory of Groups. Transactions of the American Mathematical Society, 7, 3 1906, pp. 391-400). It is curious that the challenge of algebraic correspondences is associated in category theory with what is termed the theory of motives. A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a category whose morphisms preserve this structure. When two given objects are isomorphic, a "particularly nice" representative may be selected in each isomorphism class. A current problem in this approach is that of having 'enough' morphisms. An as yet unfulfilled hope is that development of the category of motives would lead to a universal Weil cohomology.
"symbolist" theory of correspondences: As indicated by J E Cirlot (A Dictionary of Symbols, Dover Publications, 2002), this is founded on the assumption that: "all cosmic phenomena are limited and serial and that they appear as scales or series on separate planes; but this condition is neither chaotic nor neutral, for the components of one series are linked with those of another in their essence and in their ultimate significance. It is possible to marshal correspondences by forcing the components of any given scale or scales into a common numerical pattern: for example, it is not difficult to adapt the colour-scale from seven to eight colours, should one wish to equate it with the scale of temperaments laid down by modern character-study..."

For science in general, and mathematics in particular, progress in knowledge -- of which the algebraic variant is a generic feature -- has involved the progressive construction of a model for understanding the world that specifically disproved the validity of the premises of previous eras and notably the symbolist theory of correspondences central to those worldviews.

Classical Greece, the Gnostics and the Kabbalists are recognized as having founded much of their philosophy on the symbolist theory of correspondences -- dating back to the Egyptian "Emerald Tablet" of Hermes Trismegistus, a form of Rosetta stone in its own right. It was highly regarded by Renaissance alchemists, and a significant influence on the thinking of Isaac Newton that science has been slow to acknowledge. The "Isaac Newton" universally hailed as an exemplar of the scientific method -- notably his insights from a falling apple -- is in fact a cherry-picked version of the real Newton of larger, richer and more integrative perspective. He in fact made a translation of the Tablet which begins:

Tis true without lying, certain and most true.
That which is below is like that which is above and that which is above is like that which is below to do the miracles of one only thing.
And as all things have been and arose from one by the meditation of one: so all things have their birth from this one thing by adaptation.

With regard to the symbolist variant, Nathalie Wourm (The Smell of God: scent trails from Ficino to Baudelaire. 2003) notes:

This is one aspect of a theory which runs through much of European history from the Renaissance onwards, with fluctuating intensity and with fundamental variations. It has been referred to, principally, as the theory of signatures, the theory of universal analogy, and the theory of correspondences, and is originally derived from Plato's philosophy of Ideas. The most common thread of the doctrine is that there are correspondences between the material and the spiritual worlds and that the material world can therefore be read like a book, revealing the secrets of the spiritual world.

Curiously the theory of signatures, as with that of correspondences, has both a symbolist variant and a mathematical variant:

"symbolist" theory of signatures: According to this theory (or "doctrine of signatures"), dating notably to Paracelsus, the therapeutical powers of plants were suggested. by the similarity of configuration of the affected human part. Similarity is understood as the outside manifestation of hidden relationships. Plants (as well as animals and minerals and perhaps even phenomena) were understood to have clues or "signatures" in their shapes, forms and actions indicate their purpose (The Doctrine of Signatures (Doctorine of Correspondences): the union of form and function, 1998).According to Jakob Böhme (De signatura rerum): 'Because whatever lies inside, works steadily towards its revelation; and that is the language of nature, by virtue of which every thing speaks from within its qualities, always revealing itself on its own.' Prior to the development of chemistry, it explained the active principles of medicines. The doctrine remains fundamental to homeopathy. In a discussion of similarity and similitudes, Michel Foucault (The Order of Things: an arthaeology of the human Sciences, 1970)t cites Paracelsus:

It is not God's will that what he creates for man's benefit and what he has given us should remain hidden . . . And even though he has hidden certain things, he has allowed nothing to remain without exterior and visible signs in the form of special marks -just as a man who has buried a hoard of treasure marks the spot that he may find it again (p. 25).

Philosophy of Jonathan Edwards: a study in divine semiotics

"algebraic" theory of signatures: This is especially associated with homology, a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. Although there are many theories of cohomology, cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. [see also below]

A similar pattern obtains with respect to the notion of a "theory of equivalences":

"symbolist" theory of equivalences: This is most recently associated with Eric Voegelin's theory of equivalences between various sets of historical symbolisations. Two symbolisms are there considered equivalent, despite differences of individual form, if they refer recognizably to the same structures in reality. For Voegelin, the basis of a comparative study of symbols becomes a search for the constants of experience. What is permanent in the history of mankind is not the symbols but man himself in search of his humanity and its order. "For a comparative study, if it goes beyond registering the symbols as phenomena and penetrates to the constants of engendering experience, can be conducted only by means of symbols which in their turn are engendered by the constants of which the comparative study is in search. The study of symbols is a reflective inquiry concerning the search for the truth of existential order" (Eric Voegelin, Equivalences of Experience and Symbolization in History, in: Ellis Sandoz (editor), The Collected Works of Eric Voegelin, 1990, 12, pp. 115-116)

In considering the legacy of symbolism in art Charles Harrison and Paul Wood (Art in Theory 1900-1990: an anthology of changing ideas, 1996) translate a commentary by Maurice Denis (From Gauguin and van Gogh to Neo-Classicism, 1909):
We have substituted for the idea of "nature viewed through a temperament" [Zola], the theory of equivalences or of the symbol. We affirm that the emotions or states of the soul provoked by some spectacle, create in the artistic imagination signs or plastic equivalents capable of reproducing these emotionsor states of the soul without the need to create a copy of the initial spectacle; that each state of our sensibility must correspond to an objective harmony capable of being thus translated.
The linguist Roman Jakobson articulated a principle of equivalence through which poetry could be distinguished as "the projection of the principle of equivalence from the axis of selection to the axis of combination". This is held to imply that poetry successfully combines and integrates form and function, that poetry turns the poetry of grammar into the grammar of poetry, so to speak. Jakokbson notes (Two Aspects of Language and Two Types of Disturbances, 1956)
Similarity in meaning connects the symbols of a metalanguage with the symbols of the language referred to. Similarity connects a metaphorical term with the term for which it is substituted. Consequently, when constructing a metalanguage to interpret tropes, the researcher possesses more homogeneous means to handle metaphor, whereas metonymy, based on a different principle, easily defies interpretation..... Since poetry is focused upon the sign, and pragmatical prose primarily upon the referent, tropes and figures were studied mainly as poetic devices. The principle of similarity underlies poetry; the metrical parallelism of lines, or the phonic equivalence of rhyming words prompts the question of semantic similarity and contrast.
"algebraic" theory of equivalences: This is recognized in relation to process algebra, notably for regular systems. Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. They are named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Such a theory is closely associated with group cohomology -- which has a formal definition of "splendid equivalences". In mathematics an equivalence relation is a binary relation between two elements of a set which groups them together as being "equivalent" in some way. Robin Milner (Communicating and Mobile Systems: the pi calculus, 1999) has developed a calculus for analysis of the movement of a piece of data inside a computer program -- treated exactly the same as the transfer of a message -- or indeed an entire computer program-- across the internet. Its most prominent feature is the notion of a name and is dependent on the concept of behavioural (or observational) equivalence, and the use of a new theory of types to classify patterns of interactive behaviour. Equivalence is clearly of importance in data mining in response to user questions.

But with regard to the symbolist theory of correspondences, Wourm's elaborates as follows:

The journey through minds of the theory of correspondences has obscure beginnings, but is a consequence of the Platonic hierarchy of body and soul, of a sensible world and an ideal world, and of the principle of the inherence of the non-corporeal in the corporeal. The theory relates to a quest for the spiritual meaning which is contained in each sensible object, positing that there exists a conduit between the divine and the earthly. Deriving partly from Aristotle and Plotemy's cosmology, the foundations of the theory of correspondences appear in the works of Marsilio Ficino, the fifteenth century Florentine who began the modern tradition of Neoplatonism. Ficino elaborates the idea of a world soul inherent in the cosmos, and of a system of analogies and influences between the celestial, the natural and the human worlds, which is accessible to our understanding.

It is fortunate for world culture that the rise of science, enabling the emergence of sophisticated search engines, has not resulted in the deletion of any reference to ways of knowing which science believes it has superceded. Arguably the information sciences transcend in objectivity the sciences whose production they document! Indeed, as argued by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999), it is very probable that "deprecated" metaphors natural to other cultures, such as those of the East, may in future drive and condition the formulation of theories fundamental to the sciences. Presumably such "mining" might be extended (by the East?) to "deprecated" metaphors of the West as well.

Relating different ways of knowing

But the argument here is that the continuing influences of such contrasting theories of correspondences are indicative of different ways of knowing (and managing "moonshine") -- irrespective of the degree to which they consider each other to be irrelevant. Most perspectives may indeed be judged "wrong" from another perspective -- and this may indeed be a way of distinguishing perspectives. But the challenge between disciplines, between belief systems and between cognitive frameworks is how better to provide coherence to insights arising from such contrasting views (as in the X-ray diffraction pattern metaphor).

As noted by Lee Irwin (Daoist Alchemy in the West: The Esoteric Paradigms), Daoist Five Element (wuxing) cosmology is based on a theory of correspondences very similar to theories developed in the Greco-Roman world and subsequently passed onto Medieval Europe. Antoine Faivre and Karen-Claire Voss (Western Esotericism and the Science of Religions, Numen, 42, 1, 1995, pp. 48-77) suggest that modern Western esotericism is a form of thought identifiable by the simultaneous presence of four to six components including the idea of universal correspondences and living Nature. A very influential exponent of the theory of correspondences was the mystical philosopher, theologian and scientist Akel Ismail Kahera (Gardens of the Righteous: sacred space in Judaism, Christianity and Islam. Cross Currents, September 2002). Emanuel Swedenborg, who had derived the insight from Plotinus and notably applied it to interpretation of the Bible. Expressing a perspective praised by Wolfgang Goethe, he noted in 1758 that:

The whole natural world corresponds to the spiritual world -- not just the natural world in general, but actually in details. So anything in the natural world that occurs from the spiritual world is called a correspondent. It is vital to understand that the natural world emerges and endures from the spiritual world, just like an effect from the cause that produces it.

In a review of a contemporary biblical study by social anthropologist Mary Douglas (Leviticus as Literature, 1999), Don Handelman (Journal of Ritual Studies, 18, 2, 2004, pp. 162-168) remarks:

Douglas argues that the cosmos of Leviticus (in my terms) is held together through analogical or concrete thinking. That is, through analogies added to and layered onto one another, thereby bringing macrocosm and microcosm into alignment.... Douglas's template of analogical thinking is taken from the ancient Chinese, especially from the Confucian patterning of correspondences and its intimate relationship to the idea of the exemplar, the perfect sage, the exquisite model for action, indeed, for praxis. [more]

Irrespective of judgements about the "occult", despite the value attached to some of its proponents and sympathizers (W B Yeats, Arthur Schopenhauer, etc), given its preoccupation with what is conventionally hidden, it might asked how much more hidden the Monster could have been -- and remains to ordinary modes of cognition. It might well be said to be "occulted".

The tragedy for science is that the very sense of elegance and beauty, which purportedly drives many of its proponents, derives to a considerable degree from the aesthetic way of knowing associated with the symbolist theory of correspondences. Indeed, as many authors note, the theory has had a significant influence on many aspects of culture from which people continue to derive meaning. It might even be said to constitute the essence of a cultured understanding through its recognition and appreciation of an implicit interdependence of all things in the universe, and the existence of multiple relationships between various aspects of nature's kaleidoscopic richness. Examples include:

poetry: Charles Baudelaire (Correspondences / Correspondances, 1857) came to believe that the artist's unique ability to represent truth un-didactically, through symbols and metaphors, was of immense importance; Octavio Paz described analogy as a poetics that expressed the correspondence between the celestial and terrestrial realms; this poetics was nourished In Latin America by Emmanuel Swedenborg's theory of correspondences and the Baudelarian concept of synesthesia, and was seen to function as an aesthetic counter-measure to the fractures of modernity and its resulting alienation.
music (John Covach, Schoenberg and the Occult: some reflections on the "musical idea", Theory and Practice: Journal of the Music Theory Society of New York State, 17, 1992, pp. 103-118); the Wagnerian synthesis of art forms was a major Symbolist aspiration, encouraged by Baudelaire's theory of "correspondences"
architecture, as noted by Akel Ismail Kahera (Gardens of the Righteous: sacred space in Judaism, Christianity and Islam. Cross Currents, September 2002):

Finally, we may also speak of sacred geometry as having an inner reality transcendent of outer form, which has remained throughout history the basis for sacred structures. A theory of correspondences underlies sacred geometry, proportions, harmonic relationships, beauty and order, forms of crystal, and natural objects. All are part of a universal continuum and a structure of created existence.

Pointers to complementarity

At a time when global society is claimed to be challenged by a "clash of civilizations", it is appropriate to note that the science-culture disassociation noted above was given prominence long before by scientist C P Snow (The Two Cultures, 1959). In the synthesis offered by biologist E.O. Wilson (Consilience: the unity of knowledge, 1998), not only does the gap between the two cultures of the Sciences and the Humanities continue to exist today, its very origin remain unexplained.

One attempt to address this condition had been made by analytical psychologist Carl Jung for whom synchronicity was the theory of correspondences in practice. As noted by Roderick Main (Religion, Science, and Synchronicity. Harvest: Journal for Jungian Studies, 46, 2, 2000, pp. 89-107):

No less significant for the development of the concept of synchronicity was Jung's extensive research into the esoteric traditions of the West. The ancient Greek conception of `the sympathy of all things', the medieval and Renaissance theory of correspondences, and above all the alchemical understanding of the unus mundus (one world) and of the relationship between microcosm and macrocosm also provided acausal connections between events (see Jung, 1952, pp. 485-98). At times Jung presents his theory of synchronicity as simply an up-dating of these esoteric views: `Synchronicity', he writes at the end of his 1951 Eranos lecture, `is a modern differentiation of the obsolete concept of correspondence, sympathy, and harmony' (Jung, 1951, p 531).

Ironically the reception accorded the Moonshine Conjecture by some "mainstream" mathematicians can be compared with that accorded the "fringe" views of Robert Temple (The Sirius Mystery, 1987) whose comment on the unhealthy current situation is nevertheless well expressed, if bluntly:

In my opinion, a mind is healthy when it can perform symbolic acts within mental frameworks which are not immediately obvious. A mind is diseased when it no longer comprehends this kind of linkage and refuses to acknowledge any basis for such symbolic thinking. The twentieth century specializes in producing diseased minds of the type I refer to -- minds which uniquely combine ignorance with arrogance. The twentieth century's hard core hyper rationalist would deride a theory of correspondences in daily life and ritual as 'primitive superstition'. However, the rationalist's comment is not one upon symbolic thinking but upon himself, acting as a label to define him as one of the walking dead.

A more conciliatory view is expressed by Karen Armstrong (A Short History of Myth, Melbourne, Canongate, 2005) who addresses the continuing role of myth in industrialized societies and its long-demonstrated functions:

Another peculiar characteristic of the human mind is its ability to have ideas and experiences that we cannot explain rationally.... imagination is the faculty that produces religion and mythology. Today mythical thinking has fallen into disrepute; we often dismiss it as irrational and self-indulgent. But the imagination is also the faculty that has enabled scientists to bring new knowledge to light and to invent technology that has made us immeasurably more effective.... Mythology and science both extend the scope of human beings. Like science and technology, mythology...is not about opting out of this world, but about enabling us to live more intensely within it..

Variations on any "theory of correspondences"

But, despite statements such as those of Jung, that the symbolist theory of correspondences is "obsolete", it should be noted that it continues to be influential as such in a variety of domains. For example the Swedenborgian variant continue to be promoted through Swedenborgian groups, as the "science of correspondences" (cf Geoffrey S. Childs, Correspondences: a key to distinctiveness, Journal of the Correspondences Committee, 1990-1992, and bibliography) [the Wikipedia entry lists notable people influenced by these views in the past].

But, more intriguing is the degree to which surrogates have emerged that are not apparently related to either the algebraic or the symbolist variants. The theory of correspondences in its "obsolete" sense, can fruitfully be understood as having gone underground -- or even been forced underground by the dominance of cognitive styles in mainstream science and religion through the first half of the past century. "Underground" may however mean incorporated into the body of esoteric knowledge characteristic of the many rosicrucian, masonic and other secret societies.

A fruitful set for further study might therefore include:

Metaphor and analogy: These are variously distinguished and widely used, both as fundamental to cognitive processes (notably in the hardest of sciences) and for purposes of communicating complex insights to a wider audience. It is for example interesting that a new source of motivation for work on the foundations of algebraic geometry, in general, and on intersection theory, in particular, resulted from the solution of an outstanding problem in number theory (the Riemann hypothesis for fields of algebraic functions in one variable over a finite field of constants) by André Weil in 1940 using an "analogue of the theory of correspondences". The work of George Lakoff, with various collaborators, has given considerable credibility to this mode of reasoning. Its role in governance may be fundamental (cf Documents relating to Metaphor for Governance). One approach to analogy of historical interest, because of its relationship to the symbolist theory of correspondences, is the theory of universal analogy of Charles Fourier (1772-1837). An unfortunately narrow approach to similarity and analogical reasoning by Western cognitive scientists is discussed below.
Isomorphism: This is of technical significance in mathematics but is more generally recognized as a kind of mapping between objects, which shows a relationship between two properties or operations. According to Douglas Hofstadter: "The word 'isomorphism' applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where 'corresponding' means that the two parts play similar roles in their respective structures." (Gödel, Escher, Bach, p. 49) From a mathematical perspective, I. E. Segal (Equivalences of Measure Spaces, American Journal of Mathematics, 73, 2, 1951, pp. 275-313) clarifies the varieties of isomorphism, relatings weak equivalence of measure spaces to varieties of isomorphism of function-spaces over measure spaces..
Homology: This is a form of analogy or isomorphism distinctly characteristic of various disciplines. In the case of anthropology, homology refers to analogy between human beliefs, practices or artifacts due to genetic or historical connections. In biology, the analogy to which homology refers is that of structures arising from shared ancestry. In mathematics, homology (and cohomology) is a procedure to associate a sequence of abelian groups or modules with a given mathematical object. In sociology, homology is a form of structural 'resonance' between different elements making up a socio-cultural whole.
Cobordism: In mathematics, this is a relation between manifolds, based on the idea of boundary. Two manifolds M and N are cobordant if their union is the complete boundary of a third manifold L; L is then called a cobordism between M and N. By this means an equivalence relation on manifolds is obtained. Cobordism theory has become part of the apparatus of extraordinary cohomology theory.
Correlation: In probability theory and statistics, correlation indicates the strength and direction of a linear relationship between two random variables. It is an indication of the departure of two variables from independence. There are a number of coefficients that measure the degree of correlation. From a psycho-social perspective, "correlative thinking" has been noted as characteristic of premodern cultures, as discussed in more detail below.
General systems: In search for invariants across specialized disciplines, the Society for General Systems Research was founded in 1954 (see Yearbook of the Society for General Systems Research), later renamed International Society for the Systems Sciences. Essentially the initiative aimed at overcoming the growing isolation of specialized disciplines. Considerable attention has been given to isomorphism between distinct disciplines or between systemic levels, notably as subsequently developed through living systems theory. The commonalities and contrasts with cybernetics also merit attention.
Isophor: Combining isomorphism and metaphor, this term has been used by Kathleen Forsythe to depict a feeling response to one thing in terms of another. An isophor is then something that is "experienced emotionally and, (that) as such, define(s) the experience of understanding." (Isophor: poiesis of experience, 1987 ). Elsewhere Forsythe (Cathedrals of the Mind, 1986) argues that: "Analogy and its poetic expression, metaphor, may be the "meta-forms" necessary to understanding those aspects of our mind that make connections, often in non-verbal and implicit fashion, that allow us to understand the world in a whole way." With the isophor there is no separation between thought and action, between feeling and experience. The experience itself is evoked through the relation. She suggests that the experience of one thing in terms of another, the isophor, is the means by which domain is mapped to domain and that consciousness of this meta-action, when recognized, lies at the heart of cognition. Forsythe has postulated the development of an epistemology of newness in which learning is the perception of newness and cognition depends on a disposition for wonder leading to this domain of conception-perception interactions. She argues that the notion of metaphor is commonly understood to mean the description of one thing in terms of another -- presupposing an objective reality. This objectivity may be questioned especially if, as suggested by Maturana, (objectivity) is placed in parentheses.
Design: In all its forms, including sacred geometry, design is dependent on appropriate correspondences. This is most notably to be seen in use of design in marketing and communications.
Parallelism in rhetoric: As a rhetorical and poetic device (notably in Hebrew biblical poetry and Chinese poetry), two or more parts of the structure may be given a similar form so as to give the whole a definite pattern. Roman Jakobson pioneered The secular study of parallelism in poetic-linguistic traditions around the world was pioneered by Roman Jakobson (The Poetry of Grammar and the Grammar of Poetry, 1987).
"Magic": This term may be variously understood. Those skilled in any art or science are notably valued when their solutions to a challenge are described as "magic". Loosely defined, a social (or romantic) occasion is typically most highly valued if it is "magical". A more precise understanding of what makes such occasions magical was the focus of "natural magic" or "sympathetic magic", notably promoted by Marsilio Ficino (B. Copenhaver. Natural magic, hermetism, and occultism in early modern science, 1990) and contrasted with "demonic magic". As understood by its current practitioners (notably neo-pagans and wiccans), the "magical art of correspondences", based on an underlying theory of "correspondences", is held to be the basis of magic itself. These correspondences are considered to be hidden relationships among entities within the universe -- especially between human beings and the external world. They are understood as
- links that unite seemingly disparate realms;
- a window on the structure of the cosmos;
- a lens between the macrocosm ('as above') and microcosm ('so below');
- based on analogical reasoning, which seeks similarities of pattern -- a "right-brain" approach (artistic, poetic, holistic) -- rather than "left-brain" cause-and-effect reasoning which scientific materialism insists is alone valid.
Divinatory systems are considered especially powerful tools in finding and analyzing correspondences, notably with the aid of "tables of correspondences". In contrast to the pathetic conflation of magic with sorcery (as in the Encarta encyclopedia), under any focus on demonic magic, careful attention can fruitfully be given to the distinction between demon and daimon. The latter (cf Kairos -- the opportune moment for self-referential re-identification, 2007) been held to be intimately related to the still voice of conscience (as notably consulted by Socrates prior to his death), the artists muse, and duende as highly valued in Spanish culture.
Symbolism: Seemingly disassociated from the (past) "symbolist movement" in the arts and the inspirations it derived from the "symbolist theory of correspondences", symbolism is increasingly recognized to have a far greater presence in contemporary culture than most realize. As noted by symbolism.org: " More than an ancient, esoteric discipline, it really serves as the centerpiece for an entirely new modern philosophy and understanding of the world. While symbols seem to increasingly 'migrate' to contemporary culture - defining such things as leading brands, products, films, events and celebrities - few understand the dynamics and cycles of their movement, their overall language and what they tell us. In effect, we are much more busy creating symbols than trying to understand them".
Archetypal psychology: As developed by James Hillman, from the work of Carl Jung, archetypal psychology is a polytheistic psychology, in that it attempts to recognize the myriad fantasies and myths -- gods, goddesses, demigods, mortals and animals -- that shape and are shaped by the psychological lives of humans. The ego is understood to be but one psychological fantasy within an assemblage of fantasies.
Synaesthesia: The term synaesthesia is normally used to indicate a condition in which the stimulation of one sensory modality gives rise to an experience in another modality. This may be understood as a form of cognitive fusion. In an auditory synaesthete, for example, an auditory experience may give rise to an experience in the visual modality. Synaesthetes often experience correspondences between the shades of color, tones of sounds, and intensities of tastes that provoke alternate sensations (see Synaesthesia: union of the senses; Richard E. Cytowic, Synesthesia: phenomenology and neuropsychology -- a review of current knowledge, 1995). Ironically a symbolist poet, inspired by Baudelaire's views of synaesthesia, has been criticized for lack of scientific merit at a time when attention is being given to then phenomenon (Louis W. Marvick, Rene Ghil and the Contradictions of Synesthesia, Comparative Literature, 51, 4 1999, pp. 289-308)
Transdisciplinarity: Disciplines are defined by distinct methods which are necessarily disassociated from any interdisciplinary or transdisciplinary methodology intended to elicit connectivity between their respective cognitive styles. The systems sciences have offered one approach to this difficulty; proposals for unified science have offered another. These proposals have not however been able to encompass disciplines held to be non-scientific.
Psycho-cultural equivalence: Marisa Zavalloni (Identity and Hyperidentities: the representational foundation of self and culture, Papers on Social Representations, 2, 3, 1993) outlines based on three sets of equivalence:
- Linguistic equivalence means that in terms of identity, an adjective, a verb, a noun, anadverb are interchangeable, thus to be adventurous, adventure, to act adventurously all point to different accounts of a same underlying, referential complex.
- Psychological equivalence refers to the affective- representational circuit, namely if an identity word appears on one psychological dimension, it will necessarily appear on all the other dimensions. When one of these dimensions is in a foreground position (as the focus of consciousness) the other will be activated as background thinking.
- Affective equivalence means that the different feelings that are activated in association with the psychological dimensions comprising the affective-representational circuit, are experienced simultaneously either in foreground or background positions. These feelings are all part of the original affect which was triggered when a feature of the external world became encapsulated in an identity word. As a postcolonialist critic, Gayatri Chakravorty Spivak (Can the Subaltern Speak? 1988) introduced a more complex notion of affective equivalence in clarifying the position of Deleuze and Foucault:
  Their suggestion, summarized, is that, since capital decodes and deterritorializes the socius by releasing the abstract [cf. Nietzsche's slave logic] as such, capitalism manages the crisis by way of the generalized psychoanalytic mode of production of affective value, which operates via a generalized system of affective equivalence, however spectacular in its complexity and discontinuity.
Mutuality of affection: A degree of equivalence in affective and romantic relationships -- promoted as the ideal of courtly love -- is a major concern in interpersonal relationships. Victor S. Johnston (Why We Feel: the science of human emotions, 1999) describes a set of feelings particularly adjusted to the paradoxical challenges of survival: an elaborate emotional monitoring system for the mutuality of affection, protection, and concern. If the reciprocity is broken by recklessness, a "pop out feeling" indicates the loss of the "investment". An extreme example is unrequited love, namely love that is not reciprocated, even though reciprocation is usually deeply desired.

More generally, Daniel Goleman (Social Intelligence: the new science of human relationships, 2006) summarizes a wide range of research on social neuroscience indicative of the manner in which identity is established and sustained by cyclic psychosocial processes. Whereas the circulatory system is a closed loop, the emotions are an open loop system sustained by cognitive processes that allow others to help manage individual emotions more appropriately. Goleman uses the term "looping", which readily recalls the anxiety that people may feel to be "kept in the loop" -- perhaps to be usefully understood as a felt sense of equivalance. Empathy creates a feedback loop as people work towards a "fit" between their own perceptions and the reality of another -- such that looping enables a person to feel within themselves the distress expressed by another. On the other hand, looping too tightly -- excessive mutual entrainment -- can be experienced as suffocating in a relationship. He distinguishes between "positive" and "negative" (or toxic) loops.
Value-centred "equivalence":
- Value equivalence (in media preferences): The concepts of trust and credibility may be and framed as 'value equivalence' -- namely a comparison of value differences between user and media affecting whether or not alternatives to the available, dominant mass media will be sought and used. Value equivalence may then explain user preference, trust, selection, and uses of both mass and alternative media. (T. B. Christie. The role of value equivalence in news media selection, American Association for Public Opinion Research, 2003)
- Moral equivalence: This is used to challenge assumptions of moral superiority, notably in comparing acts of aggression between states or peoples. In some cases it may then be argued that the two parties could be viewed as "equally" culpable in a struggle for (moral) supremacy. Critics have for example claimed that the intervention of the USA in various arenas should be considered as morally equivalent to some interventions by the Soviet or Nazi regimes. Similar judgments have been made with respect to the Israeli actions against Palestinians, in comparison with some Nazi acts against the Jews. The term "romantic equivalence" has been similarly used with regard to efforts to compare the rights of Israelis to live in peace and of Palestinians to regain all of their land. The applicability of moral equivalence in the case of the USA was notably challenged by Jeane Kirkpatrick (The Myth of Moral Equivalence, Imprimis, 15, January 1986, 1) following her period as the first woman US Ambassador to the United Nations. Such views were effectively echoed by a subsequent US Ambassador to the United Nations, Madeleine Albright, the first woman to become US Secretary of State. When asked by an interviewer with regards to the effect of sanctions against Iraq: "We have heard that half a million children have died. I mean, that's more children than died in Hiroshima...is the price worth it?". Albright replied: "I think this is a very hard choice, but the price -- we think the price is worth it." (We Think the Price is Worth It, Fair, 2001) Such views regarding Iraq (and elsewhere) continue to be echoed by the leadership of the Coalition of the Willing. Albright was notable for her failure to take any action with regard to the genocide in Rwanda. Righteousness necessarily precludes any possible recognition of moral equivalence, no matter the degree of atrocity for which the righteous are responsible.
- Retributive justice: From this perspective there is an expectation that punishment should in some way be equivalent to the crime. Proportional punishment, scaled relative to the severity of the offending behaviour, is thereby considered as a morally acceptable response to crime. This is not necessarily interpreted to mean that the punishment has to be precisely equivalent to the crime.
Spiritual equivalence: Some interfaith groups promote initiatives that assume a degree of equivalence between the deepest insights of different religions or spiritual traditions -- although such equivalence is profoundly deplored by others as anathema or blasphemy (even to the point of being punishable by death). As consistent with a general "law of equivalence of form", a "spiritual equivalence of form" may also be understood in terms of the degree of similarity and dissimilarity of intentions. Others, such as the Asymmetric Threats Contingency Alliance, recognize a degree of equivalence between spirituality and science. A spiritual equivalence between jaguars and shamans is sometimes made explicit in the widespread belief that, under the influence of hallucinogens, some shamans transform into jaguars in body as well as spirit.

Focusing on John Locke, the process philopher Greg Moses (American Pluralisms: pragmatism, Locke and Cruse, 1995) argues with regard to spiritual equivalence:
What the contemporary mind stands greatly in need of is the divorce of the association of uniformity with the notion of the universal, and the substitution of the notion of equivalence. Sameness in difference may be a difficult concept for us -- it is. But the difficulty is historical and traditional, and is the specific blight of the modern and Western mind.
How much of the contemporary mind is still preoccupied with uniformity as the best fulfillment of unity? Note the continuing complaint that diversity is less important than universal human values.

As an early advocate, and bureaucratic casualty, of what today is called multicultural education, Locke fought valiantly for cultural approaches which embrace difference, because of the deeper unity that is signified:

What we need to learn most is how to discover unity and spiritual equivalence underneath the differences which at present so disunite and sunder us, and how to establish some basic spiritual reciprocity on the principle of unity in diversity. If for instance, one difference that divides us is racial, then it will be important to select representative expressions from across race lines in order to discover spiritual equivalences between them. Having investigated such equivalences, we would be better equipped to unify ongoing differences and mediate against further conflicts.

Correlative thinking

The term "correlative thinking" was a characterization of Chinese thinking by Joseph Needham (History of Scientific Thought, 1956). It referred to a general propensity to organize natural, political/social, and cosmological information in highly ordered arrays or systems of correspondences. His characterization was very influential as a subsequent focus of sinological studies..

A very helpful exploration of many of the above issues has been provided by Steve Farmer, John B Henderson and Michael Witzel (Neurobiology, Layered Texts and Correlative Cosmologies: a cross-cultural framework for pre-modern history. Bulletin of the Museum of Far Eastern Antiquities, 72, 2000 [2002], pp. 48-90). This study combines neurobiological and textual evidence to develop a cross-cultural model of the evolution of correlative systems of thinking. Despite the focus provided by Needham, the study specifically argues that:

... claims that correlative thought was in some way unique to China have seriously impeded comparative studies; known by other names, correlative tendencies were no less prominent (and were sometimes more extreme) in premodfern India, the Middle East, the West, and Mesoamerica than in China....Our model pictures the growth of "high correlative" -- multileveled reflecting cosmologies, nested hierarchies, abstract systems of correspondences, and similar developments -- as byproducts of exegetical processes operarting in layered textual traditions over extended periods; the origins of primitive correlative thought and related animistic ideas seen at the earliest levels of those traditions, "worked up" abstractly in later strata, are tied in our model to neurobiological data.

The authors argue that:

Correlative structures show up world wide in premodern magical, astrological, and divbinational systems; in the designs of villages, cities, temples, and court complexes; in hierarchical and temporal cosmologies; and in many similar phenomena. The idea that reality consists of multiple "levels", each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general; thracing this idea from its primitive origins to its modern decline is one of the major challenges faced by specialists in premodern thought.

However, whilst the study is extremely valuable in clarifying the nature of correlative thinking cross-culturally, it assumes that such thinking has indeed long been in decline -- presumably on the basis that it is disparaged by currently dominant fashions in mainstream thinking. The authors note, for example, an academic caricaturization of such thinking as the "twaddle of idiots". However their study in no way recognizes the continuing interplay between "symbolist" and "algebraic" notions of correspondences. Nor does it recognize the considerable importance currently attached in some countries to notions of placement, design and explanation specifically dependent on correlative thinking. Whether approved by mainstream thinking or not, failure to take account of the principles of feng shui, for example, is recognized (at the highest official level) as having disastrous financial consequences in contexts as modern as Singapore and Hong Kong [more].

As noted above, Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999), has strongly argued that it is very probable that "deprecated" metaphors natural to other cultures, such as those of the East, may in future drive and condition the formulation of theories fundamental to the sciences. Such "mining" might indeed be extended (by the East?) to "deprecated" metaphors of the West -- in which "fragmentation" is widely deplored, but without any mainstream methodology able to address the cultural schizophrenia of the non-relationship between "symbolist" and "algrebraic" correspondences. This is indeed an apt metaphor of any "clash of civilizations" in a period when the worldwide, cross-cultural attractiveness of Lord of the Rings and the Harry Potter series (drawing on such correlative metaphors) is met with disapproving astonishment by an academic world that has little to match the coherence they offer to those who fund its research.

In striking contrast to mainstream academic schizophrenia, the US military is clearly already somewhat sensitive to the strategic advantages that may derive from correlative thinking, as explored by Susan M. Puska (New Century, Old Thinking: the dangers of the perceptual gap in U.S.-China relations. Strategic Studies Institute, U.S. Army War College, 1998). The long tradition of its influence on Chinese policy has been traced by James Miller (The Development of Correlative Thinking about the Body Politic (in: Envisioning the Daoist Body in the Economy of Cosmic Power, Daedalus, Fall 2001).

In demonstrating the premodern commitment to linkages, Farmer et al. highlight an interesting example that:

Claims have been made that all of Vedic philosophy dependended on what sinologists would immediately recognize as "correlative thought" -- although indologists do not use that phrase -- emerging in increasingly complex and abstract forms in successive strata of tightly linked Brahmanas, Aranyakas, Upanishads, and Sutras in Vedic traditions; indeed, as Renou argued decades ago, the original meaning of upanisad was "connection" or "equivalence" -- which we could just as correctly translate as "correlation"! -- and the aim of the whole of Vedic thought may be expressed as the attempt to formulate upanisads.

As discussed in the main paper, these "premodern" insights do indeed have a contemporary relevance highlighted by the studies of Rg Vedic languages by Antonio de Nicolas (Meditations through the Rg Veda: four-dimensional man, 1978). Given the extensive focus of Farmer et al. on neurobiology, the subsequent work of de Nicolas (The Biocultural Paradigm: the neural connection between science and mysticism, Experimental Gerontology, 33, 1997, 1/2), in collaboration with Maria M. Colavito (The Heresy of Oedipus and the Mind/Mind Split: a study of the biocultural origins of civilization, 1995), are suggestive of expanding the focus of the study by Farmer et al. The biocultural paradigm notably relates these Vedic languages to 5 epistemologically invariant styles (maia, mythos, right brain mimesis, left brain mimesis, and logos), themselves associated with 5 features of the brain (reptilian, limbic, right and left hemisphere, and the interpreter module).

In the current era of increasing emphasis on faith-based governance, irrespective of civilization and the pathetic inadequacies of interfaith dialogue, there is surely a case for responding to integrative clues such as indicated by Farmer et al:

One of the most common syncretic methods harmonized textual conflicts by posiiting the simulataneous truth of conflicting ideas on different "levels" of reality -- in the process generating new cortical "maps" reinterpreted as religious or metaphysical realities. The results of the repeated use of such methods, applied allegorically to poetic as well as to religious and philosophical texts, were the familiar bifurcations of reality universally associated with scholastic traditions.... Despite the cultural differences dividing these concepts, the integrative processes that generated them were basically the same. Similar reconciliative ends were often achieved by assigning conflicting ideas or traditions to different cyclical stages (or "phases") in cosmic history, or to emergent cosmological "types" in linear models of time; in both cases the correlartive structures of these two basic classes of temporal cosmologies (which were often syncretically fused) were normally tightened with each exegetical act.

Given the close links that Farmer et al show to exist between neurobiological and correlative systems, their claim for the relevance of computer simulations in exploring the dynamics of the development of correlative thinking as non-linear dissipative structures bears careful consideration. They develop this discussion in a separate paper (Steve Farmer, John Henderson, Michael Witzel and Peter Robinson, Computer Models of the Evolution of Premodern Religious and Philosophical Systems. 2002).

It is nevertheless extraordinary that the authors -- perhaps wisely -- avoid any specific references to the need for correlative thinking at this time, other than in a concluding generality that highlights the challenges of "differing personal attitudes" of researchers that determines the "path dependencies" of subsequent research. They conclude that:

Recognition of the neurobiological grounds of correlative systems, coupled with cross-cultural studies of how layering processes affected those systems' later growth, can help combat persistent myths concerning "great divides" supposedly separating major world civilizations.

Similarity and analogical reasoning

"Correlative thinking" is the term that has been used to denote a style of "premodern" reasoning through which correspondences between disparate domains "were" established, notably as characteristic of China. On the other hand, "analogical reasoning" is the term that is used in the "modern" exploration of similarity and analogy, understood as fundamental to processes of human cognition, notably recognition, classification and their implications for scientific creativity and successful learning -- especially where there is a need to integrate theory and data from diverse domains.

Curiously in a classic compilation of 19 contributions on the matter, edited by Stella Vosniadou and Andrew Ortony (Similarity and Analogical Reasoning, Cambridge University Press, 1989), there is no reference to "correlative thinking" and how it has been used for thousands of years, whether in China or elsewhere. It would appear to be a modern discovery by Western cognitive scientists, for almost no reference is made to the role that "correspondences" or "equivalences" have played in the development of Western science from the Renaissance.

It would be interesting to confront this approach with that of "correlative thinking", especially in terms of the insights it offers in practice. But, as with the noted interest of the military in "correlative thinking", it is appropriate to note that the compilation on analogical reasoning arose from a workshop funded by the Army Research Institute which clearly sees the merit of spanning both domains.

It could be argued that the military has been especially sensitized to the strategic relevance of such thinking following the analysis by Scott A. Boorman (Protracted game: a Wei-Ch'i interpretation of Maoist revolutionary strategy, 1971). He notably indicated the advantages of strategic thinking inspired by the game of "go" (Wei-Ch'i in Chinese), compared to that inspired by the game of "chess", as applied to the Vietnam conflict [more | more]. This analysis is echoed in the International Bulletin of Political Psychology (Vol.10 No.13 Apr 13, 2001) comparing Vladimir Putin's judo-influenced strategy with that of the 'weight-machine' mindset of the USA.

"Pathological" correspondences?

Delusions: Readily to be understood as "pathological" are beliefs in "correspondences" judged to be inappropriate. Delusions typically occur in the context of neurological or mental illness. Those who hold them may be, partly as a result, confined to psychiatric institutions. The issue of the qualifications of who makes such an assessment and how their criteria are determined is another matter. Many people were so confined in Soviet Russia because of the "inappropriateness" of their ideological associations. The main criteria for a belief to be considered delusional are: certainty (held with absolute conviction), incorrigibility (not changeable by compelling counterargument or proof to the contrary), impossibility or falsity of content (implausible, bizarre or patently untrue). A major difficulty occurs with the application of such criteria to individuals or groups whose innovative insights subsequently come to be held as correct -- where it is the "conviction" of the majority that comes to be recognized as having been deluded, where the "incorrigibility" was a pathological characteristic of the previously dominant views, and where the "implausibility" is subsequently demonstrated to be plausible. The case of Galileo Galilei offers a useful example. The question is what extant insights, framed as deluded ("moonshine") by mainstream views, will at some future time offer an analogous demonstration of delusional dimensions to currently dominant thinking?

Given that delusions are of particular diagnostic importance in psychotic disorders and particularly in schizophrenia, it might be asked to what degree modern civilization is to be diagnosed as "schizophrenic" in the light of the virtually complete separation between the "algebraic" and "symbolist" correspondences and equivalences noted above. This would suggest that the very absence of any sense of equivalence between those two sets of "correspondences" is likely to be considered a pathological symptom by the future.

Mad politicians: It is typical of political discourse for those of one political persuasion to perceive, and to caricature, those of some other persuasion as to some degree "mad" in the light of the inappropriate conclusions they draw and for the inappropriate relationships they detect between issues in society. This is exemplified by the constituencies centred on "Davos" and those centred on "Porto Alegre" (All Blacks of Davos vs All Greens of Porto Alegre: reframing global strategic discord through polyphony? 2007). Here of course there is the question of whether the conclusions are an indication of pathological thinking or whether the "contradictions" detected are indicative of pathological relationships between psychosocial conditions.

Mad scientists: Also interesting is the case of "scientists" considered in some way to be "mad". Curiously whilst Wikipedia has a whole entry on mad scientists, and a separate list of mad scientists, in both cases these deal specifically and solely with fictional archetypes -- "Doctor Strangeloves". No reference is made to the delicate matter of determining which non-fictional "scientists" are to be considered "mad" -- specifically because of the inappropriateness of the correspondences which they believe merit consideration. However Wikipedia does have an entry on fringe science which is presumably an attractor for those who indulge in correspondences that might appropriately be considered pathological -- by mainstream science. But how, and why, is that to be distinguished from protosciences? What proportion of scientific geniuses would be judged to be psychologically deeply disturbed -- and why is so little effort made to document the inappropriate correspondences they find meaningful? Which of them is indeed to be caricatured as "nutty as a fruitcake"?

As noted earlier, Isaac Newton (President of the Royal Society) has been variously caricatured as borderline mad for his bizarre attempts to reconcile mathematics and theology. This raises interesting questions about how "mad" should now be considered those scientists who are believers in "creationism" or "intelligent design".

Given the role of mathematics in the design of places of worship, and their symbolic significance, it is puzzling that "mathematical theology" is not seen as a potential response to some of the profound issues that violently divide faiths such as Islam and Judaism, as discussed elsewhere (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000). Such explorations might be considered especially relevant given the numerology associated with "end times" scenarios and the significance attached to "72 virgins" by fundamentalist suicide bombers -- and the fatal consequences of acting on such beliefs. If correlative thinking is to be judged as the "twaddle of idiots", one wonders how the future will judge the "scientists" who currently design the weaponry for any "clash of civilizations".

As one designer of weaponry, Leonardo da Vinci might be caricatured as "mad" for the relevance he saw in painting such works as Mona Lisa -- although how this criterion is to be reconciled with the ability of scientists to appreciate beauty is problematic given its exclusion from scientific methodology. How to justify the Mad Scientists Collective -- a group of Silicon Valley techies who engage in art projects at Burning Man -- or the International Society of Mad Scientists? How pathologically symptomatic was the indulgence of world-renowned physicist Richard Feynman in bongo playing, for which he was reputed? If such activity is to be considered "re-creation", how is the process of "re-creation" to be related to the science for which he was reputed? Scientists, as scientists, do not "re-create" and the process is not held to be part of the scientific method. If it is to be considered as a valued indication of their "humanity", then maybe other processes, deprecated within mainstream science, are also essential to being "human". Possibly it is the ability to reconcile such seemingly disparate domains which is the basis for a more integrative science of the future -- as possibly anticipated by such as Leonardo da Vinci and Isaac Newton?

As noted elsewhere (Avoiding Dialogue with Alternative Worldviews at any Cost, 2005), the existential timidity in the face of "extreme dialogue" is unworthy of a civilization "reaching for the stars" and potentially dependent on fusion energy. The latter is famously dependent on the craziest "Theories of Everything", as illustrated by the much-quoted statement by Niels Bohr in response to Wolfgang Pauli: "We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough." To that Freeman Dyson added:

"When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!" (Innovation in Physics, Scientific American, 199, No. 3, September 1958)

"Pathological religion": The early understanding in Vedic correlative thinking -- caricatured as the "twaddle of idiots" -- that the Upanishads were an articulation of correlations, may be usefully constrasted with current understanding of religion as establishing a connection with the divine. Although the Abrahamic religions notably stress the manner in which man is made "in the image of God" (Genesis 1:26), the "correlative" characterististics of this "connection" would appear only to be currently represented in figurative representations of a bearded male deity (or in the obscure understandings of mysticism). It might then be asked whether the "connectivity" emphasized by such religion should be considered as a pathologically limited understanding of a more complex "correlative" understanding of the nature of such a relationship.

This raises the issue of the nature and the extent of connectivity in the pattern of correspondences perceived as characteristic of cult thinking. What kinds and degrees of connectivity and correlation within spiritually-oriented groups are to be considered pathological?

Agreement vs Disagreement: The above examples raise the more general issue of the degree to which:

agreement is to be considered pathological as a consequence of any inappropriate relationships on which it is grounded. This is exemplified in the case of perceptions of cults and groupthink (Groupthink: the Search for Archaeoraptor as a Metaphoric Tale missing the link between "freedom fighters" and "terrorists", 2002 ). To what extent does this apply to the problematic relationships between disciplines (Dynamically Gated Conceptual Communities: emergent patterns of isolation within knowledge society, 2004)? How "pathological" is a community defined by a set of mutually acceptable correspondences -- considered questionable by others? However this raises the more general question of whether any dominant or mainstream mindset will be considered by the future as having been characterized by dependence on such pathological correspondences -- as the current mainstream arrogantly frames the deprecated correlative thinking of the past.
disagreement is to be perceived as exemplifying pathological rejection of correspondences considered "normal" in favour of others that are held -- from such a "normal" perspective -- to be "abnormal" or "extremist". But to the extent that contrasting "types" and "modes" of thinking are held to exemplify the diversity of the human community, are such types effectively defined by perceptions of the "pathological" nature of the correspondences held to be valid by the other? Does this provide a "scientific" explanation for the arguments of such as Richard Dawkins (The God Delusion, 2006) suggesting that any views that are "unscientific" are necessarily based on attaching "pathological" credence to correspondences and correlations that science (currently) considers to be meaningless?

Is it the case that those who disagree with the "positive" correspondences recognized by "my community" are necessarily to be seen as cultivating "negative" pathological correspondences (cf Being Positive Avoiding Negativity Management challenge of positive vs negative, 2005)? It would appear that this conclusion is held to be self-evident in the discourse in support of the intervention of the Coalition of the Willing in Iraq to "spread democracy" -- just as it is evident in the views held by some fundamentalists regarding the legitimacy of subjecting apostates to physical sanction for rejection of their community.

To what extent should agreement and disagreement be considered part of a process -- requiring careful consideration to be given to how correspondences are labelled pathological and by whom during the course of that process?

Moonshine? As noted elsewhewhere (Postmodernism and cognitive discontinuity, 2006), in exploring the significance of catastrophe theory for any "cognitive feel", and given Rene Thom's generalization of morphogenesis to include semantic dimensions, there is a case for noting speculation on intersection between the controversial work on morphogenetic fields and quantum gravity. Writing as a physicist, Alan D. Sokal (For Transgressing the Boundaries: towards a transformative hermeneutics of quantum gravity, Social Text, 1996) provides a helpful introduction to the language used:

Finally, an exciting proposal has been taking shape over the past few years in the hands of an interdisciplinary collaboration of mathematicians, astrophysicists and biologists: this is the theory of the morphogenetic field. [Rupert Sheldrake, et al]. Since the mid-1980's evidence has been accumulating that this field, first conceptualized by developmental biologists, is in fact closely linked to the quantum gravitational field: (a) it pervades all space; (b) it interacts with all matter and energy, irrespective of whether or not that matter/energy is magnetically charged; and, most significantly, (c) it is what is known mathematically as a "symmetric second-rank tensor". All three properties are characteristic of gravity; and it was proven some years ago that the only self-consistent nonlinear theory of a symmetric second-rank tensor field is, at least at low energies, precisely Einstein's general relativity. Thus, if the evidence for (a), (b) and (c) holds up, we can infer that the morphogenetic field is the quantum counterpart of Einstein's gravitational field. Until recently this theory has been ignored or even scorned by the high-energy-physics establishment, who have traditionally resented the encroachment of biologists (not to mention humanists) on their "turf". However, some theoretical physicists have recently begun to give this theory a second look, and there are good prospects for progress in the near future.

The author later revealed the article to be a hoax (A Physicist Experiments With Cultural Studies, Lingua Franca, May/June 1996), to the embarassment of many exploring this intersection, and reinforcing the view of sceptics (cf Sokal Hoax, The Sceptics Dictionary).

Of relevance to the discussion regarding the validity of any "correspondences" is the discontinuity in cognitive space that the disagreement regarding the hoax highlights -- ironically implied by Sokal's original title "transgressing the boundaries". The implication is that for physics there are boundaries that should not be transgressed -- an implication seemingly contrary to the basic premiss of complexity theory regarding the dynamic interconnectedness of everything.

An entry in the FreeDictionary on the Sokal Affair points to limitations in any comments by a qualified physicist on philosophical issues on which he is not comparably qualified (and makes no claims to be):

Mathematician Gabriel Stoltzenberg has written a number of essays with the stated purpose of "debunking" the claims made by Sokal and his allies. He argues that Sokal and company do not possess a sufficient understanding of the philosophical positions that they criticize and that this lack of understanding renders their criticisms meaningless. Defenders of Sokal have responded that postmodernists have a vested interest in denying the validity of his criticisms, which could not be accepted without serious harm to many careers and incomes.

In passing, Flemming Funch (Quantum Physics and Elections, 2004) usefully notes with regard to the hoax that:

Actually, the crux of the matter seems to be that Sokal believes in one finite objective reality, so therefore he considers all other views unscientific, and he tried to prove that point by satirizing them.... Maybe the joke is that mutually exclusive views on the world can all be right, because you do essentially get back what you start off trying to prove.....For the first group to consider themselves right, they have to consider the second group wrong, as there can only be one objective reality. Whereas the opposite isn't particularly the case. Anyway, I choose to bet on the models that explain the most possible phenomena in the world, rather than the models that have to suppress and ridicule all the stuff that just doesn't fit into them.

This points again to the challenge of bridging such cognitive discontinuity. Funch cites Mara Beller (The Sokal Hoax: At Whom Are We Laughing?; Quantum Dialogue: the making of a revolution, 2001) who argues that much of what Sokal was saying (and which he himself considered utterly ridiculous) had been said before by much more respected scientists than himself, like Bohr, Born, Einstein, Heisenberg, Pauli and Wheeler, who had even gone a good deal further in relating theoretical physics concepts to sociology, psychology and politics. Sokal later developed his thesis (Fashionable Nonesense: Intellectual Impostures, 1997). Beller however concluded::

In an exchange several months after his New York Review of Books [3 October 1996] article, Weinberg admitted that the founders of quantum theory had been wrong in their "apparent subjectivism," and declared that "we know better now". What exactly do we know better now? Do we know better that one should not infer from the physical to the political realm -- and if yes, why? Or do we know better that the "orthodox" interpretation of quantum physics -- the one that confidently announced the final overthrow of causality and the ordinary conception of reality -- is not the only possible interpretation, and that, ultimately, it might not even be the surviving one? .... The opponents of the postmodernist cultural studies of science conclude confidently from the Sokal affair that "the emperors ... have no clothes". But who, exactly, are all those naked emperors? At whom should we be laughing?

Sokal's hoax highlights good criticism of transdisciplinary borrowings in terms of the criteria of the conventional scientific method. He effectively rejects as improper the possibility of explanations that rely to a high degree on isomorphism and metaphor -- however much the role of metaphor has been fundamental to scientific creativity. With respect to the value of such borrowings and their relevance to cognitive discontinuity, even among scientistis, it is worth noting the remark of Kenneth Boulding (Ecodynamics; a new theory of societal evolution, 1978):

Our consciousness of the unity of the self in the middle of a vast complexity of images or material structures is at least a suitable metaphor for the unity of a group, organization, department, discipline, or science. If personification is only a metaphor, let us not despise metaphors - we might be one ourselves. (p.345)

Number mysticism -- pathology or remedy?

What might be considered a pathological separation between two forms of correspondence, the algebraic and the symbolist, is addressed through a variety of forms of number mysticsm -- which may itself be considered pathological. Curiously these variously seek to interrelate cognition based on "number" (as exemplified by number theory) with cognition based on "description" (as exemplified by text). The relationships are understood as a means of transferring significance between the two domains through a particular "scientific art" of transformation. Although typically deprecated (as "pathological") by both mainstream religions and sciences, curiously they have been traditionally valued precisely because of their assistance in detecting appropriateness. Some of the terms through which forms of this art are recognized include:

numerology: as the commonest form, this was practiced by early mathematicians, including Pythagoras; it is no longer considered to be part of mathematics. Numerology and numerological divination are typically deprecated as pseudomathematics by most modern scientists. However in science a form of "'numerology" is recognized in the coincidental resemblance of certain large numbers which continue to intrigue some of the most eminent mathematical physicists. It was such coincidental resemblance that gave rise to the explorations in group theory associated with "monstrous moonshine" (as described by Mark Ronan, Symmetry and the Monster: one of the greatest quests of mathematics. 2006 and discussed in the main paper). Extensive studies of sacred literature, may focus on its numerological significance -- notably Biblical numerology. Numerology may significantly inform superstition and thereby influence even major political decisions to ensure that they are undertaken in a manner considered auspicious rather than inauspicious. Building designers may typically avoid designations involving thirteen.

There is a widespread tendency to formulate insights, proposals or principles in point form, namely as made up of a specific number of items usually presented as a list. Other than preferences for small sets, the cognitive implications for the comprehension of sets with different numbers of elements have been poorly explored (cf Representation, Comprehension and Communication of Sets: the role of number, International Classification, 1978-79). Given current recognition of "modules" for processing of information by the brain (using scanning techniques), it is possible that comprehension of sets of a particular number of elements may trigger particular patterns in the brain that could explain qualitative preferences between sets of a different size. This would be consistent with brain responses to particular styles of music.
gematria (deriving from the Greek word for geometry) is a form of numerology that establishes correspondences between numbers and letters -- understood generically in relation to different systems including Greek, Latin and Arabic. However it is most closely associated with that involving the Hebrew alphabet. Words and sentences are then read as numbers, assigning numerical instead of phonetic value to each letter of the Hebrew alphabet. In this way words can then be compared and contrasted with other words having the same numerical values -- with which distinct qualities are associated. It has been used to suggest that sacred Hebrew texts were written with a profound understanding of mathematical principles. In its mystical form, correspondences may, for example, be recognized between polyhedra and theological principles. Gematria is fundamental to the transmission of kabbalistic knowledge and its associated practices. As a central feature of Jewish mysticism, the Kabbalah is recognized as a secret wisdom tradition of many variants -- disseminated selectively through graded spiritual hierarchies. However considerable prominence has now been achieved by the association of celebrities with the Kabbalah Centre (which prominently figures Isaac Newton on its website). Obviously it is unclear to what extent the many Jewish mathematicians have been consciously or unconsciously influenced by this tradition. Publicity was given in 2006 to the capacity of the world's most foremost kabbalist, Michael Laitman, to provide solutions to global crises within the context of a media event by Dropping Knowledge that deliberately configured 112 similar personalities such as to ensure continuing popular participation via the web (cf Enabling a Living Library: reconciling "free voices" and "intellectual propriety", 2006).
`ilm al-huroof (or `ilm al-jafr) is the science of Islamic numerology based on Arabic and the Abjad system of numerals -- as originally used by Arab mathematicians. As with the relation of gematria to Jewish mysticism, it is fundamental to the Sufi mystical understanding of the relation between numbers and letters (Secrets of Numbers and Letters: Abjad Ilm Huroof) and to certain Naqshbandi Sufi meditation practices -- although the practice of numerology as such is not approved by Islam [more].
magic squares are well-known as a feature of modern recreational mathematics although they have had considerable significance to many cultures over millennia. Such patterns of numbers have been widely used as a guide to ordering semantic and symbolic content -- notably for use in magical talismans for healing purposes. Their properties have been extensively studied in relation to the content of the classic Chinese divinatory systems used in governance (cf 9-fold Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics of magic squares, cubes and hypercubes, 2003).
encryption for purposes of confidential communication is dependent (ironically given its correspondences to the previous applications) on development of highly sophisticated mathematical transformations between letters and numbers -- done in such a way as to render a text meaningless to anyone without a special key.

In response to the crises of the globe, it is unfortunate that neither the sciences dependent on algebraic correspondences, nor the relational insights carried by symbolist correspondences, seems to have much of significance to offer in practice.

It is curious that secrecy, achieved in large part by encrypting correspondences, is becoming increasingly significant to the "clash of civilizations". And yet it is puzzling that insights supplying a bridge between the "clashing civilizations" of numeracy and literacy themselves seem to have so little to offer despite claims to the highest wisdom necessitating the most profound secrecy.

The irony is all the greater given the value attached to such secret knowledge by the wise in the faith-driven cultures in deepest conflict -- the Jews and the Arabs. Greater still, given that their overt practices in separating the sexes are criticized by another Abrahmanic culture whose leadership is frequently associated with freemasonry lodges where such separation is just as strictly institutionalized -- but covertly, although freemasonry is amongst the last redoubts of symbolist correspondence thinking and the study of sacred numbers. One might well ask which aspects of contemporary attitudes to number mysticism are the most pathological.

Varieties of connectivity

The various forms of correspondence, and the vigorous views regarding their merit in the organization of any collectivity, may usefully focus attention on the varieties of connectivity as highlighted by the following examples and contexts:

academic disciplines
- natural sciences, exemplified by the standards promoted through the Skeptical Inquirer by the Committee for Skeptical Inquiry
- social sciences, typically scorned as "unscientific" by the natural sciences, and exemplified by the issues raised in the Sokkal Affair
- other "disciplines", as deliberately recognized by Paul Feyerabend (Against Method: Outline of an Anarchistic Theory of Knowledge, 1975)
commerce, exemplified by the adage "the client is king", irrespective of any judgements about the client's sense of connectivity
aesthetics: exemplified by the variety of aesthetic preferences in designed relationships, irrespective of judgements about taste
elective affinities and affective relationships: exemplified by romance, irrespective of judgements regarding the viability of such relationships
faith and belief: exemplified by religious belief, irrespective of judgements as to their foundation -- especially highlighted by faith-based science (creationism, intelligent design, etc)
ecophilosophy/ecopsychology: exemplified by a profound experiential relationship with nature (Gregory Bateson, Mind and Nature: a necessary unity, 1980; David Abram, The Spell of the Sensuous: perception and language in a more-than-human world, 1996; Alan Drengson, Ecophilosophy, Ecosophy and the Deep Ecology Movement: An Overview)

Especially interesting, again illustrated by the Sokkal hoax, are issues of:

deliberate misconnectivity: as practiced by Sokkal and in any commercial transaction dependent on bluff -- or more generally in any game, including the strategic bluff that is fundamental in warfare, and the camouflage techniques employed by predator and prey in nature
inadvertent ignorance: as exemplified by the pronouncements of scientists -- subsequently discredited by the advancement of knowledge; especially interesting are cases of groupthink, notably those within the intelligence community (cf Groupthink: the Search for Archaeoraptor as a Metaphoric Tale: missing the link between "freedom fighters" and "terrorists", 2002)
dependence on authority and accredited expertise: as exemplified in the case of mathematical proofs requiring many years of education to comprhend, and hundreds of pages to elaborate
superstition as explanation: as exemplified by the criticism of the Skeptical Inquirer, which has nevertheless been unable to stem the widespread belief in what it rates as superstition (astrology, UFOs, etc) -- where others are satisfied by the quality of connectivity they see as evidence
placebo explanation: as exemplified in medication, where what is believed to offer a remedy does indeed offer therapeutic benefit, or is felt to do so; the placebo effect has been applied metaphorically to a variety of situations
pre-logical axes of bias: as exemplified by supposedly "rational" debates between "schools of thought" within academic disciplines, even the natural sciences (cf W T Jones, The Romantic Syndrome: toward a new method in cultural anthropology and the history of ideas, 1961) -- possibly to be substantiated by genetic disposition in favour of particular styles of explanation

Metaphorically extreme forms of connectivity might be illustrated by contrasting an elephant and a migrating bird (or flying squirrel). The connectivity considered viable for an elephant in travelling from A to B, is quite distinct from that considered viable by a migrating bird. The former, like the "monstrous" komodo dragon, is unable to travel between continents. How subtle may be the pattern of connectivity that enables meaningful travel within a collectivity? (cf Walking Elven Pathways: enactivating the pattern that connects, 2006)

A trap is a function of the nature of the trapped
(Geoffrey Vickers. Freedom in a Rocking Boat :
Changing values in an unstable society, 1972)

---

Moonshine Anecdotes

Those involved in the Atlas project, which was central to the process through which the Monster sporadic group was detected, were notably fond of word and number games -- to the point of seeing patterns in the names of the participants in the project. It is therefore intriguing to look at the name chosen by two of their members for the key paper announcing the outrageous conjectured implications of the Monster (John Conway and Simon Norton, Monstrous Moonshine, Bull. London Math. Soc., 1979). Given the 26 exceptional sporadic groups, and the 26 dimensional bosonic string theory through which the truth of the Monstrous Moonshine Conjecture was finally demonstrated in 1998, it is curious that:

the letters of Monstrous + Moonshine total to 18, which equals the number of countably infinite families of finite simple groups (to which the Monster and 25 other sporadic groups are exceptions)
the initials of Monstrous and Moonshine, the thirteenth letter of the alphabet, total to 26
the letters of Monster + Moonshine + Conjecture total to 26
the closest packing of spheres (12) around a central sphere, with a second such pattern, totals 26

Given the uncontroversial, Apollonic, beauty of symmetry, it might be expected that psychocultural dynamics would engender a Dionysian counter-part onto which negativity could be comfortably projected. To a large extent this role has been well-performed by the proponents and practitioners of the symbolist theory for the benefit of the fundamentalists of convention, whether rationalist or religious. There is therefore a curious symmetry to the facts:

that a culmination of a mathematical quest into the fundamentals of symmetry, exemplifying the algebraic theory of correspondences, should have given rise to what has long been termed a Monster by its discoverers (and contrasted with a Baby Monster)
that, almost from the origins of the mathematical quest, there emerged a prime exemplar of Dyonisian symbolism at the centre of of one of its most influential "laboratories" -- the Hermetic Order of the Golden Dawn -- made famous by such as W B Yeats. The irony is that the individual was variously dubbed the "Great Beast" and the "Monster", but produced and maintained one of the major tabular classifications of the symbolist theory of correspondences. From the perspective of archetypal psychology, critical thinking and cognitive hygiene, there is however a real challenge to distinguish and assess the various qualities and processes in play for any meaningful understanding -- especially given the appreciation widely accorded to mnemonic devices embodying those insights.

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