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Annex 1 of Periodic Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order (2009)
This document tentatively explores a possible periodic organization of the 64 main categories of the Mathematics Subject Classification (MSC) in the light of arguments presented in the main paper. It then offers insights from various authors concerning the role of metaphor in relation to "doing" mathematics and to comprehending it. These arguments are seen as a justification for considering the possibility of associating distinct sets of metaphors with each of the main subject classes of the MSC  or even of detecting what metaphors have already been used in this way.
Various insights from the recent compilation (Denis H. Rouvray et al., The Mathematics of the Periodic Table, 2005) are then briefly presented to highloight the extent to which mathematics is in process of reframing understandings of "element" and their periodicity in any ordering. This evolving understanding of an "element"  due to new possibilities of distinguishing it within mathematical abstractions  is relevant to any effort to the distinction of any fundamental "category" (as in the MSC). In contrast with past assumptions regarding the concreteness of "chemical elements", there is a shift from assertion of the nature of the reality constituted by "elements" to hypothesizing their nature in terms of new abstractions of every more generic insights. As noted, this occurs in a period in which the cognitive role of metaphor in relation to mathematical understanding is of increasing significance.
Contrasting examples of comprehensive periodic classifications from other cultures are then presented, namely the I Ching of Chinese culture and the AllEmbracing Net of Buddhist culture. The former has been the subject of extensive mathematical commentary and the latter is relevant in therms of formal logical distinctions.
Although the implications are not explored here, it is appropriate to note that the Mathematics Subject Classification clusters at its heighest level 64 mathematical disciplines (although only 63 are indicated below). The I Ching is ordered in terms of 64 hexagrams. The AllEmbracing Net distinguishes 62 explicit views, although 2 further views may be considered implicit. The argument might be made that such a degree of correspondence between otherwise disparate approach to order derives from constraints on human cognitive capacity which merits exploration towards a possible Periodic Table of Ways of Knowing.
Such seeming coincidences may indeed be interpreted in terms of some underlying order in objective reality. More intriguing is the extent to which such patterns may primarily signal a certain limit in the capacity of the mind to order any disparate set of entities. This has been the theme of previous explorations (Representation, Comprehension and Communication of Sets: the Role of Number, 1978; Patterns of Nfoldness; comparison of integrated multiset concept schemes as forms of presentation, 1984; Examples of Integrated, Multiset Concept Schemes, 1984). The first noted the potential implications of the most cited study in psychology by George A. Miller (The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychological Review, 63, 1856, pp. 8197). This would tend to affect the manner in which clusters and periods were well defined and appropriately bounded. Of course the following 8x8 patterns could then be said to result from (7+1)x(7+1).
In the spirit of Gregory Bateson (Angels Fear: towards an epistemology of the sacred, 1988), the following exercise is based on the assumptions that:
The MSC2000: Public Version (May. 2009; in process of revision) is broken down into over 5,000 two, three, and fivedigit classifications, each corresponding to a discipline of mathematics (e.g., 11 = Number theory; 11B = Sequences and sets; 11B05 = Density, gaps, topology). . Dave Rusin has provided A Gentle Introduction to the Mathematics Subject Classification Scheme (2000). The top level, two digit subjects are presented in the following table (corresponding to 64 mathematical disciplines, according to Wikipedia; they represent a rough total of approx. 490 3digit topics and 5900 5digit topics  in the absence of a counting facility on the MSC site).
Fig.
1: Redistribution of the 2digit Mathematics Subject Classification
into tabular form (NB: lst digit = column; 2nd digit = row; cells coloured yellow indicate methods cited in research on periodic tables) [for each cell: a link is provided to the msc listingl; approximate counts are given for 3digit topics and 5digit topics] 

.  0  1  2  3  4  5  6  7  8  9 
9  .  $K$ theory [msc] (11:74) 
.  Difference
and functional equations [msc] (2:32) 
Calculus of variations and optimal control; optimization [msc] (8:79) 
.  .  .  .  . 
8  General
algebraic systems [msc] (3:27) 
Category
theory; homological algebra [msc] (7:94) 
Measure
and integration [msc] (5:55) 
.  .  Global analysis, analysis on manifolds [msc] (9:145) 
Computer science [msc] (8:102) 
Optics, electromagnetic theory [msc] (2:40) 
.  . 
7  .  Nonassociative rings and algebras [msc] (4:79) 
.  Dynamical
systems and ergodic theory [msc] (13:195) 
Operator theory [msc] (12:177) 
Manifolds and cell complexes 
.  .  .  Mathematics education [msc] (5:62) 
6  Order,
lattices, ordered algebraic structures [msc] (6:66) 
Associative
rings and algebras [msc] (15:133) 
Real
functions [msc] (5:75) 
.  Functional analysis [msc] (14:217) 
.  .  Fluid mechanics [msc] (23:129) 
Geophysics [msc] (23) 
. 
5  Combinatorics [msc] (5:80) 
Linear
and multlinear algebra, matrix theory [msc] (43) 
.  Partial
differential equations [msc] (17:211) 
Integral
equations [msc] (16:40) 
Algebraic topology [msc] (8:122) 
Numerical analysis [msc] (20:148) 
.  Astronomy and astrophysics [msc] (19) 
. 
4  .  Algebraic
geometry [msc] (16:196) 
.  Ordinary
differential equations [msc] (11:148) 
Integral
transforms, operational calculus [msc] (23) 
General topology [msc] (9:107) 
.  Mechanics of deformable solids [msc] (17:156) 
.  Information and communication, circuits [msc] (4:52) 
3  Mathematical
logic and foundations [msc] (7:140) 
Commutative rings and algebras [msc] (14:108) 
.  Special
functions [msc] (5:70) 
Abstract
harmonic analysis [msc] (43) 
Differential geometry [msc] (5:93) 
.  .  Relativity and gravitational theory [msc] (6:42) 
Systems theory; control [msc] (5:81) 
2  .  Field
theory, polynomials [msc] (9:56) 
Topological
groups; Lie groups [msc] (6:76) 
Several
complex variables and analytical spaces [msc] (20:225) 
Fourier
analysis [msc] (3:59) 
Convex and discrete geometry [msc] (3:63) 
Statistics [msc] (14:112) 
.  Statistic mechanics, structure of matter [msc] (3:71) 
Biology and other natural sciences [msc] (5:45) 
1  History,
biography [msc] (1:38) 
Number theory [msc] (20:305) 
.  Potential
theory [msc] (4:40) 
Approximations
and expansions [msc] (41) 
Geometry [msc] (14:103) 
.  .  Quantum theory [msc] (7:97) 
Game theory, economics, social and behavioral
sciences [msc] (6:90) 
0  General [msc] (2:30) 
.  Group
theory and generalizations [msc] (14:174) 
Functions of a complex variable [msc] (8:89) 
Sequences,
series, summability [msc] (9:43) 
.  Probability theory and stochastic processes [msc] (10:109) 
Mechanics of particles and systems [msc] (14:108) 
Classical thermodynamics, heat transfer [msc] (2:32) 
Operations research, mathematical programming [msc] (2:65) 
With regard to the above presentation:
This simplistic exercise suggests the possibility that a useful overarching map of mathematics in the form of a periodic table might be:
Such considerations were applied in the development of the Functional Classification in an Integrative Matrix of Human Preoccupations (1982)  still used to interrelate the preoccupations of international organizations, perceived world problems and advocated global strategies. As noted above this was inspired by the Mendeleev Periodic Table, proposals for its generalization (by Edward Haskell), and the matrix proposal by the founder of the International Society for Knowledge Organization (Ingetraut Dahlberg, ICC  Information Coding Classification,1982; see Matrix organization of subject fields).
In exploring the possibility that a mathematically interesting array of the subjects of the Mathematics Subject Classification (MSC) might indeed take the form of a "selfreferential" periodic table, the cells of the above presentation have been coloured yellow where the associated disciplines are cited as being of relevance to the mathematics of the Periodic Table of Chemical Elements (as cited in Denis H. Rouvray and R. Bruce King, The Mathematics of the Periodic Table, 2005). Clearly this evokes the question as to whether the limited number of cells not so cited are not in some way associated with that unique generalization of science. The relation between the uses of such disciplines in that context could be significant to any more coherent configuration of the cells.
Of further interest, in the light of the arguments of George Lakoff and Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000), is the extent to which any reordering highlights the metaphorical commonalities important to recognition of different syles and stages of learning. Of great interest is what each row, column or cell is "for", why it is uniquely vital as a mode of knowing, and the nature of the pattern of knowing that they constitute together. Does the pattern predict the emergence of as yet unknown modes of knowing  in the empty cells?
Perhaps primarily as an indication of human capacity to make distinctions considered reasonable, the MSC identifies some 490 3level topics and 5900 5digit topics. This might be compared with the Periodic Table in which 339 nuclides occur naturally, although more than 3100 nuclides are currently known, including those that are radioactive or have only been created artificially.
The emphasis on the cognitive dimension, and the processes and modes of learning, also suggests that in effect such a periodic table (informed by mathematical insight) could be understood as a "periodic table of metaphors". This has a notable justification in that the originating insight for many mathematical innovations typically takes metaphorical form. Such a table therefore provides a vital link to the process of "doing" mathematics in contrast with focusing on the use of what has been formed by others in the past.
As indicated above (Origin of mathematics and the periodic table  in human cognition?), there is increasing interest in a cognitive understanding of mathematics. The topic has been variously explored but it is useful to distinguish some possible "flavours", since some may be far from implying others.
Mathematics as metaphor: The cognitive linguist George Lakoff has described mathematics as metaphor, and he views most theory in science, and, also in the social sciences as well, as, likewise, forms of metaphor. The job of any theoretician is then to develop new mathematics, which will also entail developing new metaphors for how people behave in their relationships, notably in economic transactions.
Freeman Dyson introduces a muchcited collection of essays on this question (Yuri Ivanovich Manin. Mathematics as Metaphor, 2007) with the comment:
Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. Manin is a bird. I happen to be a frog, but I am happy to introduce this book which shows us his bird'seye view of mathematics. [more]
Manin himself declares therein:
Considering mathematics as a metaphor, I want to stress that the interpretation of the mathematical knowledge is a highly cre ative act. In a way, mathematics is a novel about Nature and Humankind. One cannot tell precisely what mathematics teaches us, in much the same way as one cannot tell what exactly we are taught by "War and Peace"
For Kenneth C. Bausch (The Emerging Consensus in Social Systems Theory, Kluwer Academic, 2001):
Mathematics itself is a marvelously complex and simple metaphor (organized complexity) of our world and lives. It is complex in the scope of the realities it encompasses and the contradictions that it manages. It is simple in its draconioan selectivity6. It repidiates Byzantine, too ornate, and luxuriant metaphors.... Today, as we have been observing, mathematicians are developing tools to manage even the complexity of human existence. With the aid of computer technology, they are working with ideas like dissipative structuring, chaos theory, fractal geometry, complexity theory, order from noise, synergy, information theory, transdisciplimnary unified theory, autopoiesis, fuzzy logic, infinite sets, and chaotic logic. They may soon be lodeling psychic and social behavior in ways tha honor humanistic values. (p. 273)
On the question of mathematics as metaphor, William Byers (How Mathematicians Think: using ambiguity, contradiction, and paradox to create mathematics, 2007) states:
When mathematics is seen as a metaphor, it brings to the fore the central role of understanding on the part of both the learner and the expert. Metaphors are not purely "logical" entities. Speaking or reading a metaphor doesn't make that metaphor come alive for you. Grasping a metaphor requires a discontiunuous leap....At this point in the discussion it is enough to insist that these creative discontinuous insights be considered as part of the subject of mathematics. A metaphoric description of mathematics will inevitably include a discussion of "doing" mathematics. (p. 70)
Byers quotes the mathematician Anna Sfaard to the effect that:
To understand a new concept I must create an appropriate metaphor. A personification. Or a spatial metaphor. A metaphor of structure. Only then can I answer questions, solve problems. I may even be able to perform some manipulations on the concept. Only when I have the metaphor. Without the metaphor I just can't do it.
For Lorna B. Hanes (The Poetry of Infinity: exploring mathematics as metaphor in the work of Jorge Louis Borges, 2001). many of the short stories of Jorge Luis Borges have both specific mathematical objects and abstract mathematical concepts embedded in their very structure.
As a physicist, Andrew May (Metaphors in Science, Physics World, December 2000):
What is a scientific theory if not a grand metaphor for the real world it aims to describe? Theories are generally formulated in mathematical terms, and it is difficult to see how it could be argued that, for example, F = ma "is" the motion of an object in any literal sense. Scientific metaphors possess uniquely powerful descriptive and predictive potential, but they are metaphors nonetheless. If scientific theories were as real as the world they describe, they would not change with time (which they do, occasionally). I would even go so far as to suggest that an equation like F = ma is a culturally specific metaphor, in that it can only have meaning in a society that practices mathematical quantification in the way that ours does. Before I'm dismissed as a loopy radical, I should point out that I'm a professional physicist who has been using mathematical metaphors to describe the real world for the last twenty years!
Gladys Sterenberg (Investigating teachers' images of mathematics, Journal of Mathematics Teacher Education,11, 2, April, 2008 pp. 89105):
Research suggests that understanding new images of mathematics is very challenging and can contribute to teacher resistance. An explicit exploration of personal views of mathematics may be necessary for pedagogical change. One possible way for exploring these images is through mathematical metaphors. As metaphors focus on similarities, they can be used to express alreadyheld perceptions about the nature of mathematics. In addition to providing a way of talking about current views of mathematics, the analogous dimensions of metaphors can prompt new ways of thinking about these images. In this article, I consider the use of metaphors as a strategy for explicating elementary teachers' views of mathematics. I claim that the investigation of metaphors of mathematics helped create a shared communicative space and enhanced the quality of the discussion with the teachers. In particular, our exploration of the metaphor mathematics is a language encouraged a consideration of the humanistic dimensions of mathematics and contributed to a varied reimaging of mathematics.
I. d C. Marques. Mathematical metaphors and politics of presence/absence. Environment and Planning D: Society and Space, 2004)
Thomas Mormann. Mathematical Metaphors in Natorp's NeoKantian Epistemology and Philosophy of Science, 2005)
David E. Thompson, Mathematical Metaphors: Problem Reformulation and Analysis Strategies, NASA/TM2005213452, 2005
This paper addresses the critical need for the development of intelligent or assisting software tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certain solution strategies requires extensive symbolic manipulation of the original mathematical model. These manipulations can be as simple as term reordering or as complicated as discovery of various symmetry groups embodied in the equations, whereby Backlundtype transformations create new determining equations and integrability conditions or create differential Grobner bases that are then solved in place of the original nonlinear PDEs. Several examples are presented of the kinds of problem formulations and transforms that can be frequently encountered in model representation for fluids problems. The capability of intelligently automating these types of transforms, available prior to actual mathematical solution, is advocated. Physical meaning and assumptionunderstanding can then be propagated through the mathematical transformations, allowing for explicit strategy development.
Finn Heather Upham (Metaphorical Mathematics: a cognitive construction of the real line. CUMC / CCEM, 2005)
On the ocasion of the 8th International Symposium of the Austrian Association for Semiotics, Solomon Marcus (Metaphor as Dictatorship, in J. Bernard, ed., World of SignsWorld of Things, OGS, Vienna,. 1998, 87108) showed in various fields of recent development of science, notably mathematics, how the choice of starting metaphors determines to a large extent the problems which are investigated, the concepts which are introduced, and the way to approach the object matters.He stressed the idea that the dictatorship of metaphor is much more relevant than what usually is called the cognitive or creative function of metaphor.
The I Ching offers a striking example of a highly detailed formal coding system. This has continued to be an inspiration to mathematicians of East and West  in the latter case to be understood as part of the history of binary coding basic to modern computers. Aside from traditional concern with the mathematics of "magic squares" (Bent Nielsen, A Companion to Yi Jing Numerology and Cosmology: Chinese Studies of Images and Numbers from Han (202 BCE  220 CE) to Song (960  1279 CE), 2003; Andreas Schoter, The Yijing as a Symbolic Language for Abstraction), in past decades there has been continuing mathematical interest in the I Ching (Chris Lofting, Mathematics and the I Ching, 2002; Tony Smith, I Ching, Genetic Code, and Hyperdimensional Physics) as discussed previously and in relation to the associated Tao Te Ching (Hyperspace Clues to the Psychology of the Pattern that Connects  in the light of the 81 Tao Te Ching insights, 2003).
The alternative orderings below are suggestive of ways in which any pattern of knowing calls for the possibility of being reconfigured. This is, for example, now the case with interactive variants of the Periodic Table. The representation can be reodered according to various properties or to highlight various patterns of relationship.
Fig. 2: Classical
Chinese Arrangements of 64 Hexagrams in Squares (described in detail by Steve Marshall, Yijing Hexagram Sequences, 2005, with reference to archival material Yijing Dao, Archive of Yijing scans from Chinese and other sources, 2006) Placing the cursor on each hexagram below displays a version of the I Ching hexagram quality Clicking on any hexagram below opens a page describing the hexagram condition from a policy perspective (as discussed with respect to Strategic Patterns in terms of Knowing, Feeling and Action, 2008) 

Figure 2a: Strategies ordered by Fu
Xi pattern 
Figure 2b: Strategies ordered by Jing
Fang pattern (see also J.M. Berger, Eight Palaces Circular Arrangement, 2006; also traditionally named as Eight Houses) 
Figure 2c: Strategies ordered
by King
Wen pattern (the order of the hexagrams in the I Ching) 
Figure 2d: Strategies ordered
by Mawangdui pattern (see Edward L Shaughnessy. I Ching  Mawangdui texts, 1997) 
Because of its formal binary (periodic) structure, interest has also focused on the manner in which it provides a framework for genetic codes as summarized by M.Alan Kazlev and Christián Begué (The I Ching and the Genetic Code, 2005).
Of prime relevance to the argument regarding any periodic table is the manner in which metaphor has been intimately associated with the coding system, from its most fundamental level to the 64 hexagram combinations. Each of the latter is explained through a distinct metaphor, as with the various ways in which any one of them may relate to another.
Fig. 3: Representation of the set of 64 hexagrams
to highlight transformative interrelationships 7 Equivalent maps are available for: Dialogue / Vision / Conference / Policy / Network / Community / Lifestyle The version below is that for Dialogue: tentative (nontraditional) names become evident if the cursor is placed over each. Clicking on them provides access to descriptions of them (as explained below). See also animated versions 
Instructions: Putting the cursor on any hexagram in the above diagram will bring up in a window the significance of the hexagram. Any line leading away from the selected hexagram will link to a second hexagram  signifying another approach into which the first will tend to transform. The text also includes a number and the name of a metaphor. Clicking on any hexagram will bring up the text corresponding to that metaphor  which explains the significance of that particular approach. Note that in this version, hexagrams are to be read with the "top" on the outer side of the circle  the "bottom" towards the centre. Commentaries:

Of great relevance to explorations of any ordering of ways of knowing is that formulated in the Brahmajala Sutta. This is considered to be one of the Buddha's most important and profound discourses, weaving a net of sixtytwo cases capturing all the philosophical, speculative views on the self and the world (Bhikku Bodhi (Tr). The Discourse on the AllEmbracing Net of Views; the Brahmajala Sutta and its commentarial exegesis. Kandy, Buddhist Publications, 1978).
Despite the text's almost mechanical precision in classifying and distinguishing some views, all of them need to be considered connotatively rather than denotatively, especially as possible metaphors for more comprehensive levels of meaning than is apparent. This is especially the case given the antiquity of the text and the difficulties of translation from a nonwestern language.
Fig. 4: AllEmbracing Net of
Views Adaptation of an appendix in the English translation by Bhikku Bodhi (1978), .providing a checklist of the views in the order in which they are discussed in the original text and its commentaries. 

1. Speculations about
the past (Pubbantakappika)
1.1.2 Based on recollection of up to 100,000 past lives 1.1.3 Based on recollection of up to 10 aeons of world contraction and expansion 1.1.4 Based on recollection of up to 40 such aeons 1.1.5 Based on reasoning
1.2.2 Polytheism held by beings who were gods corrupted by play 1.2.3 Polytheism held by beings who were gods corrupted by mind 1.2.4 Rationalist dualism of an impermanent body and an eternal mind
1.3.2 View that the world is infinite 1.3.3 View that the world is finite in vertical direction but infinite across 1.3.4 View that the world is neither finite nor infinite
1.4.2 Held by one fearful of clinging 1.4.3 Held by one fearful of being crossexamined 1.4.4 Held by one who is dull and stupid
1.5.2 Based on reasoning 2.1 Percipient Immortality (Sannivada), with the self immutable after death, percipient and:
2.1.2 Immaterial 2.1.3 Both material and immaterial 2.1.4 Neither material nor immaterial 2.1.5 Finite 2.1.6 Infinite 2.1.7 Both finite and infinite 2.1.8 Neither finite nor infinite 2.1.9 Of uniform perception 2.1.10 Of diversified perception 2.1.11 Of limited perception 2.1.12 Of boundless perception 2.1.13 Exclusively happy 2.1.14 Exclusively miserable 2.1.15 Both happy and miserable 2.1.16 Neither happy nor miserable 
2.2 Nonpercipient Immortality (Asannivada),
with the self immutable after death, nonpercipient and:
2.2.2 Immaterial 2.2.3 Both material and immaterial 2.2.4 Neither material nor immaterial 2.2.5 Finite 2.2.6 Infinite 2.2.7 Both finite and infinite 2.2.8 Neither finite nor infinite
2.3.2 Immaterial 2.3.3 Both material and immaterial 2.3.4 Neither material nor immaterial 2.3.5 Finite 2.3.6 Infinite 2.3.7 Both finite and infinite 2.3.8 Neither finite nor infinite
2.4.2 Annihilation of the divine sensesphere self 2.4.3 Annihilation of the divine, finematerialsphere self 2.4.4 Annihilation of the self belonging to the base of infinite space 2.4.5 Annihilation of the self belonging to the base of infinite consciousness 2.4.6 Annihilation of the self belonging to the base of thingness 2.4.7 Annihilation of the self belonging to the base of either perception nor nonperception
2.5.2 Nibbana here and now in the first jhana 2.5.3 Nibbana here and now in the second jhana 2.5.4 Nibbana here and now in the third jhana 2.5.5 Nibbana here and now in the fourth jhana 
As prviously discussed with regard to its patterning principles (Patterning: Interrelating incompatible viewpoints, 1995):
Of relevance to the general argument of this document, with respect to a related Buddhist text (The Anguttara Nikaya (Gradual Collection" or "Numerical Discourses"), the translator of the above, Bhikkhu Bodhi, wrote:
In Anguttara Nikaya, persons are as a rule not reduced to mere collections of aggregates, elements and sensebases, but are treated as real centers of living experience engaged in a heartfelt quest for happiness and freedom from suffering.
The following was an early effort to position mathematical disciplines elicited in the study of periodicity in the table of chemical elements and in that of other domains as cited in the main paper (categories, mathematics itself, genetic codons, I Ching, etc).. The cells of the MSC table above could have been mapped into it. However further insight is required into the entailment of its disciplines, only partially reflected in the relationship between the 2digit subect codes (the cells of the above table) and the "see also" links in the MSC itself. Another possibility is the representation of the relevant relationships on a torus rather than as below, as discussed separately (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006).
Fig. 5: Tentative sketch of concentric periodic table [abandoned] 
This work is licenced under a creative commons licence. 