- / -
Introduction
Monstrous connectivity
-- Monstrous Moonshine: a collective knowledge creation quest
-- Implications of the connectivity of Moonshine
-- Possible application to contemporary challenges
-- Decrypting complexity
-- Cognitive psychology and comprehension
-- Representation of the 26 sporadic groups
-- Patterns of conceptual integration
Cognitive challenges to understanding
-- Intimate understanding
of mathematically defined relationships
-- Cognitive science of mathematics
-- Source of intimate knowing in mathematics
-- Possible resonant insight into the gross structure of
the Monster
-- Mirroring within the Monster
-- Eliciting the beauty of the fine structure of the Monster
Given the beauty of symmetry as an attractive indicator of truth (cf Ian Stewart, Why Beauty Is Truth: the history of symmetry, 2007), and given the "monstrous" complexity of the highest forms of symmetry (cf Mark Ronan, Symmetry and the Monster: one of the greatest quests of mathematics, 2006), is it possible that the governance challenges of "globalization" call for a form of marriage between "beauty" and such a "monster" -- but of a quite unexpected order of complexity? Why is it assumed that the knowledge required is not of such an order of complexity?
Mathematicians exploring forms of symmetry now accept that any proof of a theorem relating to them may be hundreds of pages in length. That for the so-called "enormous theorem" is some 15,000 pages in length -- and far beyond the capacity of any single individual, however specialized. Unforeseen, "outrageous" implications of work have been specifically labelled "moonshine" by mathematicians challenged to explain them. This evolution in knowledge creation and substantiation, in domains potentially fundamental to the future management of complexity, raises questions about how knowledge emerging from such a process is to be rendered credible to more than a a small group of experts -- themselves challenged in that respect. Specifically how is that unforeseen connectivity, possibly vital to issues of governance, to be comprehended?
If the proof of mathematical conjectures, potentially vital to global governance of complexity, is recognized as acquiring such characteristics, what if proof of an essential insight were in future to take years to read and understand -- possibly a lifetime -- and how would the correctness of the proof be confirmed? How could it be credibly communicated? The length and complexity of the following argument is but a trivial example of the challenge !
The current psychosocial context is characterized by a multiplicity of belief systems, disciplines and cognitive frameworks. It is conjectured here -- in the light of the fundamental nature of the symmetries of the newly discovered Monster of symmetry -- that such conceptual systems are each coherent cuts or slices through the multidimensional complexity of such symmetry, effectively functioning as a Rosetta stone. Furthermore, it is argued that their complexity can best be communicated through symbols and metaphors with mnemonic characteristics. One thread in what follows is how such different cognitive styles interweave to create knowledge and ensure its comprehension and credibility -- despite its complexity.
This exploration arose from the "outrageous" possibility that the formulation of the special theory of relativity was to some degree influenced by patent office procedures and mindsets (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). To what extent can "frames of reference" be considered as finite simple groups fundamental to higher orders of symmetry?
Whilst symmetry is intimately associated with truth, its beauty and elegance can also be fruitfully understood dynamically as a form of strange attractor for all human endeavour -- not only that of mathematicians ! High degrees of symmetry are therefore very powerful attractors and may be fruitfully associated with the drivers constituted by the highest human values (Human Values as Strange Attractors: coevolution of classes of governance principles, 1993). However, whilst Ian Stewart relates the beauty of symmetry to truth, elsewhere (Fearful Symmetry: is God a geometer? 1992) he emphasizes symmetry breaking as necessary to create structure -- with a completely symmetrical system having no structure, and arguably therefore no "truth". If the world is as complex as it appears, then lack of symmetry offers an alternative understanding of "truth" -- as with the beauty associated with the "harmony of imperfections", so valued by the Japanese.
Could the generation of value through socio-economic "development" be associated with some new understanding of the "velocity" of a frame of reference relative to other frames, with "acceleration" indicative of "development" plus "knowledge creation" (R&D)? Rather than appropriate policy as a typical form of cognitive property for which claims are made and upheld, should a degree of emphasis be shifted to the dynamics of the art of policy-making as a creative process (cf Poetry-Making and Policy-making: arranging a marriage between Beauty and the Beast, 1993)?
Inspired by the mathematical quest for the Monster of symmetry, this exploration is fundamentally about the relationship between connectivity (and coherence), conjecture, comprehension, credibility and communication -- namely to what degree a conjecture "holds" and for whom. This would appear to go to the root of current challenges of governance -- whether evidence-based or faith-based -- and in what it is then appropriate for people to express belief, as a "credo". It highlights the challenging relationship between "conjecture" and "projecture" in determining the nature of reality -- in a context in which much is conditioned by simplistic "projection". The length of this document also highlights related challenges of concision in explicating complexity.
-- Monstrous Connectivity -- |
The exciting pursuit of understanding the mathematical beauty of higher forms of order by mathematicians has been notably undertaken of more than a century through the progressive elaboration of a form of "periodic table" of elements of symmetry -- a table of finite simple groups (Ronald Solomon, A Brief History of the Classification of the Finite Simple Groups. Bulletin of the American Mathematical Society, 2001).
These groups are objects in mathematics that measure symmetry in nature, as helpfully explained by Richard Elwes (An Enormous Theorem: the classification of finite simple groups, 2006). The "classification theorem" of such groups, also known as the "enormous theorem" (requiring over 15,000 pages to "prove"), states that the finite simple groups can be classified completely into 5 groups. One of these groups is however made up of exceptions to the regularity of the others. These exceptional sub-groups are known as sporadic groups.
Attention has long been focused on these 26 exceptional sporadic groups. The largest of these, incorporating 20 of the others, has been named the Monster -- the most exceptional finite symmetry group in mathematics. It is a giant snowflake in 196,884 dimensions composed of more elements than there are supposedly to be elementary particles in the universe (approx. 8 x 10^{53}). Its size is defined by:
2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 |
The significance of the Monster is briefly well-summarized by Marcus du Sautoy (Patterns that Hold Secrets to the Universe) and in a more technical manner by Richard E. Borcherds (What is the Monster? Notices of the A.M.S., 2002). The history of its discovery is recounted by Mark Ronan (2006). It is indeed suspected that the Monster is built in some subtle way into the structure of the universe. In musical terms, the challenge of its exceptional nature might perhaps be appropriately, but very crudely, compared with the well-known diabolus in musica. With regard to such groups, Ronan quotes one mathematician (John Conway) as saying:
The trouble is that groups behave in astonishingly subtle ways that make them psychologically rather difficult to grasp.
Given the challenge of such symmetry to comprehension, is it possible that folk intuitions of "taming monsters" through music are indicative of powerful truths -- as discussed separately in relation to a possible periodic table of beliefs (Systematic Visual Representation of Musical Possibilities on an Orbifold, 2007)? Beliefs are perhaps to be understood as the most generic forms of cognitive property. The patterns through which they are articulated might lend themselves to a form of classification -- as with the set of finite groups of mathematics (John Conway, et al. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, 1985; Robert Wilson, et al. Atlas of Finite Group Representations, version 3).
Curiously, given the fundamental role of "light" evoked in the earlier exploration (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007), the report of the discovery of a completely unexpected intimate connection between the mathematical "monster" and modular functions by John Conway and Simon Norton (Monstrous Moonshine, Bull. London Math. Soc., 1979) introduced a light metaphor that continues to be considered appropriate (Terry Gannon, Monstrous Moonshine: the first twenty-five years, 2004; Moonshine Bibliography).
Given the role of an orbifold in ordering musics (Dmitri Tymoczko, The Geometry of Musical Chords, Science, 313. 5783, 7 July 2007, pp. 72 - 74), it is intriguing that orbifolds have a recognized role with respect to the connectivity of such "moonshine" (Michael P. Tuite, Monstrous Moonshine from Orbifolds, 1992) -- indeed it has been John Conway who developed the notation by which orbifolds are characterized.
As explained by Mark Ronan:
The term Moonshine... has a variety of meanings. It can refer to foolish or naive ideas, but also to the illicit distillation of spirits... It gave an impression of dabbling in mysterious matters that might be better left alone, but also had the useful connotation of something shining in reflected light. The true source of light is probably yet to be found, but there were further strange connections to come later... The Monster's connections with number theory -- the Moonshine connections -- had suggested it was a more beautiful and important group of symmetries than first realized.... The Moonshine connections between the Monster and number theory have now been placed within a larger theory, but we have yet to grasp the significance of these deep mathematical links with fundamental physics. We have found the Monster, but it remains an enigma. Understanding its full nature is likely to shed light on the very fabric of the universe. (p. 203-229)
The truth of the "outrageous" Monstrous Moonshine Conjecture was finally proven by Richard E. Borcherds (What is Moonshine? Proceedings of the International Congress of Mathematicians, 1998) -- making surprising use of a theorem from string theory. For this he was awarded the Fields Medal. In 2002 he described, what had been considered meaningless in relation to "moonshine" in the following terms:
The term 'moonshine' roughly means weird relations between sporadic groups and modular functions (and anything else) similar to this. It was clear to many people that this was just a meaningless coincidence. [more]
One of those 'weird' connections between distant and apparently unrelated fields of mathematics, for example, is that the divisors of the order of the Monster are precisely the 15 supersingular primes, which are intimately related to the j-function of number theory. Other striking coincidences, or maybe deep connections, link the Monster group to the Lorentzian geometry of general relativity, the multidimensional space of string theory, and the enigmatic properties of the number 163 in number theory. In commenting on the relation of the latter to the Monstrous Moonshine Conjecture, Titus Piezas III (Ramanujan's Constant (e^{π√163}) and its Cousins, 2005) remarks: "Ramanujan would have loved this".
However, as noted by Ronan:
The Moonshine mystery itself is still unresolved, despite Borcherd's proof! ... there are facts about the Monster and Moonshine that we don't understand.... The method leading to its discovery, brilliant though it was, gave no clue to the Monster's remarkable properties... (pp 226-227)
Curiously, given the 26 exceptions to the periodic table of symmetry (noted above), bosonic string theories are 26-dimensional. Its fundamental implications are indicated by M. Thomas (Monster Sporadic Group encoding of the Schwarzschild metric, 2004):
The classic construction of the Monster Sporadic Group involves the 24 dimensional Leech lattice and a 2 dimensional orbifold which are related to the 26 dimensional bosonic string. It will be shown that there is an astrophysical model encoded with this largest of the finite sporadics with implications in quantum gravity and a possible physical mechanism for exploration of the 26 dimensional bosonic string.
As a further indication of its fundamental significance, Frank Dodd (Tony) Smith, Jr. (Monster = Group of Lattice Bosonic String Theory, 2007) presents an Orbifold Lattice Monster, namely a physically realistic lattice bosonic string theory (in which strings are interpreted as world-lines) containing gravity and the Standard Model -- constructed through a 12-step process. He offers the following "remarkable characterization" of the Monster -- as the largest sporadic finite simple group:
The Monster is the automorphism group of the smallest nontrival string theory that nature allows ... Bosonic 26-dimensional space-time ... "compactified" on 24 dimensions, using the orbifold construction V[flat] ... or more precisely, the automorphism group of the vertex operator algebra with the canonical "smallness" properties.
A tragic dimension to these exciting fundamental preoccupations, as specifically noted previously in relation to Einsteinian relativity (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007), is the failure to apply them to contemporary challenges (cf And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000).
No "group theorists" have ever applied their skills to the design and construction of interdisciplinary, interfaith or intersectoral groups -- or their frames of reference -- more appropriate to the dynamic challenge of the world problematique, especially in the light of Ross Ashby's Law of Requisite Variety. Indeed no "group theorists" have applied their skills self-referentially to a representation of the groups composing the relational world of mathematics (or the cognitive property they own). It is however unfortunately the case that their skills are best recognized institutionally in "defence" research, surveillance -- and especially cryptography.
The decades of insight applied by an array of mathematicians to the challenge of interrelating finite groups in a classification scheme is now being consolidated. It could be considered a powerful metaphor of an enterprise, as yet to be undertaken, to classify the belief systems currently inspiring action to tear the connective tissue of global society. How structurally different are the patterns of finite groups and those underlying belief systems -- typically defined by sociology in terms of the "groups" imbued by them, as discussed elsewhere (Tuning a Periodic Table of Religions, Epistemologies and Spirituality: including the sciences and other belief systems, 2007)?
The question here is what possibilities of psychosocial relevance are suggested by the Monster and the quest for it -- together with the Moonshine conjectures, and their elucidation?
Few would deny the increasingly apparent complexity of the world and the struggle that the strategic challenges constitute for governance. The complexity sciences have been developed in recent decades as one tool of possible relevance.
As noted, it is through the insights of group theory and number theory that the Monster was discovered. These insights are typically applied to the challenges of cryptography in the service of defence and security. The question is whether the application of these skills could be "inverted" in some way (cf From ECHELON to NOLEHCE: enabling a strategic conversion to a faith-based global brain, 2007).
One interesting possibility is the implication of the Monster in service of a form of elegant simplification -- a "decrypting" -- of this complexity, even such as to highlight a degree of inherent "beauty". The symmetry properties of the Monster would be the key to such an operation -- transforming global complexity into a "snowflake". Furthermore it is precisely such properties that are the key to any mnemonic considerations essential to comprehending such simplicity in complexity -- perhaps to be termed "simplexity" (Jeffrey Kluger, Simplexity: the simple rules of a complex world, 2007). These are also essential to any capacity to communicate the significance more widely -- just as they are the key to rendering communications incomprehensible through encryption. This may be understood as an orderly collapsing of complexity, a process of enfolding comparable to folding (or closing) petals in certain types of origami.
Potentially even more significant however is the possibility of using symmetry structures and isomorphisms within the Monster as a means of providing "translations" between the category patterns characteristic of different schools of thought -- namely between different cognitive frameworks. The Monster might then serve as a form of Rosetta stone in interrelating what amounts to different "scripts" represented on distinct "forms" -- in the light of the discussion by Michael Schiltz (Form and Medium: a mathematical reconstruction, Image [&] Narrative, 6, 2003) of how the form/medium is "the image for systemic connectivity and concatenation", as described in the work of Humberto Maturana and Francesco Varela.
It is to be expected that neither specialists in group theory, nor those in number theory, would have any interest in the psychological dimensions of comprehension. Conway's recognition that groups like the Monster are "psychologically rather difficult to grasp" is an exceptional admission that is not considered relevant to the actual challenge of the quest for the Monster, nor for understanding it. Curiously mathematicians are happy, informally, to recognize and admire an attribute termed "brilliance" but the implications of its absence are simply deplored. Comprehension capacity is not a feature of mathematics -- despite the kinds of arguments formulated elsewhere (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007).
However, as carefully reviewed in detail by Eduard Prugovecki (Historical and Epistemological Perspectives on Developments in Relativity and Quantum Theory, 1992):
The founders of relativity theory and of quantum mechanics were as concerned with the epistemological aspects and mathematical consistency of these theories, as they were with their empirical accuracy as reflected by experimental tests. In fact, some of them gave to epistemological scope and soundness preference over immediately apparent agreement with experiment, since they were acutely aware that all raw empirical data are submitted to a considerable amount of theoretical analysis and interpretation, before they are eventually released for publication. Of necessity, all such interpretations reflect the experimentalists' conscious or subconscious biases. Hence, the outcome is prone to various kinds of errors, ranging from systematic ones, due to the faulty design of apparatus or erroneous analysis of the raw data, to the subtle ones, due to misinterpretation or unwarranted extrapolation....
Unfortunately, after the Second World War this attitude towards epistemology and foundational issues in quantum physics became reversed2, as leading physicists of the post-war generation obviously decided that, contrary to the opinions of their great predecessors, it was legitimate to secure 'the adaptation of the theory to the facts by means of additional artificial assumptions'. Thus, soon after the 'triumph' of renormalization theory, Dirac (1951) felt compelled to point out in print that: 'Recent work by Lamb, Schwinger and Feynman and others has been very successful... but the resulting theory is an ugly and incomplete one.'
In the psychosocial domain there is a real challenge to formulating, representing and comprehending simple sets. This has been discussed in some detail elsewhere (Representation, Comprehension and Communication of Sets: the role of number, International Classification, 1978-1979). The "sets" in question may be formulations of principles, values, strategic elements, or the like -- whether for governance, as essential to religious doctrine, or in the organization of any enterprise. Many "models" elaborated and used by academics, and/or for purposes of management, are defined in terms of sets of concepts, perhaps presented as a list or a matrix. Governance, for example, is conducted through a set of ministries and/or departments -- possibly in the light of a set of electoral commitments or principles. Such sets may be composed of sub-sets.
The branch of mathematics known as group representation theory does not consider the cognitive challenge of comprehending a group as it may be variously represented. This is to some degree addressed more generally through category theory which deals, at a higher order of abstraction with mathematical structures and relationships between them -- notably the equivalence of categories and isomorphism of categories. The issue of how understanding of such abstractions is to be enabled is not however their concern.
However the 15,000 page proof is clearly an embarrassment as Ronan implies:
The proof of the Classification has come a long way from the time when a handful of experts believed in it, to the point where is it being written for future generations of mathematicians to understand. This is the role of the great Revision project, which will form a basis on which we can continue to strive for a better understanding of it all. (pp 214-5)
The work on this Revision project was initiated in 1982 by Daniel Gorenstein and has been continued, since his death in 1992, by Richard Lyons and Ronald Solomon and is expected to be completed in 2010.
One way of representing the sporadic groups is in terms of their constituent prime number factors -- as indicated above for the Monster group. They could then be presented as in Table 1.
Table 1: Sporadic group prime number factors
(and powers) Cells of table contain factor powers (e.g. 3^{N}) Shaded rows indicate those groups that are not subgroups of the Monster group Note that the groups together use primes in the first 20 prime numbers, excepting 53 and 61 |
|||||||||||||
Sporadic group | Prime factors | Order / Size | |||||||||||
. |
. | 2 |
3 |
5 |
7 |
11 |
13 |
17 |
19 | 23 | Other |
Total | . |
Mathieu | M_{24} | 10 | 3 | 1 | 1 | 1 | . | . | . | 1 | . | 17 | 244823040 |
M_{23} | 7 | 2 | 1 | 1 | 1 | . | . | . | 1 | . | 13 | 10200960 | |
M_{22} | 7 | 2 | 1 | 1 | 1 | . | . | . | . | . | 12 | 443520 | |
M_{12} | 6 | 3 | 1 | . | 1 | . | . | . | . | . | 11 | 95040 | |
M_{11} | 4 | 2 | 1 | . | 1 | . | . | . | . | . | 8 | 7920 | |
Janko | J_{1} | 3 | 1 | 1 | 1 | . | . | . | 1 | . | . | 7 | 175560 |
J_{2} | 7 | 3 | 2 | 1 | . | . | . | . | . | . | 13 | 604800 | |
J_{3} | 7 | 5 | 1 | . | . | . | 1 | 1 | . | . | 15 | 50232960 | |
J_{4} | 21 | 3 | 1 | 1 | 3 | . | . | . | 1 | 29, 31, 37, 43 | 34 | 86775571046077562880 | |
Higman-Sims | HS | 9 | 2 | 3 | 1 | 1 | . | . | . | . | . | 16 | 44352000 |
McLaughlin | Mc | 7 | 6 | 3 | 1 | 1 | . | . | . | . | . | 18 | 898128000 |
Held | He | 10 | 3 | 2 | 3 | . | . | 1 | . | . | . | 19 | 4030387200 |
Suzuki | Suz | 13 | 7 | 2 | 1 | 1 | 1 | . | . | . | . | 25 | 448345497600 |
Rudvalis | Ru | 14 | 3 | 3 | 1 | . | 1 | . | . | . | 29 | 23 | 145926144000 |
O'Nan | ON | 9 | 4 | 1 | 3 | 1 | . | . | 1 | . | 31 | 20 | 460815505920 |
Lyons | Ly | 8 | 7 | 6 | 1 | 1 | . | . | . | . | 31, 37, 67 | 26 | 51765179004000000 |
Conway | Co_{1} | 21 | 9 | 4 | 2 | 1 | 1 | . | . | 1 | . | 39 | 4157776806543360000 |
Co_{2} | 18 | 6 | 3 | 1 | 1 | . | . | . | 1 | . | 30 | 42305421312000 | |
Co_{3} | 10 | 7 | 3 | 1 | 1 | . | . | . | 1 | . | 23 | 495766656000 | |
Fischer | Fi_{22} | 17 | 9 | 2 | 1 | 1 | 1. | . | . | . | . | 31 | 64561751654400 |
Fi_{23} | 18 | 13 | 2 | 1 | 1 | 1 | 1 | . | 1 | . | 38 | 4089470473293004800 | |
Fi_{24} | 21 | 16 | 2 | 3 | 1 | 1 | 1 | . | 1 | 29 | 48 | 1255205709190661721292800 | |
Harada-Norton | HN | 14 | 6 | 6 | 1 | 1 | . | . | 1 | . | . | 29 | 273030912000000 |
Thompson | Th | 15 | 10 | 3 | 2 | . | 1 | . | 1 | . | 31 | 33 | 90745943887872000 |
Baby Monster | B | 41 | 13 | 6 | 2 | 1 | 1 | 1 | 1 | 1 | 31, 47 | 69 | 4154781481226426191177580544000000 |
Monster | M | 46 | 20 | 9 | 6 | 2 | 3 | 1 | 1 | 1 | 29,
31, 41, 47, 59, 71 |
95 | 808017424794512875886459904961710 757005754368000000000 |
Total | . | 363 | 165 | 70 | 37 | 23 | 11 | 7 | 7 | 10 | 19 | 712 | . |
The "sets" characteristic of different psychosocial systems were analyzed as an exercise for a meeting of the Forms of Presentation group (Geneva, 1980) of the Goals, Processes and Indicators of Development (GPID) project of the United Nations University (Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980). The table summarizing the results is presented below as Table 2 (with links to the "annexes" providing the source information). These results were later published with related papers (Patterns of Conceptual Integration, 1984). The presentation of Table 2 raises useful questions concerning the challenge to comprehension by comparison with Table 1 above.
Table 2: Concept set prime number factors
(and powers)tentative Cells of table contain range of factor powers: N - M (e.g. from 3^{N} to 3^{M} ) Parentheses Indicate less significant factors/powers in the concept scheme |
||||||||||
Prime |
2 |
3 |
5 |
7 |
11 |
13 |
17 |
Other factors |
Max. set |
Annex name |
Annexe # | . | . | . | . | . | . | . | . | . | . |
0-3 |
0-1 |
0-1 |
. |
. |
. |
. |
. |
30 |
||
0-2 |
0-1 |
0 |
0-1 |
. |
. |
. |
. |
12 |
||
0-(6) |
0-(3) |
0-(1) |
0-(2) |
. |
. |
. |
. |
10^{3} |
||
0-(4) |
0-1 |
0-1 |
0-1 |
0-1 |
. |
. |
. |
16 |
||
0-6 |
0-3 |
0-2 |
0-2 |
0-1 |
0-1 |
0-(1) |
51, 111 |
111 |
||
0-5 |
0-1 |
0-1 |
. |
. |
. |
. |
. |
64 |
||
0-4 |
0-2 |
0-2 |
0-1 |
. |
. |
. |
23 |
10^{2} |
||
0-9 |
0-6 |
0-4 |
0-1 |
0-1 |
0-1 |
. |
29 |
10^{3} |
||
0-3 |
0-3 |
0-1 |
0-1 |
. |
. |
. |
. |
360 |
||
0-2 |
0-3 |
0-1 |
0-1 |
. |
. |
. |
. |
9 |
||
0-2 |
0-1 |
0-1 |
0-1 |
. |
. |
. |
. |
12 |
||
0-3 |
0-3 |
0-1 |
0-1 |
. |
. |
. |
. |
360 |
||
. |
. |
. |
. |
. |
. |
. |
. |
10^{2} |
||
0-2 |
0-1 |
0 |
0-1 |
. |
. |
. |
. |
7 |
||
0-1 |
0-1 |
0-1 |
0-1 |
. |
. |
. |
. |
10^{2} |
||
0-3 |
0-2 |
0-1 |
0-1 |
0-1 |
. |
. |
. |
(96) |
||
0-3 |
0-2 |
. |
. |
. |
. |
. |
. |
9 |
||
0-4 |
0-2 |
0-2 |
0-1 |
0-(2) |
0-1 |
0-(1) |
31 47 |
720 |
||
0-4 |
0-1 |
0-2 |
0-1 |
0-1 |
0-2 |
. |
19 23 31 |
92 |
||
0-6 |
0-2 |
0-2 |
0-1 |
0-1 |
0-1 |
. |
19 23 29 37 41 |
64 |
||
0-3 |
0-1 |
. |
. |
. |
. |
. |
. |
. |
||
0-3 |
0-1 |
. |
. |
. |
. |
. |
. |
. |
\
-- Cognitive Challenges to Understanding -- |
There is an easy assumption that cognitive capacity is not determined or predisposed by mathematically defined relationships -- such as those which are the preoccupation of group theorists and number theorists. And yet claims made by such mathematicians with regard to the Monster recognize the degree to which it may in some way be fundamental to the structuring of the universe. It is part of the source of their excitement.
Indeed, if such relationships are so fundamental, then they should be intimately associated with the understanding that any individual has of relationships of the most fundamental kind. This understanding may not be well articulated in the mathematical sense -- typically quite the contrary. It may also be achieved through other "ways of knowing", as indicated by the theory of multiple intelligences (cf Martin Gardner, Frames of Mind: the theory of multiple intelligences, 1983; Darrell A. Posey, Cultural and Spiritual Values of Biodiversity, 1999 ). It may not be readily communicable -- or perhaps only to a degree and to those relying on the same kind of intelligence. Those with kinesthetic intelligence, for example, may well have a better understanding of how to solve what might otherwise be described as very complex equations regarding the dynamics in which they so skillfully engage.
The point could be made differently by recognizing the extent to which a living body necessarily has a deep operational understanding of biochemistry and microbiology, for example -- beyond anything that has been articulated by the relevant disciplines. Without it that body -- whether human, animal, or plant -- would not be able to live and survive. Humans have "known" about DNA long before that structure was articulated by Crick and Watson. One effort to document this kind of knowing is that of Jeremy Narby (The Cosmic Serpent: DNA and the origins of knowledge, 1999).
Such indications make the point that the Monster should not be understood as a complex set of relationships only known -- and most recently at that -- to a particular set of mathematical disciplines.
The question is how any such deep understanding might be related to an intimate form of knowledge of the Monster. How might that knowledge be embodied? How might it be expressed? Clearly any answers would be dependent on the degree or manner of understanding. It is however to be expected that traces of that understanding would be expressed in:
Such arts might be understood as articulating patterns that are in some way isomorphic with the gross or fine structure of those increasingly recognized as fundamental to the structure of the universe. Why would it be expected to be otherwise? The isomorphism, as with the structure of any antenna, should enable a particular form of cognitive resonance with the patterning of fundamental structures, such as those of the Monster.
Tyger, tyger, burning bright In the forests of the night, What immortal hand or eye Dare frame thy fearful symmetry? (William Blake, 1757-1827. The Tyger) |
Especially interesting is the knowing of mathematicians prior to, or external to, what can be successfully articulated through conventional formalizations. Mathematicians, as any account of their activity indicates, are driven by strange forms of excitement and insight that typically they can only share with other mathematicians (cf Jacques Hadamard. The Psychology of Invention in the Mathematical Field, 1945; Philip J Davis and Reuben Hersh, The Mathematical Experience, 1981). In fact even that may be problematic in that such excitement may be seen as detracting from doing "serious" mathematics. Efforts to communicate it may be framed disparagingly as "pop-maths", as with a review of the exercise by Mark Ronan.
Consider for example the prime numbers -- fundamental to discovery of the Monster -- as described in the concluding paragraph of a work by Marcus du Sautoy (Music of the Primes: why an unsolved problem in mathematics matters, 2003), Professor of Mathematics at the University of Oxford:
In Andrew Wiles's words, the proof of the Riemann Hypothesis will allow us to navigate this world in the same way that the solution to the problem of longitude helped eighteenth-century explorers to navigate the physical world. Until then, we shall listen enthralled by this unpredictable mathematical music, unable to master its twists and turns. The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle. We still await the person whose name will live for ever as the mathematician who made the primes sing.
One possible response on the identity of that "person" is in terms of a more profound truth that to be alive is itself to be able to make the "primes sing". In this sense every living being is a "mathematician". Whether as a consequence their names will "live for ever" (with a Fields medal?) seems poorly to frame the real challenge. Living entities are "mathematicians at work" -- whatever their different capacities, skills or cognitive styles. "Doing mathematics" might therefore be understood as one method of acting on the Delphic injunction: "Know Thyself".
The issue here is where does such intimate, intuitive sense of intimations of deeper knowing come from? The question was well-recognized in the mathematical classic by Philip J Davis and Reuben Hersh (The Mathematical Experience. Boston, 1981; The Companion Guide To "The Mathematical Experience" Study, 1995). With their support, the question has however been carefully addressed in terms of the insights of cognitive science by George Lakoff and Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000) who endeavour to ground mathematics in the realities of human biology and experience.
They argue that the ability to construct conceptual metaphors is neurologically based, and enables humans to reason about one domain using the language and concepts of another domain. Conceptual metaphor is thus both what enabled mathematics to grow out of everyday activities, and what enables mathematics to grow by a continual process of analogy and abstraction. Their approach has evoked criticism from many mathematicians (with a preference for classical music, metaphorically speaking), notably anxious to question the mathematical competence of the authors. However one critic emphasizes the argument developed here:
Perhaps the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). (Tom Siegfried, 2001).
Most challenging is the question of what renders possible the profound, untutored understanding of the natural geniuses such as Évariste Galois, Srinivasa Ramanujan and Niels Henrik Abel -- of which there are many examples in mathematics. What is the nature of their "brilliance" as recognized by others? But how is it that they themselves experience such "brilliance"? Research on autistic savants provides some indication -- but presumably only the tip of the iceberg. Conversely of course is the question of why such excitement is completely meaningless to so many -- perhaps attuned to other ways of knowing they find more meaningful.
Just as the branch of mathematics known as group representation theory does not consider the cognitive challenge of comprehending a group as it may be variously represented, care should be undertaken in recognizing the scope of a discipline such as mathematical psychology. This is primarily concerned with the use of mathematics to model psychological behaviour. On the other hand more might be expected of a discipline such as the "psychology of mathematics", but this is primarily embedded within preoccupations with mathematics education. As Lakoff has demonstrated, it is more specifically the cognitive science of mathematics which addresses issues of relevance here. This is the study of mathematical ideas using the techniques of cognitive science. Specifically, it is the search for the foundations of mathematics in human cognition.
On the assumption that insight into the structure of the Monster is both possible and potentially even a daily reality -- embodied (as it is thought to be) into the very structure of the universe -- how might traces of it be recognized? Is there a sense, intuitive or otherwise, through which the gross structure of the sporadic groups is recognized?
One set of clues would naturally be expected to be associated in some way with the gross structure of the Monster in relation to the 25 other sporadic groups. The set of such groups might, for example, be suggestively reclustered in various ways:
The factorized representation of the Monster, with high order exponents, suggests that in terms of cognition there is a high degree of mirroring -- mirrors within mirrors -- perhaps to be understood as an effect of symmetries. Consideration of mirror symmetry was one approach used to the detection of sporadic groups, as Ronan notes with respect to those of Fischer:
Fischer's groups are very large, but the way to understand them is to look at the complex arrangement of mirrors.... The largest group, Fi_{24}, has more than a million million million million symmetries, but the number of mirrors is less than a third of a million... (p. 163)
It is therefore relevant to note a collection of discourses from the Sufi Inayat Khan (The Palace of Mirrors, 1935-1976) which introduces comments on mirroring and reflection as follows:
The book deals with the "mirroring" faculty, a specific quality of mind, which often passes unobserved under the "rust" of the usual, emotional and logical, processes. It is only recently that this faculty has come to be recognized on a wider scale under different technical terms, such as synchronicity, gestalt, non-sequential thinking or self-remembering
Each discourse starts with reference to aspects of this understanding, where the Palace of Mirrors could be a way of understanding the factoral mirroring within the Monster:
Mirroring has long been used to reflect on the insights of self-reflexiveness through which internal and external realities mirror each other (see Mark Pendergrast, Mirror, Mirror: a history of the human love affair with reflection, 2004; Paul Demiéville, The Mirror of the Mind, In: Peter N Gregory, Sudden and Gradual; approaches to enlightenment in Chinese Thought, 1991) -- as discussed in Mirrors of My World (2002). In Buddhism, for example, the mirror is one of a group of eight auspicious symbols relating to right thought on the eightfold path, reflecting things as they really are. In a famous passage Nichiren Daishonin (On Attaining Buddhahood, 1999) states
Even a tarnished mirror will shine like a jewel if it is polished. A mind which presently is closed by illusions originating from the innate darkness of life is like a tarnished mirror, but once it is polished it will become clear, reflecting the enlightenment of immutable truth.
A quite different line of reflection is associated with uncertainty and speculation (Louis Bachelier, Theory of Speculation: the origins of modern finance, 2006). Mirror divination (catoptromancy), or mirror scrying, has long been common to many cultures. The Tibetan art of mirror divination has even given rise to an official Divination Mirror of the State Oracle of Tibet.
Ironically, and potentially of relevance to understanding how mathematicians formulate conjectures in the knowledge creation process, a study of mirror divination by Katherine Swancutt (Representational vs conjectural divination: innovating out of nothing in Mongolia, Journal of the Royal Anthropological Institute 12, 2006, 2, pp. 331-353) distinguishes between "representational" and "conjectural" forms of divination. Swancutt demonstrates how conjectural divinations initiate processes of innovation wherein repeated questioning leads to combinatory thought which imposes novel combinations on people, who perceive the need for innovation, access an innovation, and finally recursively posit that innovation's conceptual origins.
The theme of what may be hidden in the mirror of higher dimensions, as an attractor for popular interest, has been the subject of a valuable exploration by physicist Lawrence M. Krauss (Hiding in the Mirror: the mysterious allure of extra dimensions, from Plato to string theory and beyond, 2005).
Another set of clues might be provided through alternative understanding of the operational "slices" that make up the structure of the Monster and through which its structure was determined. If the Monster is typically only understood partially, whether intuitively or through particular ways of knowing, then it might be expected that these would be more attuned to particular slices. This suggests that slices might be understood as forms of engagement with reality -- the categories or modalities through which reality is articulated within a particular cognitive framework. In this sense the various examples of "concept sets" indicated in Table 2 suggest how slices might be determined in terms of factors -- given the crude comparability with Table 1.
The coherence of such "slices" may be usefully explored in relation to the Poincaré sections common to nonlinear dynamics, as in the work of Helwig Löffelmann (Visualizing Local Properties and Characteristic Structures of Dynamical Systems, 1998; Visualizing Poincaré Maps together with the Underlying Flow, 1998).
Much is made of the skill of number theorists in recognizing "interesting numbers" -- and deriving considerable pleasure from such recognition. Less publicized is the analogous pleasure in "interesting shapes", or "interesting patterns" -- presumably derived by topologists and those mathematicians with skills in spatial representation and complex geometries. To the extent that the relations of number and group theory can be transformed into such representations, there is clearly scope for seeking some form of geometrical analogue to the different sporadic groups in terms of axes of symmetry. Ronan describes the Monster as a snowflake in 196,884 dimensions. A difficulty is that any "geometry" may be described analytically rather than graphically (cf Alexander V. Ivanov and S. V. Shpectorov, Geometry of Sporadic Groups, 1999). An, noted earlier, exception is the visual representation of a hyperbolic plane.
Earlier reference was made to the discussion elsewhere (Systematic Visual Representation of Musical Possibilities on an Orbifold, 2007) of the use of an orbifold (by Dmitri Tymoczko) as a means of ordering musics, especially in the light of a recognized relation of orbifolds to the Monster (Michael P. Tuite, Monstrous Moonshine from Orbifolds, 1992) and Conway's own involvement in orbifold notation.
Of interest therefore is the possibility of generating music -- as "interesting sounds" -- from the factors describing individual sporadic groups as a method of obtaining another form of insight into them. As noted above, this would be consistent with the work of Ernest G McClain (The Myth of Invariance: the origins of the Gods, Mathematics and Music from the Rg Veda to Plato, 1976; The Pythagorean Plato: prelude to the song itself, 1978). who notably focuses on the implication of such factors. His work was used as one of the examples in Table 2 (Tonal patterns of Rg Veda poetry). As noted by McLain (The Myth of Invariance, 1976) with respect to the study by Antonio de Nicolas (Meditations through the Rg Veda: four-dimensional man, 1978) :
The four Rgvedic "languages" de Nicolas defines have their counterparts in the foundation of all theories of music. His "language of Non-Existence" (Asat) is exemplified by the pitch continuum within each musical interval as well as by the whole undifferentiated gamut -- chaos - - from low to high. His "language of Existence" (Sat) is exemplified by every tone, by every distinction of pitch, thus ultimately by every number which defines an interval, a scale, a tuning system, or the associated metric schemes of the poets, which are quite elaborate in the Rg Veda.
The "language of Images and Sacrifice" (Yajna) is exemplified by the multitude of alternate tone-sets and the conflict of alternate values which always results in some accuracy being "sacrificed" to keep the system within manageable limits. The "language of Embodied Vision" is required to protect the validity of alternate tuning systems and alternate metric schemes by refusing to grant dominion to any one of them". (21, p. 3). The embodiment of Rg Vedic man was understood... as an effort at integrating the languages of Asat, Sat and Yajna to reach the dhih, the effective viewpoint, which would make these worlds continue in their efficient embodiment (17, p. 136).
In the light of the work by Dmitri Tymoczko (The Geometry of Musical Chords, Science, 7 July 2007), might sporadic groups correspond to quite different styles of music or tuning system? In the light of the role of the "organ" as a musical instrument that metaphorically inspired conventional approaches to"organization", is there a possibility that the orbifold approach might lead to an "organ-ization of knowledge" sensitive to musical harmony? Such was indeed the implication of the magnum opus of Nobel Laureate Herman Hesse (The Glass Bead Game, 1943).
Given the recognized potential of sonification (discussed elsewhere) in enabling the human mind to recognize patterns that are otherwise challenging, it is therefore interesting to consider how factorized sporadic groups could be represented through parameters of sound and music beyond the indications of Tymoczko and McLain. McLain explores a number of possibilities. Can particular musical properties (tone, rhythm, beat, etc) be significantly associated with:
Is it to such patterns, implicitly associated with sporadic groups, that music enthusiasts worldwide have long been attracted?
A classic approach to such matters is through number symbolism (cf Marie-Louise von Franz, Number and Time, 1974) which despite numerous reservations (regarding numerology) remains a major factor, even in stock market trading. One effort to integrate the implications of such insights is reflected in a study associated with the work on Table 2 (Distinguishing Levels of Declarations of Principles, 1980) which endeavoured to highlight the comprehension challenges and possibilities associated with each of the numbers from 1 to 20.
Especially interesting in the technical distinctions between the sporadic groups is the concept of a "cycle" which has many different connotations in mathematics. But from the perspective stressed here, of great interest is how an individual may comprehend and identify with sets of interlocking cycles that might be expressed musically (cf Emergence of Cyclical Psycho-social Identity: sustainability as "psyclically" defined, 2007). Extreme examples of efforts to express complex integrated wholes through music are works like:
Given the central role they have played in a culture over an extended period, and the mathematical interpretations to which they have led, several classical Chinese texts explicitly concerned with a representation of the whole might also be considered as offering insights into how the Monster can be comprehended (9-fold Magic Square Pattern of Tao Te Ching Insights: experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2007; Mapping Songlines of the Noosphere: use of hypergraphs in presentation of the I Ching and the Tao te Ching, 2006; Hyperspace Clues to the Psychology of the Pattern that Connects: in the light of the 81 Tao Te Ching insights, 2003; 9-fold Higher Order Patterning of Tao Te Ching Insights, 2003).
-- Connectivity, Comprehension and Credibility -- |
This exploration cannot presume to take a position in the passionate debates about whether particular approaches to mathematics are to be preferred as correct or incorrect -- whether they reflect appropriate or inappropriate understandings of mathematical objects and operations. In this respect the challenge is exemplified by the response by mathematicians to the exploration by Lakoff and Núñez (2000). The concern here is more with why people with considerable cognitive skills sustain such debates without any self-referential concerns about the significance of such debate for their variously preferred preoccupations, methods and capacities. This is especially relevant when others, apparently without such skills, engage in "doing mathematics" which are viable in their own terms -- even though employing quite distinct cognitive styles.
As one such style, this approach here is in sympathy with a more anarchistic view of science and knowledge, as notably articulated by Paul Feyerabend (Against Method, 1975). This has been discussed elsewhere (Beyond Method: engaging opposition in psycho-social organization, 1981), and provided the context for the explorations of Patterns of Conceptual Integration (1984) summarized in Table 2 (above), and for the experimental clarification of the cognitive implications of sets of size ranging from 1 to 20 (Distinguishing Levels of Declarations of Principles, 1980). It was also associated with specifically acknowledging various understandings of the range of cognitive styles (Systems of Categories Distinguishing Cultural Biases, 1993).
Philip J Davis and Reuben Hersh (The Mathematical Experience, 1981) specifically address the issue of the cognitive styles through which mathematicians "do mathematics" -- following on the earlier inquiries by Jacques Hadamard (1945). These suggest a variety of creative and ordering processes -- some implying a high degree of incoherence -- which have remarkably little to do with how mathematics is portrayed to outsiders, especially in its final formulations. This is notably true of Einstein as discussed elsewhere (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). Lakoff and Núñez (2000) stress the role of metaphor in providing any necessary coherence during these processes.
With respect to the Monster, these issues touch upon the credibility of the "weird" connectivity labelled "Moonshine" -- and considered pejoratively to be so by many at the time. The fundamental issue may then be framed in terms of the nature of "connectivity" in relation to "credibility". As Gregory Bateson (Mind and Nature: a necessary unity, 1979), the challenge is giving credibility to the "pattern that connects" (cf Frederick Steier and Jane Jorgenson, Patterns That Connect Patterns That Connect, Cybernetics and Human Knowing, 12, 2005, 1-2, pp. 5-10).
In a quite different context, how is it that no popular newspaper can risk excluding astrological information to which clearly many attach a form of credibility -- extending its relevance to stock market trading and political decisions at the highest level. Whilst this may be readily dismissed as "superstition", the degree of involvement in astrology of many early mathematicians should not be forgotten. The same may be said of Taoist disciplines associated with feng shui. A more interesting approach is why different degrees and styles of credibility "work", and are preferred, under different conditions -- despite being repeatedly disparaged by those with other preferences.
Various authors have (variously) noted the manner in which the genetic coding for the amino acids in the set (described above) are neatly reflected in the total pattern of hexagrams of the I Ching -- a seemingly quite unrelated approach to encoding patterns of change (cf Katya Walters. Tao of Chaos -- DNA and the I Ching: unlocking the code of the universe, 1996; Johnson Yan, DNA and the I Ching: the Tao of life, 1993: Martin Schönberger, The I Ching and the Genetic Code: the hidden key to life, 1992). The correspondence is a natural consequence of the coding conventions adopted.
With respect to the enormous theorem of 15,000 pages (noted above), how credible is its proof compared with that of the Moonshine Conjecture? Especially when the only person who really understood their interconnectivities has died? How adequate is an explanation that takes x-years of education to understand or prove its truth? Put bluntly, who is kidding whom?
The challenge of credible explanations maybe related to two mapping paradoxes as identified by P. Hughes and G. Brecht (Vicious Circles and Infinity; an anthology of paradoxes, 1978):
As a basis for further comment on the relation between theories, models and conjectures -- of different degrees of credible connectivity -- it is appropriate to compare several examples of concept sets. These might be subject to the strictures and insights of category theory, notably with respect to degrees of equivalence and isomorphism -- an issue that was an early preoccupation of "general systems research". As with the earlier study (Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980), no effort has been made here to adjust the content of the examples to highlight the nature of any equivalence, correspondence or isomorphism. The question highlighted by the following table is the quality and credibility of any "moonshine connectivity" -- whether within each set of quadrants or between the various 4-fold sets. Comments are offered below the table.
Table 3: Comparison of credible connectivity in select 4-fold frameworks | |||
AQAL framework (Ken Wilber) | Fundamental interaction (Physics) | ||
Upper-Left Quadrant:
"I" Interior-Individual Intentional (e.g. Freud) |
Upper-Right Quadrant:
"It" Exterior-Individual Behavioral (e.g. Skinner) |
Strong
interaction Range 10^{-15 }(localized) Relative strength 10^{38} |
Gravitation Range infinite Relative strength 1 |
Lower-Left Quadrant:
"We" Interior-Collective Cultural (e.g. Gadamer) |
Lower-Right Quadrant:
"Its" Exterior-Collective Social (e.g. Marx) |
Weak
interaction Range 10^{-18} (localized) Relative strength 10^{25} |
Electromagnetic
interaction Range infinite Relative strength 10^{36} |
Archetypal classical elements (symbols) | States of matter | ||
Fire | Earth | Plasma | Solid |
Water | Air | Liquid | Gas |
Metaphors in "doing mathematics" (Lakoff/Núñez) | Epistemological mindscapes (Maruyama) | ||
Object collection | Object construction | I-mindscape (heterogenistic, individualistic, random) | H-mindscape (homogenistic, hierarchical, classificational) |
Using a measuring stick | Moving along a path | S-mindscape (heterogenistic, interactive, homeostatic) | G-mindscape (heterogenistic, interactive, morphogenetic) |
Connectivity-Credibility | Rg Vedic languages | ||
inner-individual inspiration, dreams |
outer-individual logic, directives, conviction |
Language of Non-Existence
(Asat) |
Language of Existence (Sat) |
inner-collective cultural mindsets archetypes, myths |
outer-collective patterns, spin, aesthetics |
Language of Images
and Sacrifice (Yajna) |
Language of Embodied Vision (Rta) |
Some comments:
Ways of understanding connectivity are then well illustrated, at least metaphorically, by phase transitions between states of matter. Their complementarity is notably well illustrated by that of the classical elements. The epistemological mindscapes, and the biocultural paradigm, point to the need for different "languages" to navigate the complexity of psychosocial circumstances. This points to the need for skills in shifting between languages as argued elsewhere (Walking Elven Pathways: enactivating the pattern that connects, 2006; En-minding the Extended Body: enactive engagement in conceptual shapeshifting and deep ecology, 2003).
As noted in the annex, non-western metaphors may encourage a shift to a 4-valued logic. As stated by N. Katherine Hayles (Complex Dynamics in Literature and Science, 1991), regarding the predominance of binary logic:
If order is good, chaos is bad because it is conceptualized as the opposite of order. By contrast, in the four-valued logic characteristic of Taoist thought, not-order is also a possibility, distinct from and valued differently than anti-order. The science of chaos draws western assumptions into question by revealing possibilities that were suppressed when chaos was considered merely as order's opposite.... chaos represents not just hitherto unrecognized phenomena but an unjustly neglected set of values.
Such insights are a reason to look more closely at the I Ching (as noted earlier) as a Taoist device for interrelating complementary understandings of order and disorder. The contemporary relevance to governance is indicated by the manner in which the chaos of popular protest is seen as necessarily requiring suppression by the forces of "law and order". There is no cycle of knowledge creation through which other options may be imagined.
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 -- "algebraic" and "symbolist". 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. The same is true of the associated "theory of signatures".
The degree of connectivity between "theories of correspondences" and their surrogates is explored in an annex (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). The concern is the fractured cognitive connectivity and how it may be addressed.
An interesting overview of the varieties of symbolist correspondence is that of Emanuel Swedenborg. This is summarized by Aubrey Cole Odhner (Search for the Ancient Word, through myth and correspondence, 1971):
He outlines four kinds of correspondences: harmonica, allegorica, typica, and fabulosa. The harmonic correspondences are those which have a relationship between their functions, like light, intelligence, and wisdom; the allegorical correspondences are like the Biblical parables; typical correspondences are, as I understand them, like the archetypes of Jung -- prototypes or prophetic parables: as the story of the near sacrifice of Isaac is prophetic of the crucifixion of the Lord; fabulous correspondences would be like those involving myths and poetry.
The contrast between the "algebraic" and "symbolist" approaches to connectivity is essentially a reflection between contrasting operational preferences for "quantitative" or "qualitative" criteria of credibility and communicability of coherence. Unfortunately the sense that both are required -- that poesis is a necessary precursor to any autopoesis -- is effectively suppressed (cf Poesis as a prerequisite for autopoiesis -- in psychosocial systems? 2007).
Potentially of great value in addressing this dysfunctional separation are the collection of contributions regarding complex dynamics from both frameworks (N. Katherine Hayles, ed, Chaos and Order: complex dynamics in literature and science, 1991), and notably that of Eric Charles White (Negentroy, Noise and Emancipatory Thought). As Hayles herself asks in introducing contributions that address the question::
In what sense are the correspondences between literature and science merely metaphorical, and in what sense do they go beyond metaphor? ... beyond a metaphoric connection to assert a deeper congruence.
-- Dynamics of Knowledge Creation -- |
The following exploration is based on the assumption that a preferred degree of correspondence within one cognitive style may necessarily be perceived as inappropriate from another, but the challenge is to recognize the conditions under which each form of correspondence may be appropriate. The various flavours of "correspondences" explored in the annex might therefore be tentatively juxtaposed in a framework such as Table 4.
The question however is how best to embody in any such tabular representations a range of dimensions relating to knowledge creation in pursuit of connectivity such as best to ensure its wider credibility.
The challenge of reducing high degrees of complexity to a simple four-fold table is that any such reduction is best described as being cognitively "slippery". The nature of such slipperiness is partly to be understood by comparison with the choice of a musical tuning system. Each tuning system offers advantages and disadvantages and in opting for one the challenge of the choice may be forgotten -- even though alternatives may be favoured in other cultures. What meaningful process might then be associated with "category tuning systems"? Given the challenge in the musical case, how is "goodness of fit" to be determined in any such attunement?
In this respect, a particular challenge is the possibility of equivalents to issues of discernment that might be metaphorically described as: tone deafness (no "ear" for music), lack of visual taste, an uneducated palette, or a lack of aesthetic "feel".
One insightful approach to understanding knowledge creation is that of Marcus Berliant and Masahisa Fujita (Knowledge creation as a square dance on the Hilbert cube, 2006). Their model incorporates two key aspects of the cooperative process of knowledge creation by "myopic" agents:
(i) heterogeneity of people in their state of knowledge is essential for successful cooperation in the joint creation of new ideas, while
(ii) the very process of cooperative knowledge creation affects the heterogeneity of people through the accumulation of knowledge in common.
It is interesting to consider their method (which relies on symmetry, even mirroring) as also being potentially applicable to distinct cognitive styles or processes (as implied by Table 3), whether or not they are associated with single individuals (the focus of the model), groups, movements, cultures or "civilizations". The authors note :
The model with only two people is very limited. Either two people are meeting or they are each working in isolation. With more people, the dancers can be partitioned into many pairs of dance partners. Within each pair, the two dancers are working together, but pairs of partners are working simultaneously. This creates more possibilities in our model, as the knowledge created within a dance pair is not known to other pairs. Thus, knowledge differentiation can evolve between different pairs of dance partners. Furthermore, the option of switching partners is now available. We limit ourselves to the case where N is divisible by 4. This is a square dance on the vertices of the Hilbert cube. When the population is not divisible by 4, our most useful tool, symmetry, cannot be used to examine dynamics.
Their analysis identifies points of relatively unproductive equilibria and conditions for sustainable productivity:
We have seen that once the agents reach the bliss point (where the growth rate is highest), achieved from large initial homogeneity by cycling through all partners as rapidly as possible, they break into groups of 4.... This dance pattern allows them to remain at the highest productivity forever.
They offer explanations for why 4 is the "magic number" by placing the model in a more general context. There is a strong case for integrating their work with the management cybernetics of Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994; Gunter Nittbaur, Stafford Beer's Syntegration as a Renascence of the Ancient Greek Agora in Present-day Organizations, Journal of Universal Knowledge Management, vol 0, 1, 2005), as well as with the work of Edward Haskell (Generalization of the structure of Mendeleev's periodic table, 1972) with respect to engendering entropy and negentropy in a coaction cycle as discussed elsewhere (Psychosocial Energy from Polarization within a Cyclic Pattern of Enantiodromia, 2007). Also of relevance is the focus of Lakoff and Núñez (2000) on an innate human ability, called subitizing, namely to count, add, and subtract up to about 4 or 5 -- and how much larger numbers are then handled through metaphorical constructions.
It is assumed here that "slipperiness" in relation to grasping, ordering or representing parameters about knowledge creation is -- self-reflexively -- inherent in the process of knowledge creation. An interesting metaphor is that of the plasma "snake" instability in a tokamak for nuclear fusion (cf Enactivating a Cognitive Fusion Reactor, 2006). Some of the relevant parameters, notably in the light of Table 3, are:
Self-reflexively, the "conjecture" that there is a significant connectivity between these items, such as to constitute a higher degree of coherence, may be immediately rejected outright as "incredible", considered "weak" or potentially "strong". This exemplifies the challenge of knowledge creation with respect to any "pattern that connects" -- and the extent to which it is "moonshine" by any conventional standards.
Echoing the above dance metaphor, Ronan (2006) remarks:
The Monster...first arose as an enormous collection of operations: something to be studied, something to be constructed (did it really exist?), something to be understood. The fact that mathematicians can get a vague but but increasingly precise view of something that is 'out there' may come as a surprise to some people. We are not usually thought of as creative artists, yet in some mays what mathematicians do has a lot in common with what artists achieve....
A choreographer may know exactly what effect is needed, but fitting the steps to the music is what it is all about. In fact, let me compare an abstract group to a dance. The group can be represented in many ways, by many different dancers.... Another good analogy is music. A piece of music can be accompanied by words, movement, or dance, or can simply be appreciated on its own. It is the same with groups. (pp 50-51)
This implies that there may well be some form of "mapping" between the algebraic and symbolist theories of correspondences (or signatures) -- across the schizophrenic divide to which proponents of each subscribe. Is it possible that the algebraic theories are generic formalizations of the symbolist variants, hygienically occluding cognitive implications and their operational role in engendering those formalizations and rendering their connectivity comprehensible?
It is tragic that the subtlety of Ronan's comment has not been reflected in assessment of the claims of non-scientific cognitive styles, whether over past centuries or in the current "two culture", "two civilization", binary era. There are many repugnant, dysfunctional examples of the attitude of "mainstream" science to "fringe" science -- perhaps symbolically exemplified by a famous review by the editor of Nature questioning whether a book was "fit for burning".
Table 4: Approach to a synthesis of Table 3 (discussed below) | ||
complexity compressed into a simple explanatory formula | algebraic theory of correspondences explanation at a high level of abstraction
(reducing its comprehensibility and (intellect) (Collective-Outer) |
metaphor,
analogy explanation expressed in response to circumstance through familiar patterns (increasing its comprehensibility and credibility to others) (intuition) (Individual-Outer) |
complexity expressed associatively inviting participation | symbolist theory of correspondences explanation through an orderly array of accessible symbols, implying
an underlying unity and eliciting various styles of interaction (including
rituals and dreams); inclusive and participatory |
marketing concepts and operations emphasis on dynamic engagement with a chaotic "market" of attractors |
An interesting issue is how to respond to preferences for sets of distinctions of different size -- namely how, in an orderly manner, to collapse them (conflating categories) or expand them. This relates to the consideration, noted above, of Lakoff and Núñez (2000) regarding called subitizing, namely the ability to count, add, and subtract up to about 4 or 5 -- and how much larger numbers are then handled through metaphorical constructions. Collapsing 4 conditions leads into the dynamics of 3 or 2 or the categorical absolutism of 1. Expanding to 5, as is characteristic of many alternative representations of the symbolic "elements" offers some interesting possibilities. With respect to the 4 Rg Vedic languages in Table 4, in relation to the 5 "modules" of the biocultural paradigm, Antonio de Nicolas (personal communication) remarks:
... the "languages" are descriptive of a psychosocial situation and the "bio-cultures" are embodied frames of invariant epistemologies. Each bioculture will produce for you a multiplicity of generic tables, but not in reciprocity since generic tables do not necessarily produce invariant/embodied epistemologies... Bio-cultures are the set of embodied epistemologies that give rise to the rest of the generic tables that appear in the market place. It is like a tree with multiple (5) trunks, each one in turn producing the multiple theoretical classification tables, and so on...
In this sense each quadrant then offers 5 possibilities of engaging in the knowledge creation process -- giving a total of 20 possibilities that is suggestive in the context discussed earlier.
The challenge might be expressed as developing a meaningful relationship to "moonshine". An aspect of this was discussed elsewhere (Walking Elven Pathways: enactivating the pattern that connects, 2006). The challenge might be expressed of struggling with how well potential correspondences "hold", whether "conjectures are to be considered "weak" or "strong" in the light of different preferences. In this respect the 7 "axes of bias" that may predetermine academic discourse are especially helpful (W T Jones, The Romantic Syndrome: toward a new method in cultural anthropology and the history of ideas, 1961). One extreme is of course that of Paul Feyerabend.
This challenge is especially significant in the real and slippery world of political decision-making and strategic governance. Typical issues are associated with "not knowing", including: lack of information, limited competence, uncertainty.
Several quite distinct (or potentially complementary) approaches to this challenge might be considered:
Especially in the light of the emphasis placed by Lakoff and Núñez (2000) on the use of metaphor in knowledge creation in mathematics (object collection, object construction, measuring, and a pathway), there is a case for considering a metaphor that holds a useful spectrum of relevant insights. One example is a climbing metaphor, notably in its more extreme forms of rock climbing. Some of the interesting features of such a metaphor are:
From the perspective of such a metaphor, mathematics may be considered as a form of climbing notation -- as with various forms of mapping (eg Laban dance notation, pilot map). Such mapping also implies a notion of "navigation" (Noonautics: four modes of travelling and navigating the knowledge "universe"? 2006), notably:
Understanding metaphors as cognitive devices offering a means of travel across the vastness of the knowledge universe stresses the different nature of the logic experienced in the moment of creating or being exposed to a metaphor. The distortion of knowledge space by such a devices has similarities to the bending of the space of astrophysicists as seen to be necessary for any form of hyperdimensional travel. Metaphor combines the technology of cognitive "vehicle" and a "wormhole" in a manner that justifies the term "songline". The different logic of a metaphor enables "easy travel for all" in a manner somewhat reminiscent of the London Transport advertising invitation to "Hop on a Bus". Curiously the travel association is echoed in the more traditional theory of "correspondences". This term is of considerable significance to travel between the well-mapped pathways of the Paris Metro system. It may be related to the middle eastern travel metaphor of "magic carpets" -- in which the map of the knowledge universe is effectively woven into the carpet design. Within this metaphor the challenge of empowering the carpet is dependent on (the user) weaving the pattern appropriately to constitute a vehicle.
The Monster is acknowledged to be the largest sporadic finite simple group. The Mandelbrot set is recognized as the most complex object in mathematics -- as well as being much admired for its aesthetic appeal. As enthusiastically noted by James Gleick (Chaos, Making a New Science, 1987):
An eternity would not be enough time to see it all, its disks studded with prickly thorns, its spirals and filaments curling outward and around, bearing bulbous molecules that hang, infinitely variegated, like grapes on God's personal vine.
The question to be asked is whether there is any significant relationship between these two mathematical objects.
One advantage of using the Mandelbrot set is that it may be considered as specifically addressing the issue of slipperiness in knowledge creation as discussed above. Its focus on the non-linear dynamics of complex quadratic polynomials points to its possible relevance to understanding the complex challenge of any momentary positioning -- during a knowledge creation cycle -- within the 4-quadrant patterns typical of Table 3 and its crude synthesis in Table 4.The challenge of "slipperiness" might be understood as the challenge of complex non-linear dynamic systems.
The set was discovered by Benoit Mandelbrot through recognition of repeating patterns on all scales in numerous phenomena. As a fractal it fractal corresponds to the simplest nonlinear function -- but is also as complicated as a fractal can get. It distinguishes the simplest boundary between chaos and order and is recognized as the simplest non-trivial example of a holomorphic parameter space.
Curiously the Mandelbrot set is itself occasionally dubbed the "Mandelbrot Monster". This may be due to the fact that starting in the late 1800's and into the early 1900's, a number of strange mathematical objects, recognized as "monsters" were developed by Georg Cantor, Helge von Koch, David Hilbert, Giuseppe Peano, Carl Ludwig Sierpinski and others -- later to be called fractals by Mandelbrot. These included the Koch snowflake and similar constructions (resulting from an infinite number of iterations). They had been called "monster curves", as if they were unruly beasts who needed to be locked up before they did some real damage (Dave Snyder, Benoit Mandelbrot, Fractals and Astronomy, Reflections, November, 1998).
Calling them Peano Monster Curves, Mandelbrot himself collected a series of quotations in support of this terminology. An example of a Mandelbrot "monster" is that built by an iterative process of multiple conformal mapping of the circle exterior (cf M V Entin and G M Entin, Polarizability of 2D Monster and Light Scattering, 1999). The "monstrous" nature of these objects has even been termed "pathological" in the manner in which it seriously challenges the intuition on which so many mathematicians rely (Solomon Feferman, Mathematical Intuition vs. Mathematical Monsters, Synthese, 125, December 2000, 3, pp. 317-332). Feferman's earliest source for such terminology is Henri Poincaré (Mathematical definitions and education, 1906): " Logic sometimes breeds monsters".
But no relationship seems to have been considered between the "Monster curves", exemplified by the Mandelbrot set, and the "Monster group". Indeed the only web resources that appears to have discussed the two objects within the same context are the challengingly speculative blog pages of Dan Smith (Best Possible World: Gateway to the Millennium and Eschaton), notably: Bootstrap (2003), Mandelbrot Mystery (2003), The Heart of the Matter (2005).
Given the significance of symmetry and scale in both the Monster and the Mandelbrot set, what relationships emerged from remarks such as that of N. Katherine Hayles (Complex Dynamics in Lirterature and Science, 1991):
The repetition of symmetrical configurations across multiple levels acts like a coupling mechanism that rapidly transmits changes from one scale to another.
A question for exploration is therefore whether visual representation of the Mandelbrot set as a fractal offers a way of looking at the Monster group, or thinking about that group, and the other sporadics -- if only as an approximation. This approach would then hopefully integrate some of the possibilities explored with regard to the psychosocial implications of the Mandelbrot set (Sustainability through the Dynamics of Strategic Dilemmas -- in the light of the coherence and visual form of the Mandelbrot set, 2005; Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005; Imagination, Resolution, Emergence, Realization and Embodiment: iterative comprehension ordered via the dynamics of the Mandelbrot set, 2005).
To what extent is such possible connectivity to be considered as "moonshine" in the light of current understanding?
Some possibilities for consideration are presented in Table 5.
Table 5 | ||
. | Mandelbrot set (M-set) | Monster group (M-group) |
Algebraic geometry | associated with curves, engendered by quadratic equations -- representative of non-linear dynamics (effectively dynamic objects) | associated with groups, engendered by symmetry operations that establish its non-conformity to the periodicity of other finite simple groups -- (understood as static objects) |
Packing | as typically represented, is a wonderful (as well as beautiful), achievement in packing complexity without losing access to it (by zooming). | has highly ordered features which could possibly "piggy back" on the M-set visual rendering |
Planes | is intimately associated with representation in the complex plane | lends itself to a degree of representation in the hyperbolic plane |
Emergence | emerges through numerous "iterations" accompanied by a filtering test to determine whether any particular outcome mentions inclusion or exclusion | emerges through numerous "operations" accompanied by a filtering test to determine whether they constitute a sporadic group |
Primes | structural details ("bulbs") generated through particular combinations of primes, notably as coprimes | (sub-group) features are a consequence of combinations of primes; the divisors of the order of the M-group are precisely the 15 supersingular primes (using only these in coprimes might produce a "skeletal" version of the M-set) |
Sub-structure | a pattern of "bulbs" and other secondary features | sporadic sub-groups distinguished from 4 other clusters of finite simple groups |
Symmetry | exemplifies symmetry and the nature of orderly exceptions to it; its fractal-like geometry exhibits self-similarity at multiple scales. | is the supreme example of group symmetry |
Signature | the M-set might be analyzed algebraically as a "signature" of the M-group (homology?) | the M-set might be understood symbolically as a "signature" of the M-group |
Chaos/Order | the M-set maps the complex boundary between chaos and order | the M-group emerges through exclusion of regular finite group possibilities and by excluding infinite group possibilities |
Escape | escape speeds | ...? |
2 | situated very close to the origin of the complex plane, within a radius of 2. | as the first supersingular prime, 2 (and the mirroring it engenders) is fundamental to its structure |
Comprehensibility | as visually rendered, facilitates a degree of comprehension of its complexity and coherence (possibly to be understood as a way of mapping the boundary between incomprehensibility and unvariegated order -- or between mnemonic memorability and unmemorability) | currently exceeds any reasonable capacity to comprehend its multidimensional symmetry -- despite claims that it has the elegance of a "snowflake" |
Credibility | has achieved widespread credibility both for technical purposes (eg stockmarket speculation) and as an evocative aesthetic representation (Buddhabrot, etc) | currently of very limited interest except to some with a degree of understanding of challenges in mathematics or physics |
Connectivity | as visually rendered, uses fractal properties to hold a high degree of connectivity | a challenge to understanding the nature of its connectivity |
Attractor | elegantly combines a mapping of a range of attractors with a representation which is strikingly attractive (effectively acting as its own signature?) | is highly attractive to the mathematical intuition (possibly to be compared, in an astronomical metaphor, with the role of the Great Attractor) |
Variants | has interesting formal relationships to other fractal forms | has interesting formal relationships to other sporadic (sub-) groups |
This exploration arose from the possibility that Einstein's special theory of relativity was partially a generalization of his understanding of the disciplines of cognitive property in the patent office where he worked when he formulated the theory (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). There is a sense in which the algebraic theory of correspondences, partially instrumental in the processes through which the Monster was discovered, are themselves generic formalizations of the symbolist theories that preceded them. It is clear that the mind-bending complexity of the Monster poses a major challenge to its comprehension and the credibility of any implications associated with it -- especially if it may take years to understand, depending on one's point of departure.
The question is whether comprehension of the connectivity associated with the Monster is itself not a basis for a further generalization through which the cognitive implications are incorporated in the representation.
Consider therefore the possibility of a form of "theory of relativity" with elements such as:
c = a measure of the speed of enlightenment (the learning time required to travel a certain cognitive distance)
m = a measure of the (pattern of) connectivity or number of connections in what is learnt or known, namely the connectivity within a body of knowledge (by comparison with the atomic bonds in matter, notably in the light of current recognition of matter as information)
e = a measure of the empowerment consequently associated with the combination of "c" and "m"
This is consistent with Gregory Bateson's concerns about the destructive effects on quality of "breaking the pattern that connects" -- a "quality bomb" or "information bomb" by comparison with a "nuclear bomb". One outcome is revolution.
The relationship between these factors might, if only for mnemonic purposes, then take the form of e = mc^{2}. Especially intriguing is then the "empowerment" associated with the cognitive "embodiment" of connectivity (cf George Lakoff and Mark Johnson. Philosophy in the Flesh: the embodied mind and its challenge to western thought, 1999). Such a relationship is especially intriguing in the case of the moonshine connectivities of the Monster.
Perhaps of value for communication might be naming the speed of enlightenment as an "enlight" whether:
Also of interest, if only for mnemonic purposes, is the "squaring" of "c" as a recognition of "comprehension" x "credibility" -- namely taking account, self-referentially, of collective confirmation of the comprehension -- if empowerment is to be achieved from the connectivity comprehended. Squaring "c" might mnemonically distinguish "comprehension" from "understanding", as argued elsewhere (Fundamental learning distinction: Understanding vs Comprehending? 2007).
However any understanding of "collective" raises the question of "whose" cognitive universe it is and the status of any "other" (cf Being the Universe: a metaphoric frontier, 1999). As yet another reformulation of the famous statement by Pogo: "I have met the others and them is I"? This relates to the archetypal understanding of the curvature of that universe as implied by the Ouroboros -- and to the self-referential issues explored by Douglas Hofstadter (I Am a Strange Loop, 2007; Gödel, Escher, Bach, 1979). Classically this is the challenge of the relationship between the knower and the known.
Such considerations beg the question about the origin of light that Ronan (2006) asks in relation to the Monster:
The term Moonshine... has a variety of meanings. It can refer to foolish or naive ideas, but also to the illicit distillation of spirits... It gave an impression of dabbling in mysterious matters that might be better left alone, but also had the useful connotation of something shining in reflected light. The true source of light is probably yet to be found... (p. 203)
The question is implicit in the "brilliance" of those mathematicians who enabled the discovery of the sporadic groups -- in the "moonshine" of Ronan's "reflected light". But it is also implicit in the title of the study by George Lakoff and Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000). It is presumably in some way to be understood through the metachemistry of solar dynamics. Different spiritual traditions offer their own metaphors in relation to the religious ecstasy experienced by mystics (Parabola, 23, 1998, 2). It is especially well-modelled as a form of plasma-like "cognitive fusion", transcending space and time, as exemplified in Jewish mysticism by the exceptional experience of hitlahavut (Martin Buber, The Goblet of Grace: Hitlahavut as the key to life. Parabola, 23, 1998, 2). This is variously translated as the burning ardour of ecstasy, spiritual enthusiasm or passion -- namely an inner spark or flame through which the meaning of life is unlocked, embracing God beyond time and space.
Ronan (2006) commences his book with a quotation from Freeman Dyson (Unfashionable Pursuits, Mathematical Intelligencer, 5, 1983, pp. 47-54), anticipating the discovery of the Monster:
I have a sneaking hope, a hope unsupported by any facts or any evidence, that sometime in the twenty-first century physicists will stumble upon the Monster group, built in some way into the structure of the universe.
But the challenge of comprehending the connectivity of any "Theory of Everything" incorporating cognitive dimensions (as foreseen in George Lakoff and Mark Johnson, Philosophy in the Flesh: the embodied mind and its challenge to western thought, 1999) may be better exemplified by a much earlier comment of Dyson on the classic exchange between 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 it is not crazy enough.
To that had added Freeman Dyson (Innovation in Physics, Scientific American, 199, 3, September 1958):
Such perceived degrees of craziness ("c"?) may be one crude measure of distance from accredited, conventional cognitive frameworks.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!
Michael Aschbacher. Sporadic Groups. Cambridge University Press, 1994
Louis Bachelier. Theory of Speculation: the origins of modern finance. Princeton University Press, 2006 (translated with commentary by Mark Davis and Alison Etheridge)
Stafford Beer. Beyond Dispute: the invention of team syntegrity. John Wiley, 1994
Marcus Berliant and Masahisa Fujita:
Betty J. Birner. Metaphor and the Reshaping of Our Cognitive Fabric. Zygon, 39, 2004, 11, pp. 39-48 [abstract]
David Bohm. Wholeness and the Implicate Order. Routledge, 2002
Richard E. Borcherds:
Brian Butterworth. What Counts: how every brain is hardwired for math. Free Press, 1999.
Luigi Borzacchini. Light as a Metaphor of Science: a pre-established disharmony. Semiotica, 2001, 136, November 200, pp. 151-171
Glynis M. Breakwell. Mental Models and Social Representations of Hazards: the significance of identity processes. Journal of Risk Research, 4, 4 October 2001, pp. 341 - 351 [abstract]
William W. Cobern. Distinguishing Science-Related Variations in the Causal Universal of College Students' Worldviews. 2006 [text]
John Horton Conway and Simon Norton. Monstrous Moonshine. Bull. London Math. Soc., 1979 [text]
John Horton Conway, R T Curtis, S P Norton, R A Parker and R A Wilson. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press, 1986
R. T. Curtis (Ed.). The Atlas of Finite Groups: Ten Years. Cambridge University Press, 1998 [contents]
Philip J. Davis and Reuben Hersh:
Paul Demiéville. The Mirror of the Mind. In: Peter N Gregory (Ed.), Sudden and Gradual; approaches to enlightenment in Chinese Thought. Delhi, Motilal Banarsidass, 1991
Antonio de Nicolas:
Marcus du Sautoy:
Kenny Easwaran. Explanations: Monstrous Moonshine and the Epistemological Argument for Fictionalism. AntiMeta, 31 August 2005 [text]
Richard Elwes. An Enormous Theorem: the classification of finite simple groups. Plus, 41, December 2006 [text]
Murray Faure. Relativism and Rationalism in Science and Politics. UNISA, 2007 [text]
Solomon Feferman:
E. Fischbein. Intuition in Science and Mathematics. Dordrecht, Reidel, 1987
Northrop Frye. Fearful Symmetry: a study of William Blake. Princeton University Press, 1969
Terry Gannon:
Jonathan Glover. Taming the Monster Inside Us. Vision, Summer 2001 [text]
Daniel Gorenstein, Richard Lyons and Ronald Solomon. The Classification of the Finite Simple Groups. American Mathematical Sociaty, 1994, 1996 [contents]
Robert Griess. Twelve Sporadic Groups. Berlin, Springer-Verlag, 1998
J. Hadamard. The Psychology of Invention in the Mathematical Field. Princeton University Press, 1945
N. Katherine Hayles (Ed.). Chaos and Order: complex dynamics in literature and science. University of Chicago Press, 1991
Patrick A. Heelan:
Klaus Hentschel. Philosophical Interpretations of Relativity Theory: 1910-1930. Proceedings of the Biennial Meeting of the Philosophy of Science Association, Volume Two: Symposia and Invited Papers. 1990, pp. 169-179 [abstract]
John Horton Conway and Simon P. Norton. Monstrous Moonshine. Bull. London Math. Soc. 11, 1979, pp. 308-339
P. Hughes and G. Brecht. Vicious Circles and Infinity; an anthology of paradoxes. Penguin, 1978
Alexander V. Ivanov and S. V. Shpectorov. Geometry of Sporadic Groups, Cambridge University Press, 1999 (Encyclopedia of Mathematics and its Applications) [summary]
A. Jaffe and F. Quinn. Theoretical Mathematics: toward a cultural synthesis of mathematics and theoretical physics. Bull. Amer. Math. Soc, 29, 1993, pp. 1-13
W T Jones. The Romantic Syndrome: toward a new method in cultural anthropology and the history of ideas. Martinus Nijhoff, 1961 [summary]
Anthony Judge:
Jerry Katz (Ed). One: Essential Writings on Nonduality. Sentient Publications, 2007 [summary]
Inayat Khan. The Palace of Mirrors. Sufi Publishing Company, 1935-1976
Jeffrey Kluger. Simplexity: the simple rules of a complex world. John Murray, 2007
Lawrence M. Krauss. Hiding in the Mirror: the mysterious allure of extra dimensions, from Plato to string theory and beyond. Viking Adult, 2005.
Jerome Iglowitz:
George Lakoff and Mark Johnson. Philosophy in the Flesh: the embodied mind and its challenge to western thought. Basic Books, 1999 [summary]
George Lakoff and Rafael Núñez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, 2000
Helwig Löffelmann. Visualizing Local Properties and Characteristic Structures of Dynamical Systems. Technische Universität Wien, 1998 [text]
Helwig Löffelmann, T. Kucera, and E. Gröller. Visualizing Poincaré Maps together with the Underlying Flow. In: H.-C. Hege and K. Polthier, editors, Mathematical Visualization, Algorithms, Applications, and Numerics, Springer, 1998., p 315-328.
Saunders MacLane:
Magoroh Maruyama. Mindscapes, social patterns and future development of scientific theory types. Cybernetica, 1980, 23, 1, pp. 5-25) [summary]
Ernest G McClain:
Joseph Nechvatal. Immersive Ideals / Critical Distances: a study of the affinity between artisitic ideologies based in virtual reality and previous immersive idioms. 1999 [text]
Javier Otal. The Classification of the Finite Simple Groups: an overview. Monografas de la Real Academia de Ciencias de Zaragoza, 26, 2004, pp. 89-104 [text]
Kent D. Palmer:
Mark Pendergrast. Mirror, Mirror: a history of the human love affair with reflection. Basic Books, 2004 [review]
Anthony Peressini. Confirming Mathematical Theories: an ontologically agnostic stance. Synthese, 118, 2, February 1999, pp. 257-277 [abstract]
Titus Piezas III. Ramanujan's Constant... and its Cousins, 2005 [text]
Eduard Prugovecki. Historical and Epistemological Perspectives on Developments in Relativity and Quantum Theory. In: Quantum Geometry. Dordrecht, Kluwer, 1992, chapter 12 [text]
Martin Rees. Just Six Numbers: The Deep Forces That Shape the Universe. New York: Basic Books, 2000
Mark Ronan. Symmetry and the Monster: one of the greatest quests of mathematics. Oxford University Press, 2006 [review]
Christopher S. Simons. An Elementary Approach to the Monster. Mathematical Association of America Monthly 112, April 2005 [text]
Robert D. Sloane. Outrelativizing Relativism: a liberal defense of the universality of international human rights. Vanderbilt Journal of Transnational Law, 34, 2001, 3 [text]
Ronald Solomon:
Ian Stewart:
Ian Stewart and Martin Golubitsky. Fearful Symmetry: is God a geometer? Penguin, 1992
R. L. Tieszen. Mathematical Intuition. Kluwer, 1989
Michael P. Tuite. Monstrous Moonshine from Orbifolds. Communications in Mathematical Physics, 146, 1992, 2, pp. 277-309 [abstract]
Dmitri Tymoczko. The Geometry of Musical Chord. Science, 313. 5783, 7 July 2007, pp. 72-74 [text]
Robert A Wilson. The Taming of the Monster (Plenary talk at New Zealand Mathematics Colloquium, Auckland, 2002) [text]
Robert Wilson, Peter Walsh, Jonathan Tripp, Ibrahim Suleiman, Richard Parker, Simon Norton, Simon Nickerson, Steve Linton, John Bray, and Rachel Abbott. Atlas of Finite Group Representations. Version 3 [access]
Francisco Varela and Humberto Maturana. Autopoiesis and Cognition: the realization of the living. Reidel, 1980
Francisco Varela. Laying Down a Path in Walking: essays on enactive cognition. 1997
Marie-Louise von Franz. Number and Time: reflections leading toward a unification of depth psychology and physics. Northwestern University Press, 1974
Tamito Yoshida. The Second Scientific Revolution in Capital Letters: the informatic turn. TripleC, 4, 2006, 1, pp. 100-126 [text]
A. Zea. Fearful Symmetry: the search for beauty in modern physics. Macmillan, 1968
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.. |