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21 August 2007 | Draft

Tuning a Periodic Table of Religions, Epistemologies and Spirituality

including the sciences and other belief systems

- / -

Periodic table -- precedents and parallels
Taxonomies of classification and self-referential dynamics
Precautionary comments regarding integrative initiatives
Dimensions of a general periodic structure?
Comparison with current situation
Possible future design considerations
Polarization and development of binary ordering
Fundamental learning distinction: Understanding vs Comprehending?
Fractal dimension: reconciling the uniqueness and sufficiency of each religion?
Adaptation of extended periodic table
Mode of dialogue
Playfully playing the periodic table
-- Musical metaphors
-- Sonification
-- Circle of fifths as a "periodic table"
-- Tuning systems
-- Systematic visual representation of musical possibilities on an orbifold
-- Musical embodiment
Developmental directionality?


It is obvious that religions have been a focus of numerous studies. These have included many efforts to classify religions. As usefully summarized in the Wikipedia article on major religious groups, such efforts have had different biases at different periods of time. Any such classification remains highly controversial as religions continue to compete for followers. Estimates of numbers of followers, and definitions of what is included in a particular religious group, continue to be vigorously contested. An additional dynamic arises from the fact that most religions necessarily consider themselves to be "right" and "good" in some absolute way, whilst framing others as "wrong", "misguided" or even "evil". These dynamics underlie many bloody religious conflicts -- especially in a period of increasingly faith-based governance.

As the Wikipedia article shows, tables can be produced to cluster religions in different ways. The question is whether some of the problematic dynamics could be rendered more explicable and predictable by moving beyond the simplest form of table to a periodic table -- inspired by the complexity of which it has been necessary to take account in the Periodic Table of Chemical Elements. The challenge of producing such a table is what is explored here. It should be emphasized that this is exercise is not intended to seek premature closure bur rather to look at what might (or might not) be an insightful way of organizing beliefs -- religious or otherwise -- given the nature of the dynamics between them. It should be stressed that this is not an interfaith exercise in syncretism. There is no question of seeking to amalgamate distinct religions or approaches to spirituality.

This exploration follows from a much earlier initiative by the author to produce a Functional Classification in an Integrative Matrix of Human Preoccupations (1982), partially inspired by the periodic table. This has since been used to order information on international organizations, world problems, strategies, values and human development -- for several reference publications (notably the Yearbook of International Organizations and the Encyclopedia of World Problems and Human Potential). These are now accessible online.


It is appropriate to ask why religions have not been clustered in less simplistic ways to highlight their correspondences and qualitative differences. When exploring this question at the time of the Parliament of the World's Religions (Chicago, 1993), it became apparent that any such possibility was highly contentious and of little interest to any particular religion. It is difficult to cluster meaningfully the range of perspectives which each consider themselves to be the essence of meaning and a unique channel for transcendental significance.

At that time the author was only able to trace one effort to juxtapose a spectrum of religious groups for presentation on that occasion. That was an initiative by Hinduism Today. The content is now accessible on the website of the Himalayan Academy (Major Religions of the World) but no longer in a manner that highlights their correspondences.

The following effort may therefore readily be considered a case of "fools stepping in where angels fear to tread". However the degree of violence that continues to be perpetrated in the name of religion and divinity is so horrendous that the possibility of foolishness could well be considered the least of concerns.

As noted above, it is indeed a period of increasing emphasis on faith-based governance, associated with demonisation of alternative perspectives -- as a justification for any violence perpetrated on such demons. It is also a period of increasing articulation of concern at the psychosocial damage associated with religion (cf Richard Dawkins, The God Delusion, 2006; Christopher Hitchens, God is Not Great: how religion poisons everything, 2007).

Whilst there have been numerous interfaith initiatives over past decades to remedy some of the misunderstandings, it cannot yet be said that these have given rise to a level of understanding that goes beyond some degree of mutual tolerance. On the other hand, with the rise of fundamentalism, any such tolerance is viewed with the deepest suspicion -- as is evident from the religions that choose to have nothing whatsoever to do with interfaith initiatives.

The notion of "religion" may possibly be usefully generalized to include somehow:

The construction of a periodic table is a potentially interesting way of "reframing" the cognitive challenge at this time. But:

Whether in the form of a (round) table or not, the exercise may offer a more integrative insight into the array of religions without questioning their integrity or uniqueness.

Periodic table -- precedents and parallels

There are a number of notable precedents for use of a "periodic table" for this purpose. They might be considered to include:

Clearly the extensive literature on cultures, and how they may be distinguished, might also inform the exploration -- as in the initiative of Haskell. Past and potential considerations of the social organization of knowledge, partially in the light of the periodic system, have been the subject of extensive commentary by Birger Hjørland (Social Organization of Knowledge, 2007; Bibliometric Knowledge Organization, 2007).

Taxonomies of classification and self-referential dynamics

Comprehensive studies of classification and taxonomy have been produced:

Birger Hjørland (The Periodic System, 2007) provides a valuable review of the significance of the periodic classification as an (iconic) model for knowledge classification.

Of great interest is the manner in which taxonomies, periodic or otherwise, emerge into competition with their predecessors and those with which they co-exist (whatever their incommensurability) -- a coexistence typically characterized by dynamics lacking in any elegance, dignity or mutual respect. Subsequently only to lose favour, become obsolete as historical curiosities, and then to fall into oblivion. This inexorable process continues despite the claims of their proponents for the eternal merit of the system in question. It raises questions about how world models collapse, which could be explored in the light of the recent study by Jared Diamond (Collapse: how societies choose to fail or succeed, 2005) and the earlier debates regarding paradigm change, notably in relation to the study by Thomas Kuhn (The Structure of Scientific Revolutions, 1962)

This evolutionary process might be caricatured by comparison with the mystery of "where elephants go to die". Where indeed do world models go to die? In terms of the concern here with religion, the challenging question is the mystery of the "dead gods" each offering a pattern of coherence to their followers. Francis Tremblay (Graveyard of the Gods) offers a list of thousands of dead gods, inspired by H.L. Mencken (Where is the graveyard of dead gods?) and the comprehensive profiling by Michael Jordan (Encyclopedia of Gods: Over 2,500 Deities of the World, 1993).

The divinely-enabled cosmologies of the past may indeed be understood as forms of periodic tables of qualities, values and principles with their complementary sets of deities of different generations -- valuable mnemonic devices for societies focused on complex kinship groups. Ironically, as shown by Neil deGrasse Tyson (Cosmos on the table: an astrophysicist looks at chemistry's most famous chart, Natural History, July-August, 2002), many of the elements in the periodic table of elements are named after deities.

In this light it is interesting to see a "cosmology" as a predecessor to what is now termed antiseptically a "worldview". Clément Vidal (A Minimal Philosophical Agenda: worldview construction as a philosophical method, 2007) describes a worldview as having the following seven components:

  1. A model of the world: Who are we?
  2. An explanation: Why is the world the way it is? Where does it come from?
  3. Futurology: Where are we going?
  4. Values: What is good and what is evil?
  5. Action: How should we act?
  6. Knowledge: What is true and what is false? How do we know what we know?
  7. Building blocks: What preexisting theories and models have been used to answer the questions of the other six categories?

The challenge is to learn from the questions: who was mistaken and how were they misled? To that extent the history and dynamics of such initiatives may be seen in terms of collective learning. The elements of any table might even be understood in terms of "learning modules" -- as the set of distinct questions raised in response to particular kinds of "catastrophe" (Cognitive Feel for Cognitive Catastrophes: question conformality, 2005).

The question as to "why" it is sought to produce an all-encompassing periodic table is then particularly relevant (cf Engaging with Questions of Higher Order, 2004). It raises the question of what the instigator achieves thereby. It has something to do with:

It is possibly appropriate to recognize motivations in terms of such caricatures as:

Precautionary comments regarding integrative initiatives

It is a fact that down the centuries different people have articulated integrative "models" to explain experience in an all-encompassing manner. Religions might be considered to be such initiatives as sanctioned and inspired by the divine. The periodic table is a relatively new metaphor that has inspired new approaches to organization which nevertheless echo efforts from the more distant past.

Some concerns relating to current initiatives towards synthesis have been listed elsewhere (Evaluating Synthesis Initiatives and their Sustaining Dialogues: possible questions as a guide to criteria of evaluation of any synthesis initiative, 2000). Particular concerns relate to the production and status of "models" -- in this case "periodic tables" -- in the current commercially biased academic context. Questions might include:

Some of these issues have been discussed elsewhere (Musing on Information of a Higher Quality, 1996; Future Coping Strategies: beyond the constraints of proprietary metaphors, 1992). Specifically it needs to be asked -- if such a cognitive device is all that it claims to be, and its use is subject to exclusive copyright or patent -- to what extent is the world thereby held to ransom, possibly when there is a vital need for such a device? Some implications of such issues have been raised in The Economist (16 June 2007) by recent efforts by Craig Venter to patent life. At some threshold "cost", it becomes cheaper to invent one's own model rather than "buy into" one for which there is a significant price to be paid. Ironically, this is a basis for reproduction (at least for the moment).

A religion may also be understood, to some degree, as such a device about which such questions may be asked. Access to spiritual insight may only be available under special conditions. How is this reflected in the design of any periodic table?

Dimensions of a general periodic structure?

It must be repeatedly stressed that the purpose of this exercise is to explore the possible design of such a table. Closure, premature or otherwise, is not the intention. In fact designing the process of how to design or "tune" such an arrangement is potentially of greater interest, given the many issues involved (and the reservations above). The process may be seen in terms of the challenges and controversies (in the background) surrounding the design of the table at any international summit as a prelude to dialogue at the summit. A round table is merely one of the options.

The art in this exploration would seem to be to use the complex order of the periodic table as a suggestive, metaphoric template for ordering religious and spiritual dimensions. It should be stressed as is obvious from a glance at the standard "periodic table", or alternative proposals, that the "tabular" arrangement is not as simple as might be readily assumed -- despite the underlying logic.

Consider some challenges and possibilities:

In distinguishing the periods or rows of the table, it may be useful to distinguish the following succession in some way:

Such possibilities highlight an immediate question as to the degree to which, for a particular religion tentatively associated with a column of the table, the characteristics at different periodic levels would be consistent. Alternatively would a religion have a characteristic set of more tangible practices but be associated with disciplines or principles that would more appropriately be clustered in a different group?

This issue points to the fundamental problem encountered in the development of the original periodic table, namely whether what had been assumed to be an "element", on the basis of the distinguishing capacities of the time, was not in fact a combination of two or more elements from different parts of the periodic table -- namely a "molecule" (or a superatom, as noted above). Changing metaphors, if each element is understood to be a musical "tone", is the religion effectively a "chord" -- perhaps despite its claims and pretensions to be an "organ" encompassing all tones (or all the possible music that might be played on it)?

Comparison with current situation

As with the variety of religions, there is currently every shade of worldview and preferred mode of classification. Each may be more or less successful in promoting or imposing its preferred system of order. Most intriguing is that any success in this respect, whatever the type of initiative, is increasingly problematic, as illustrated by:

The potential of these different approaches, as a source of all-encompassing order, has been bypassed or undermined by:

Given these various tools, there is relatively little felt need for an all-encompassing pattern of order, especially given the problematic dynamics engendered by placement of collective initiatives within such a framework. The issue can be caricatured by the dynamics associated with allocation of executive offices in a corporate HQ (no window, one window, two windows, corner office, with a view, etc). Avoidance of the challenges of eliciting an overarching framework nevertheless filters down into religious conflicts, conflicts between disciplines, and between religion and science. It is so much easier to invest funds in travelling to Mars or confirming the existence of the God Particle.

In the light of the precautionary comments above, some of the current issues include:

Possible future design considerations

The issue here is to explore the interplay between the design elements of a periodic table and those of the set of religions. Raising interesting questions and possibilities, some of these include (in no particular order):

Periodic table Religions
Charge: electropositive vs electronegative Proactive (proselytizing) vs Non-missionary?
Degree of charge: reactivity Intensity of response to "infidels"?
Increasing atomic number/weight Historical sequence of emergence?
Neutrality (associated with shell completion) as with the "noble gas" group Religions such as Quietism, Taoism, Zen, Agnosticism, Atheism?
Elements in the same group but of different periods Appropriation of isomorphic elements of one religion by another (Paganism by Christianity)?
Electrons in outer shell Number of active principles or values? Design of symbol?
Electrons in lower-level shells Number of sub-principles?
Isotopes Schismatic heresies?
Half-life (of isotopes) Stability (duration) of a heretical perspective?
Radioactivity Mystical insight and physical consequences (sidhis)?

More challenging and controversial is to move beyond such generalities and to consider possibilities and implications in particular cases (again in no particular order):

Polarization and development of binary ordering

It could readily be argued that belief systems have little in common with the logic of the development of the order represented in the periodic table of chemical elements. There is however a fundamental binary logic fundamental to both -- manifesting in the case chemical elements in the consequences of positive and negative electric charges and how they are expressed in various parts of the table..

In the case of religions and belief systems in general, it might be said that the fundamental doctrine and dynamics are concerned with managing the binary dilemmas of:

However in the periodic table this binary order is based on:

Accepting the co-existence of elements, the process of shell completion, from which elements of different characteristics emerge, is a valuable way of exploring the set of religions. See separate discussion (Conditions of Objective, Subjective and Embodied Cognition: mnemonic systems for memetic coding of complexity, 2007).

Especially interesting for a belief system is how the resolution of such "completion" is framed and experienced. Aspects of this are suggested by the following:

Fundamental learning distinction: Understanding vs Comprehending?

A major advantage of using the periodic table of chemical elements as a metaphor is the radical corrective it brings to superficial understandings of periodicity, especially those inspired in part by the seductive comparison with octaves of musical tones. In saying this, it is important to recall that such a parallel was first made with respect to the ordering of chemical elements -- prior to the work of Mendeleev. However it is quantum theory that has clarified the "irregularities" in Mendeleev's table -- without being able fully to explain them.

Whilst the philosophy of physics (as distinct from metaphysics) has a history of several decades, that of chemistry (as distinct from alchemy) is of recent origin. In an insightful summary of the second gathering of the International Society for the Philosophy of Chemistry, Lila Guterman (Philosophy and Reasoning Network, 1998) notes the conclusion that important facets of chemistry have not been explained by quantum mechanics -- a fact that many philosophers overlook. This is evident at the the heart of modern chemistry -- the periodic table -- never successfully explained by the laws of physics and quantum mechanics, which have failed to justify the ordering of the elements in the table, despite their valuable insights into the process. As she puts it, quoting Eric Scerri, with respect to "wrong filling":

Potassium's outer-most electron sits in the fourth shell, even though 10 of the 18 spaces in the third shell remain empty. Its neighbour, calcium, behaves the same way. But then the next element, scandium, "remembers" that the third shell is not yet full and puts its extra electrons into it. "If the shells were to fill in a sequential order, we would have a perfect quantum mechanical explanation of the periodic table," says Scerri. But they don't.

The lesson here is that the superficial regularity -- congenial to comprehension by the mind -- may obscure more fundamental processes of completion. The ability of scandium to "remember" that it has unfinished business thus offers a contrast between superficial metaphysical comprehension of any whole and development at a more fundamental level -- perhaps the essence of the experience of understanding. Comprehension may indeed see a pattern, but runs the risk of premature closure -- perhaps reinforced by groupthink. Understanding, however, embodies that pattern in application. This is a form of "maturity" that recognizes a more fundamental form of balance that needs to be kept.

Both purely esoteric approaches to periodicity, and those cited above, tend to avoid this challenge at the heart of ordering psychosocial knowledge. And yet, on the one hand, religions are notably to be distinguished by their preferences for distinct sets of principles, and on the other, in the real world, efforts to design simple periodic tables for information classification typically end up with "bulges" and "fixes" to accommodate reality. Scandium is arguably "very human" in its need to remember unfinished business at a more fundamental level. Any periodic table of religions needs this quality.

In this respect, in reviewing the work of Allan Combs, Stuart Sovatsky (The Radiance of Being, Cybernetics and Human Knowing, 1999) notes his agreement with 'recent evolutionary views of consciousness, as well as [with] systems theory in general, as seen in the grand evolutionary synthesis, [which] tend to view earlier evolutionary stages not as lost, but as continuing to express themselves at lower levels of constantly complexifying systems.'

Simply put, development of understanding is not just a matter of "higher and higher" and "up and up". There is fundamental work to be done in grounding in order that understanding can be embodied. This reflects common knowledge in some psychotherapeutic disciplines. Arguably it is also in accord with some forms of spiritual understanding, perhaps as carefully clarified by Jorge Ferrer (Embodied Spirituality, Now and Then. Tikkun, May / June 2006) through a process of what he describes as "creative interreligious hermeneutics". Ferrer's distinguishes between "disembodied spirituality" and "embodied sprituality" in terms of "sublimation" (or "heart-up") vs an "integration" engaging the whole body. He argues that many apparently embodied religious orientations conceal highly ambivalent views toward sensuality and the physical body.

This raises the question of the value of the periodic table as a metaphor indicative of how much "unfinished business" there may indeed be. In this respect, one interesting approach has been efforts to "extend" the periodic table (Jeries A Rihani, The Extended Periodic Table Of The Elements). Of particulat interest in the use of this metaphor is Ferrer's reference to a process of "completion" in his argument regarding "grounded spiritual visions":

As we have seen, most spiritual traditions posit the existence of an isomorphism between human beings, the cosmos, and the Mystery. From this correspondence it follows that the more dimensions of the person that are actively engaged in the study of the Mystery, the more complete his or her knowledge will be. This 'completion' should not be understood quantitatively but rather in a qualitative sense. The more human dimensions that creatively participate in spiritual knowing, the greater will be the dynamic congruence between inquiry approach and studied phenomena and the more grounded in, coherent with, or attuned to the nature of the Mystery will be our knowledge.

Associated with this grounding process, Ferrer stresses the emergence of: an in-the-world nature, re-sacralization of nature, social engagement, and the integration of matter and consciousness. These are all to be seen as relevant to appropriate cognitive engagement with the challenges of the times. Indeed, if the requisite "integration" is above all through pattern isomorphism and resonance, reflections on "quantum consciousness' may have much to do with understanding of orbitals, their configuration and some form of interlocking (cf Emergence of Cyclical Psycho-social Identity: sustainability as "psyclically" defined, 2007). This might be understood as creating frameworks, and "launch windows", through which any engagement with a kaleidoscopic reality takes place. The results of the major exercise by mathematicians in classifying finite simple groups within an "atlas" might be considered as offering related insights, as discussed separately (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007). Arguments here relating to simple periodicity are further emphasized in the contrast, within that classification of symmetry groups, of the exceptional sporadic groups -- and notably the discovery amongst them of the Monster of symmetry. This may exemplify the ultimate challenge to understanding.

The contrast proposed here between "comprehension" and "understanding" can be fruitfully related to that offered by Magoroh Maruyama (Polyocular Vision or Subunderstanding?, 2004). However his focus is primarily on the failure to recognize the "polyocular" complementarity essential even to adequate "comprehension" at the levesl framed above as superficial. This failure indeed results in "subunderstanding" at those levels -- perhaps to be recognized as analogous to the failure to "understand" at more grounded levels. Ironically this grounded understanding could have been appropriately termed "subunderstanding" in a sense quite contrary to that implied by Maruyama.

The gathering of the International Society for the Philosophy of Chemistry (1998), offered a second insight of relevance to any periodic table of belief systems. Again in the words of Lila Guterman, quoting Joseph Earley (a theoretician of chemical explanation):

Earley claims philosophers have been too content with a narrow view of what counts as a "thing". To most philosophers, a material object must be something like a rock -- something held together by forces, producing a tangible object that functions as one unit and resists change. But there are many dynamic objects, Earley says, that philosophers have difficulty describing... In such systems, simple components or processes can work together to generate a more complex whole....And he has found an example from chemistry that can help to define for philosophers how the components of dynamic objects must balance to produce a unified whole -- oscillating chemical reactions.

These are reactions that cycle through distinct states only to come back to the starting point and begin all over again. A well-known example is the Belousov-Zhabotinsky reaction, in which concentrations of ions oscillate regularly.... "You have something that is an autocatalytic process, where one makes two, makes four, makes eight -- something exploding. Then there's some control mechanism that shuts that down and starts you again.....

To him, balancing these processes so that the cycle repeats indefinitely produces a whole that is more than the sum of its parts. The components constitute one "thing" because they work together and the chemical reactions influence each other to achieve a balance.

Although some religions specifically define themselves as being like a "rock" for the faithful to stand on (or cling to in times of turbulence), this suggests a valuable way of giving a dynamic dimension to the invariance sought in spirituality -- avoiding psychological dependence on misplaced concreteness. This also offers scope for discussion in relation to the nature of territorial conflicts -- typical of the Abrahamic religions -- over any land held to have been "gifted by God" (a theme discussed in Thinking in Terror: refocusing the interreligious challenge, 2005).

The contrast offered here, in terms of the periodic table, between comprehension and understanding fails however to address another dimension briefly acknowledged earlier in terms of self-reference. The question was well raised by Douglas Hofstadter (Gödel, Escher, Bach, 1979) . How implicated is the creator or user of a periodic table in that device? In the case of an array of religions and belief systems, the periodic table then stands as a kind of mirror of the mind's ability to variously order reality. The relation to such a mirror has been a theme of centuries of dialogue between the "sudden" and the "gradualist" approaches to enlightenment in Chinese thought (Peter N Gregory (Ed) Sudden and Gradual; approaches to enlightenment in Chinese Thought. Delhi, Motilal Banarsidass, 1991). This dialogue was notably triggered by two very simple contrasting poems based on a mirror -- by Shen-hsiu (606-706) and Hui-neng (638-713) in the Platform Sutra [texts] and whether it needed "cleaning".

For Luis Gomez (Purifying gold: the metaphor of effort and intuition in Buddhist thought and practice):

...those who assume that the object of religious, aesthetic or intellectual apprehension is somehow innate in the apprehending subject tend to assume at the same time that the act of apprehension is direct, abrupt, effortless. The most common metaphor employed by the advocates of this type of position... is the mirror as symbol for the mind: both are innately pure, both are able to know (or reflect) clearly, passively, and integrally. The opposite view would then propose that the object of religious esthetic, or intellectual apprehension is not innate, and that the act of apprehension is indirect and gradual, the result of dedicated self-cultivation.

A different take on this challenge is offered by Bill Halpin (Engaging Emptiness: Stepping into the Mirror, 2000). This is consistent with the reflections on enactivism (Francisco Varela, Evan Thompson, and Eleanor Rosch, The Embodied Mind, 1991; George Lakoff and Mark Johnson, Philosophy In The Flesh: the embodied mind and its challenge to western thought, 1999)

Fractal dimension: reconciling the uniqueness and sufficiency of each religion?

By definition, as noted earlier, each religion is a complete and sufficient articulation of the nature of the relation between the particular individual and all-encompassing transcendental reality. The existence of other religions is necessarily profoundly offensive to this understanding and readily to be understood as a consequence of misguided insight, possibly to be appropriately framed as "untruthful", "wrong" or "evil".

The art of a fruitful exploration of a possible periodic table design is to note the complex patterns of organization of the array of chemical elements whilst not losing sight of the fundamental challenge to comprehension posed by spiritual development. Religious insights can be readily (mis)understood simplistically. The more profound insights are a lifelong challenge. The standard periodic table can also be (mis)understood simplistically and the subtleties of its current articulations in terms of quantum theory are a matter of extensive study -- whether or not this results in appropriate comprehension.

One way to look at the challenge of the sense of uniqueness claimed by each religion is the consequence of being cognitively positioned within a particular (elemental) framework from which all engagement with all-encompassing reality takes place. It is the window on the universe. Some process of this form may be the essence of belief. The integrity and coherence of such a framework may be defined by sets of principles and insights that could have some correspondence to electron shells. This organization is intuitively echoed to some degree in sacred geometry (religious architecture) and the design of rose windows and rosaries.

From within that framework the possibility of spiritual development through some form of "shell completion", and the opening up of successive (concentric) shells of "higher potential", may appear totally consistent in the quest for a "theory of everything". There is necessarily no other way and that process clearly offers greater and greater insight into "reality" whilst reaffirming a sense of identity therein.

The challenge remains of showing how each religion, or other belief system, could have a sense of its own unique adequacy as a vehicle through which to engage with the world -- an understanding from which other perspectives would necessarily appear to be of lesser adequacy. It is in this respect that the fractal organization and representation of the Mandelbrot set suggests many leads for further reflection (cf Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005)

Adaptation of extended periodic table

The challenge in what follows is to clarify ways of thinking about the relevance of the periodic table as a metaphor of requisite complexity for ordering the complex relationships between belief systems. A useful point of departure is to highlight how this facilitates understanding of the distinction from the simplistic patterning typical of currently preferred periodic orderings.

Two approaches might be considered:

In the following table the information is distributed otherwise in order to show the development of (emergent) periods over time (from left to right). Crucially the emergence of "blocks" (s, p, d, f, g) associated with each period (as a result of filling electron shells) is shown vertically (from top to bottom). Rather than distinguish the group allocation within such blocks (as presented in standard tables), the information is summarized as a range. For example, for Period 2, the s-block is split in two (given its fundamental significance), but the [6] elements in the p-block are presented as the range from Boron to Neon, with atomic numbers ranging from 5 to 10. Each of these is a member of one of the 6 groups (with corresponding elements of other periods). Cells for the s-block are shaded in common with the p-block because together they reflect the 8-fold pattern common to many octave based periodic systems, whether presented in a table or a spiral. It is the failure to take account of the greater requisite complexity, highlighted by the role of the d-, f- and g-blocks, that the table is designed to highlight. Specifically, as noted above, it is the fact that "scandium remembers" the need to fill a lower shell (Period 3, d-block), before returning to the octave pattern (period 3, p-block), that is made apparent by this presentation -- as for the later need to fill other lower shells.

Adaptation of the extended periodic table
Helium is placed next to hydrogen, instead of above neon, because it is part of the s2 group;
allocation to groups, marked [N], within p, d, f and g blocks is separately distinguished.(follow links)
symbol (range) with atomic number (range) is given for each cell .
119 169 Alkali metals
120 170 Alkaline earth metals
. B/Ne [6]
Al/Ar [6]
Boron, Carbon, Nitrogen,
Chalcogens, Halogens,
Nobel gases
. . . Sc/Zn [10]
Y/Cd [10]
Lu/Hg [10]
Transition metals
. . . . . La/Yb [14]
Ac/No [14]
Lanthanide series,
Actinide series
. . . . . . . ?/?
Hypothetical series
2 8 8 18 18 32 32 50 50 .
2 10 18 36 54 86 118 168 218 .
1 2 2 3 3 4 4 5 5 .
1 3 5 8 11 15 19 24 29 .

This method of complexifying any periodic table of belief systems clarifies the distinction between a "superficial" approach, whose apparent validity obscures the fact that it is effectively "ungrounded" -- ironically reinforced by reference to some such groups as "rare earths". In the case of belief systems, this "re-membering" process may be understood in terms of:

Any failure to provide such necessary "grounding" of the 8-fold pattern increasingly accentuates what might be understood as a "prehensile" or "grasping" quality of comprehension (cf Beyond Harassment of Reality and Grasping Future Possibilities, 1996). The more "profound" insights recognized to be associated with higher periods (to the right) within the octave pattern in the table then become increasingly tenuous intuitions rather than capable of being effectively embodied and applied. How to ground such intuitions, implicit in the 8-fold set, may then be held to be a "hidden secret".

The above presentation may potentially also be used to clarify other sets of beliefs:

As has been noted, the periodic table has a metaphysical (or perhaps "metachemical") component because its arrangement is partially subjective and cannot be tested. This does not invalidate such a table, whose role is primarily as an aid to communicating the insights of chemistry. It just means that no variant can be claimed to be 'the real periodic table'. In particular, the orientation of the above table could be rotated through 90 degrees or 180 degrees to reinforce different interpretations -- depending on the metaphoric significance attached to that part which is "up" or "down" (as explored by Geoerge Lakoff and Mark Johnson, Metaphors We Live By, 1980)

This recognition can be used to address the challenge of the different ways in which patterns are distinguished and highlighted by belief systems. The relation between the two "levels" of the s-block, and the s-block itself, are especially significant in this respect. The s-block is a reflection of the basic binary pattern, especially in Period 1:

Such issues are also reflected in how various other patterns are distinguished, notably 10-fold and 12-fold. Are the qualities of the Greek Dodekatheon then to be understood as characteristic of Periods 2 and 3 of the p-block -- as with archetypal roundtables?

Mode of dialogue

The above discussion clarifies the mode of exploration that might be considered fruitful in avoiding premature closure. The purpose is to promote dialogue about possibilities, probabilities and potentials within design constraints-- perhaps especially consistent with understandings of the probability of truth (cf V V Nalimov, Realms of the Unconscious: the enchanted frontier, 1982)). The approach might be described by three distinct metaphors, of which the first two are extensively explored elsewhere (Animating the Representation of Europe, 2004):

The exploration might be illustrated or reframed by metaphors specifically associated with the table above. Given a degree of correspondence between the atomic number and the human life span, the rows might be understood as follows for mnemonic purposes: infancy (s-block), pre-adult (p-block), adult (d-block), elder (f-block), ? (g-block). And, given the very short half-life of the hypothesized g-block elements, this might be consistent with the passing nature of the intimations of larger orderings. Within the same age-related metaphor, the periods might be related mnemonically to functions: birth (Period 1), education (Periods 2-3), employment (Periods 4-5), retirement and "wisdomhood" (Periods (6-7), ? (Periods 8-9).

The process might also be framed as one involving various meta-modelling (flying machines) experiments

As a learning process, the dialogue might be understood as a form of dance:

A periodic table might be fruitfully understood as a form of all terrain cognitive vehicle, or as a radio receptor device that may be tuned to a wide spectrum of frequencies -- an all-frequency radio. It may also be better understood as an approach to designing the seats at a roundtable:

Playfully playing the periodic table

How is such a periodic table -- effectively an "organ-ization" of knowledge and ways of knowing -- to be "played"? What unexpected resonances may then engender unforeseen forms of harmony -- capable of sustaining higher forms of psychosocial organization? (cf Some Clues to Social Harmony from Music, 1993; Liberation of Integration, Universality and Concord -- through pattern, oscillation, harmony and embodiment, 1980)

Belief systems are fundamental to the identity of many and need to be treated with great respect as the fruit of creative insight and possibly of divine revelation. So seriously are they taken that they lead to violence in the defence of what they hold to be true or to promote those truths amongst those who have failed to recognize their vital significance. They may well be suspicious of any failure to frame them otherwise than as required by tradition. The contribution of humour and playfulness to learning processes is recognized in some belief systems (Humour and Play-Fullness: Essential integrative processes in governance, religion and transdisciplinarity, 2005). But it may well be perceived as "blasphemous" -- even in the case of science.

This raises the question of how to play -- respectfully -- with a periodic table of belief systems embodying perspectives antagonistic to each other, and with which one may personally be perceived to disagree. How is a "light touch" to be introduced into explorations of ways in which a periodic table might be "tuned"?

Musical metaphors: As an example, the above "table" may be reframed with several musical metaphors:

In such terms, rather than as an element, a superatom or a molecule, a set of world models, as belief systems, might then be understood as set of instruments in a quartette, a jazz group a symphony orchestra. This raises such issues as:

Sonification: The value of a musical reframing to comprehension (if not understanding) is powerfully indicated by scientific interest in sonification as a means of pattern "re-cognition" (cf International Community for Auditory Display. Sonification Report: status of the field and research agenda. Prepared for the National Science Foundation, 1999). Indicative of related possibilities are the current explorations into "musical genetics" (cf Music Genome Project). This enables the full range of musics to be described by a set of multiple criteria (rock requiring some 90, classical music 300-450). This should enable musics to be clustered into patterns that might well be related to a periodic table.

Also of interest is the "information music" (iSIC project) of Sheridan College Institute of Technology and Advanced Learning of Canada initiated in 2001 [more more]. This is an alternative approach to remotely monitoring complex systems like communications networks through presentation of information in a synergized acoustical format that provides a holistic and uninterrupted audio (musical) model of the system under observation. Clearly it could be adapted to weather systems and, why not, to the relationships between belief systems,

Circle of fifths as a "periodic table": Curiously the circle of fifths, is recognized by some musicians as analogous in significance to the periodic table for chemists, although not necessarily as fundamental, although see the work of Ernest G McLain (The Myth of Invariance: the origins of the Gods, Mathematics and Music from the Rg Veda to Plato, Shambhala, 1978; The Pythagorean Plato: prelude to the song itself, 1978). However the Chemistry Daily (a "chemistry encyclopedia") includes (from the Wikipedia entry) a description of the organization of pitch space as follows:

The circle of fifths is one representation of pitch space, first proposed geometrically (see: Pythagoras) by Johann David Heinichen (1728), though he included the relative minor (thus the circle clockwise would read C, a, G, e...) [Lerdahl, Fred (1992). Cognitive Constraints on Compositional Systems, Contemporary Music Review 6, 1992, 2, pp. 97-121; Tonal Pitch Space, Oxford University Press, 2001, pp. 42-43]. The current major on the outside relative minor on the inside format was proposed by David Kellner (1737). M.W. Drobisch (1855) was the first to suggest a helix (ie the spiral of fifths) to represent octave equivalency and reoccurance (Lerdahl, 2001). Shepard (1982) uses a double helix of two wholetone scales over a circle of fifths which he calls the "melodic map" (Lerdahl, 2001). Michael Tenzer suggests its use for Balinese gamelan music since the octaves are not 2:1 and thus there is even less octave equivalency than in western tonal music (Tenzer, 2000). See also chromatic circle.

Tuning systems: As noted earlier, the differences between tuning systems, and the various appreciations of them, might better inform understanding of the conflictual relationships between disciplines. Daniel White (Potential Mathematical Models for the Western Musical Scale: a historical and empirical comparison, University of East Anglia, 2007) concludes his exploration through "the mine-ridden multi-dimensional maze of tuning", with indications that could as well be applied to beliefs systems:

Such insights are confirmed by those of Gavriel Segre (On the Mathematical Structure of Tonal Harmony, 2004):

No example could be given of the radical dichotomy existing nowadays among Science and Humanities as the intellectual analysis on the structure of Contemporary Music. Let us start analyzing the overwhelming confusion existing in most of the discussions concerning the concept of musical consonance.

Systematic visual representation of musical possibilities on an orbifold: A musician at Princeton University, Dmitri Tymoczko (The Geometry of Musical Chords, Science, 313. 5783, 7 July 2007, pp. 72 - 74) has recently demonstrated the possible use of advanced geometry as a tool for understanding musical structure [comment]:

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by utilizing short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries, and suggest different musical uses.

Through the nature of the mapping onto an orbifold, it can be shown that if a listener has a liking for a particular chord, or group of notes, then it can be shown how to find other, similar chords and link them together to form attractive melodies. The particular merit of the approach, in relation to the above argument regarding a musical understanding of a periodic table of beliefs, is that it covers a far broader range of harmonies and melodies than has previously been represented by music of particular (notable western) styles. It provides a framework that enables understanding of music regardless of style. It effectively positions the harmonious chords traditionally preferred in contrast to those favoured by more experimental music.

Tymoczko's use of the orbifold reinforces the point made above regarding the limitations of the earlier "octaval" levels of the periodic table in relation to unforeseen developments. As he notes:

Western music theory has developed impressive tools for thinking about traditional harmonies, but it doesn't have the same sophisticated tools for thinking about these newer chords.

Perhaps understandably, given the range of styles of music, the orbifold on which Tymoczko has indicated they can be meaningfully mapped is a very complex mathematical object -- difficult for most to either comprehend or understand. In topology and group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a particular kind of topological space (called an underlying space). A periodic table is a very simple structure in comparison.

But the significant point to be made is that if the set of styles of music can be understood as a sonification of the range of belief systems, knowing that the styles can be mapped in an orderly manner onto such a mathematical object (rather than into a periodic table) gives confidence that the coherence of the ordering of the belief systems can at least be heard -- even though it can only be partially seen. The orbifold might then be understood as the appropriate form for a Rosetta Stone interrelating belief systems. Related issues with regarded to the mathematics underlying the periodic table of forms of symmetry, and the fundamental importance of its exceptions, are discussed seaparately (Potential Psychosocial Significance of Monstrous Moonshine: the highest degrees of symmetry as a Rosetta stone for cognitive frameworks, 2007). Such explorations are potentially highly relevant to mathematical theology (cf Towards a Logico-mathematical Formalization of "Sin": fundamental memetic organization of faith-based governance strategies, 2004).

However, as noted by Tymoczko regarding spaces on the orbifold with respect to music:

This idea that you should stay in one part of space is an important ingredient of our notion of musical coherence.

Such a structure offers both a valuable justification of the notion promoted by every religion regarding its own coherence as well as pointing to the kind of complexity through which the nature of the relationship between belief systems can be understood -- a potential key to interfaith or interdisciplinary relationships (notably in the light of what has been described in the Principia Cybernetica as special political modality known as the "Belgian compromise"). It might offer a whole new level of potential significance to the annual Urbi et Orbi message of the Pope!

Musical embodiment: How indeed does one embody and express the insights of an orbifold?

As argued elsewhere (Anti-Developmental Biases in Thesaurus Design, Paper for the Conference on Conceptual and Terminological Analysis in the Social Sciences, Bielefeld, 1981), much richer approaches to thesaurus integration emerge from, and are necessitated by, such varied domains as ecosystem integration, "oscillatory" integration in multi-party political systems, education, strategy, etc. Any organic form of integration which matches the dynamism of real-world phi nomena is perhaps necessarily oscillatory.

Surprisingly, perhaps, there in fact much to be learnt from the theory and philosophy of music as guide to further investigation.It is refreshing to note how this possibility emerges from reflections on the non-Western 4,000 year-old chanted hymn of the Rg Veda of the indian tradition. A careful exploration of this work by a philosopher, Antonio de Nicolas (Meditations through the Rg Veda, 1978), using the non-Boolean logic of quantum mechanics, as articulated by Patrick Heelan (The Logic of Changing Classificatory Frameworks, 1974) opens up valuable approaches to integration. The unique feature of the approach is that it is grounded in tone and the shifting relationships between tones. it is through the pattern of musical tone that the significance of the Rg Veda is found.

Of the greatest interest is the link made by de Nicolas with P.A. Heelan's concern with the "Logic of Changing Classificatory Frameworks" in terms of the conceptual freedom of quantum logic -- which is in complete contrast to the essentially mechanistic structure of conventional thesauri. It is difficult to imagine that significant breakthroughs would not emerge from investigation of such leads in terms of thesaurus design. It might well be expected that the theory and appreciation of music would indeed help to facilitate both comprtehension and understanding (in the above sense) of these possibilities. In this respect the review by Edgar Taschdjian (Music and Mathematics, General Systems, 17, 1972) is very suggestive of future possibilities.

Developmental directionality?

As noted above, rotating the table reinforces implications about possible understandings about the direction of development. Just as with respect to significance attached to "up" and "down", so different significance can be associated with "light" (immature? spiritual?) or "heavy". Chris Lucas (personal communication) argues, for example:

I like the historical "increase in weight" argument and the relative population occurrences, but would suggest that you reverse the rows of the table, the lighter elements are at the top of the periodic table and these correspond I'd say more to spirit than to matter. As religions "congeal" they become more solid and inflexible, obsessed by rules and fixed views and this corresponds to the lower rows -- rather like gas becoming liquid and then solid as we descend from the celestial heavens

One issue in this respect, as noted by Wilber, is whether development goes "on and on". Another is how any cultural evolution might relate to the controversial issue of orthogenesis, or directed evolution (Igor Popov, The Persistence of Heresy: the concepts of directed evolution (orthogenesis), 2005). Such questions may however be reframed by considering whether a "table" is the appropriate surface on which to represent a "periodic table".

As discussed separately (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006), in contrast to the plane surface of a simple matrix, a torus holds an interesting position in the discussion of the relationship between form and medium as fundamental to advanced theories of communication. This notably featured in the work of Niklas Luhmann (Die Gesellschaft der Gesellschaft, 1997) and discussed by Michael Schiltz (Form and Medium: a mathematical reconstruction, Image [&] Narrative, 6, 2003) in relation to the calculus of indications of George Spencer-Brown (Laws of Form, 1969/1994). Schiltz notes that form/medium is "the image for systemic connectivity and concatenation", as described by Humberto Maturana and Francesco Varela. The question then becomes whether and how a periodic "table" might be meaningfully mapped onto a torus to hold the possibility of "on and on" development. Perhaps of greater relevance is how any musical representation of belief systems might be mapped onto an orbifold -- as suggested above.

Curiously, as argued by Andrew Duncan (Combinatorial Music Theory, Journal of the Audio Engineering Society, 39, 1991, pp. 427-448), in discussing the fingerboard of a fretted string instrument:

Mendeleyev's periodic table of the elements was successful because of the added dimension it introduced. Rather than a simple list of known elements, they were arranged with a second axis, so that elements adjacent vertically shared common properties. For the same reasons it is very useful to arrange musical notes as patterns on the plane: the added dimension allows us to corral some correlations.... [However] identifying notes separated by octaves has a "curling" or "looping" effect. In two dimensions, the periodicity along the neck has the effect of turning the infinite fingerboard into a cylinder... The periodicity across the neck curls the cylinder upon itself, forming a torus...In fact, we may consider the 12 x 12 torus to be tiled with smaller tori -- to wit, twelve of them, each containing one copy of each distinct note.

Duncan proceeds to represent graphically the discrete topology, or connectivity of the fingerboard:

This graph has many interesting properties. For example, we first note that it consists of two loops, each of which goes through each note exactly once. (Such a loop is referred to in graph theory as a Hamilton cycle) Each loop is the edge of a Möbius band with 1 1/2 twists, the bands for the two loops being of opposite handedness. Each loop also constitutes a trefoil knot: a fundamental way of knotting a loop in three dimensions. One may think of the graph as having its notes divided into six pairs, a pair consisting of any vertex and that vertex which is six steps away (by either cycle).

Also of related interest is the issue of the historical sequence in which "elements" are gradually discovered (cf Discovery of the Elements) and the implications for the coherence of the set of those considered credible at any one time in relation to those yet to be discovered.


There would seem to be a reasonable case for combining some of the threads explored as a means of more fruitfully interrelating beliefs systems. Specifically:


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