-- / --
The focus here is on the possible psychosocial implications of any new global understanding of the Periodic Table of Chemical Elements (originally formulated by Dmitri Mendeleev in 1869) in relation to other global frameworks. It is another approach to an argument initially developed in Periodic Pattern of Human Life: the Periodic Table as a metaphor of lifelong learning (2009). The Periodic Table is especially significant in that it is considered to be one of the most comprehensive generalizations of science.
Of particular interest is the evolving understanding of the chemical elements and the Periodic Table as recently articulated (Denis H. Rouvray et al. The Mathematics of the Periodic Table, 2005), notably in the light of the:
The concern here is not with such patterns of numerical order in their own right, nor for their biological significance, but with what they might imply for the relationships between seemingly disparate modes of cognition. This is increasingly justified by recognition of the role of metaphor in mathematical creativity -- together with challenging questions as to the degree to which mathematics (notably as applied to the Periodic Table) is itself to be considered as a metaphor.
Of particular relevance is the evolving understanding of an "element" -- due to new possibilities of distinguishing it within mathematical abstractions -- and consequently its relationship to distinction of any fundamental "category". 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 ever more generic insights. This occurs in a period in which the cognitive role of metaphor in relation to mathematical understanding is of increasing significance.
With respect to pattern, the concern here follows from earlier explorations (Representation, Comprehension and Communication of Sets: the Role of Number, 1978; Patterns of N-foldness; comparison of integrated multi-set concept schemes as forms of presentation, 1984; Examples of Integrated, Multi-set Concept Schemes, 1984). The question here is not the validity or status of specific initiatives (discussed below), but rather what such forms of order might in future imply for psychosocial organization and ways of knowing. This concern also follows from an earlier exploration (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002).
Although such frameworks are seemingly quite distinct, even the preoccupation of quite distinct disciplines, the argument here is that there is every possibility, notably in the light of the role of isomorphism in general systems theory, that some kind of relationship is to be discovered between them. Just as the elements in a Periodic Table are distinct, there is an underlying pattern relating their structure. Is it more probable -- after many millions of years -- that present patterns of cognition ("ways of knowing") would rely on forms of which there is no past trace, or rather that those patterns would be conditioned by those of the past -- of which they would then be new instances in some way?
More fundamentally the question may be how we think about what we distinguish and the ordering of it that we consider appropriate. Formally this relates to issues arising from the calculus of indications as initiated by G. Spencer-Brown (Laws of Form, 1969), otherwise known as boundary algebra.
Whilst the "elements" and their periodic organization are increasingly presented by disciplines using very sophisticated and widely incomprehensible methodologies, it is vital to recall that a unique working comprehension of them by everbody is effectively fundamental to the biological processes of their daily life. In this sense humans operate out of a profound understanding of the "elements" and their periodic order at every moment of their lives. That understanding might be said to be fundamental to sustainability of human life.
As explored here, the disconnect between such understandings of such a periodic pattern plays itself out in:
The exploration was triggered by: the seeming lack of relationship between proposals for numerical solutions describing the fundamental and atomic number sequences of the periodic table; other mathematical explorations of their periodicity; proposals for a periodic table of mathematics; the fundamental role played by the Gaussian copula with respect to the financial crisis of 2008-2009; the fundamental importance attributed to symmetry group discoveries; and the continuing quest for a Theory of Everything. The question implied by each such approach to a "comprehensive" framework is how might any such framework affect cognition, especially if there is any implication that "coherence" calls for a cognitive relationship between them -- namely some kind of Rosetta stone, with the integrative comprehension that itself implies -- and from which it originates in some way. The challenge would be all the greater if such frameworks were held to be of profound significance in alleviating uncertainty as some form of ultimate explanation or solution.
The Periodic Table is therefore understood here as the instance of a pattern and, as such, is an indication of what might be comprehended as a periodic pattern of life. But the psychosocial dynamics of how such a pattern is intuited, recognized and apprehended is as much a part of the preoccupation.
However the concern here is less with how any such formulation is held to be true in some way by universal consensus (if only amongst specialists). Rather, given their challenge to average comprehension, it is with the interrelationship between:
Briefly the issue is with how one engages with the complexity of abstract formulations -- beyond one's capacity -- especially when that complexity purportedly holds a higher degree of order and significance, whose integrity one can only partially intuit, if at all. This is a challenge in a context in which those associated with any such formulations often have their own peculiar and seemingly dysfunctional dynamics. This concern follows from earlier involvement in two complementary projects within the framework of the Encyclopedia of World Problems and Human Potential whose subject matter was organized in a periodic pattern explicitly inspired by the Periodic Table (see Functional Classification in an Integrative Matrix of Human Preoccupations, 1982). The complementary projects of relevance here, each with an integrative focus, were: Human Development Project and Integrative Knowledge Project.
Of particular interest, given the challenge of representing higher order patterns, is how to talk about them without rendering them even more meaningless to many. In this sense the relationship between the processes whereby such understanding is attempted are themselves indicative.
The challenge of the previous paragraph was the theme of earlier explorations (In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007; Conditions of Objective, Subjective and Embodied Cognition: mnemonic systems for memetic coding of complexity, 2007; Comprehension of Appropriateness, 1986). The first notably endeavoured to map the associated dynamics onto a single diagram (Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation, 2007). The assumption is that there is a need for a degree of self-consciousness and self-reflexivity in advancing and endeavouring to comprehend formulations that claim to be of a more integrative order.
A further exercise, in anticipation of the challenge of the formlations highlighted below, is to consider the following "psychosocial" processes which typically are irrelevant to the composition of such formulations from a mathematical perspective (in contrast with judgements on the quality of the mathematics) -- and deprecated in any mathematical lexicon. Terms commencing with "co-" are used to offer a provocative mnemonic set.:
In earlier explorations the following figures were used to present the interrelationships between conventional categories and processes and those which were notably characteristic of lived reality. (Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008)
|Fig. 1: Interrelating
"imaginatique" and "irresolutique"
[click for larger version in source]
| Fig. 2: Indication
of relationship between dimensions
relating to engagement with symmetry
[click for larger version in source]
Rather than use the real and imaginary axes of the complex plane as in the figures above, in the figure below they are treated as extremes on the same axis, with a second axis based on temporary/invariant. This allows any approach (of an individual or a group) to challenging formulations to be positioned within the arena -- as illustrated by the dashed smaller diagrams ("crosshairs" of identity) -- each structured to reflect its position in the arena. Learning, as collectively recognized, then involves movement into the top right quadrant. Much daily subjective experience is in the bottom left quadrant. More interesting is that the dashed constructs are not static but may move around the arena, alternating between various positions -- with the representative construct adjusting accordingly. The challenge with any definitive formulation (of the type identified below) is that it is a characteristic outcome of processes in the top right quadrant. The "co-" processes (above) are characteristic of the other quadrants, notably in imagining (intuiting) their possibility or recognizing one's ignorance when faced with learning about them.
|Fig. 3: Mapping the comprehension/communication challenge|
Formulations characteristic of the top right quadrant are effectively set in "cognitive stone". In navigating towards them through a learning process, delicate associations of connectivity are imagined between phenomena. These may be understood as "pathways" but possibly best described as "elven pathways" negotiable only by the light and swift of cognitive foot, as discussed elsewhere (Walking Elven Pathways: enactivating the pattern that connects, 2006; Climbing Elven Stairways: DNA as a macroscopic metaphor of polarized psychodynamics, 2007).
The most remarkable tale regarding such unconventional connectivity is that relating to the discovery of the Monster Group of symmetry, a feature of the discussion below (Mark Ronan, Symmetry and the Monster: one of the greatest quests of mathematics, 2006). Curiously there are many mathematical papers explaining the associated theory of "monstrous moonshine", as discussed separately (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).
The issue is what degree of connectivity is necessary to move into the top right quadrant, and are other degrees of connectivity appropriate for other purposes -- justifying any reluctance to aspire to move there. Part of the response is to be found in the nature of meaningful "correspondences" -- an issue at the heart of the controversial "moonshine", as discussed separately (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). One interesting formulation of the challenge of appropriate connectivity is the famed 10 ox-herding images of Zen Buddhism, as discussed elsewhere (Progressive integration of the shadow of non-self-reflexivity, 2007). In such terms, at its simplest perhaps, should formulations (as with those discussed below) be progressively interrelated, namely:
There are indeed traces of connectivity between the formulations that are more readily comprehensible. However these traces are most evident in a language that is (necessarily) highly abstract. In the spirit of the argument above the question is the psychodynamic challenge constituted by such abstraction.-- and the accessibility that it precludes at a time when it might be said that there is a desperate need for higher orders of connectivity.
David Corfield (Mathematical Kinds, or Being Kind to Mathematics, Philosophica, 2004) makes the point clearly:
In the case of the philosophy of science, for instance, one may either be a reductionist and take a few physical properties and their laws as fundamental or natural, or else think that some chemical or biological properties are not reducible, and so argue for a disparate array of local laws. The harder step is to attempt to patch together these local studies.... Towards the end of the century, a few were beginning to realise that set theoretic reductionism ignores distinctions between specific kinds of reasoning and went in search of local particularity...but in selecting local studies, and later in fitting them together, we needed to be guided by a larger conception of mathematics. Here the huge obstacle looms: we have no overview of mathematics as a whole. [emphasis added]
With regard to this challenge, Corfield cites an insight of Vladimir Arnol'd:
One... characteristic of the Russian mathematical tradition is the tendency to regard all of mathematics as one living organism. In the West it is quite possible to be an expert in mathematics modulo 5, knowing nothing about mathematics modulo 7. One's breadth is regarded as negative in the West to the same extent as one's narrowness is regarded as unacceptable in Russia. (1997)
and (indicative of what mathematicians are conditioned to consider as relevant), Arnol'd also notes:
All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA. (1999)
Corfield indicates that towards the end of the twentieth century several mathematicians proposed overarching schemes to organize the facts they considered most significant. He briefly discuss three of these schemes (those of Vladimir Arnol'd, Michael Atiyah, and of John Baez with James Dolan), before drawing some philosophical consequences from their attempts. He argues that we are dealing here with a more open-ended sense of conceptual growth. He illustrates his theme by discussing the elaboration of algebraic structures designed to measure symmetry.
Corfield worries about the (above-mentioned) ignorance that presents a "huge obstacle" for a philosophy of mathematics sensitive to what mathematicians have discovered. However he notes:
Where a philosopher of physics will be able to offer you some kind of sketch of the whole domain of physics, for most outside of mathematics, including most philosophers of mathematics, there is little more to work with than the idea gained from the generic view that everything is expressible set theoretically.
His succinct review of the organization schemes leads him to conclude:
So we have a series of interrelated schemes covering large tracts of mathematics, but what to make of them? There is a temptation to relate these forms of mathematical classification to types of classification in other sciences: physical particles, periodic table of chemical elements, animal species.... We saw several organisational schemes in mathematics from... Are they pointing to some ultimate classification, or are there a vast range of schemes shaped by our interests. For instance, computational complexity classes may be seen as not intrinsic characteristics, but as part of a classification due to an interest in what we can compute with our present kinds of technology. [empahsis added]
Given the concern expressed above, what is unsatisfactory about his concern is that it seemingly excludes any reference to the experiential implications of mathematics and classification -- even though he refers readers to his more extensive philosophical treatment (Towards a Philosophy of Real Mathematics, 2003), and he himself has a professional involvement with psychology. Whatever the capacity of expression "set theoretically", this is not indicative of comprehensibility or experiential relevance.
Corfield's expression of concern has been powerfully reinforced by the challenge of managing the volume of mathematical knowledge as articulated in the keynote speech to a Mathematical Knowledge Management meeting in 2003 by Michiel Hazewinkel (Mathematical knowledge management is needed) who notes:
Hazewinkel's framing of the situation is that: "We don't even know how much we know that we don't know we know"?? This is reminscent of the preoccupation of the notorious "poem" of Donald Rumsfeld, as discussed separately (Unknown Undoing: challenge of incomprehensibility of systemic neglect, 2008).
The meeting was associated with the Mathematical Knowledge Management Network, funded for a year as a project under the Knowledge Technologies action line of the European Union's Fifth Framework. That event has been associated with a series of conferences (Austria, 2001; Italy, 2003; Poland, 2004; Germany, 2005; UK, 2005; Austria, 2007; UK, 2008) assembling valuable contributions. However it is difficult to detect much preoccupation with any overarching "overview" (as stressed by Corfield) as opposed to improving "access".
A MKM Interest Group (Mathematical Knowledge Management) has since been created as a loose network to focus on the intersection of mathematics and computer science. Specifically it focuses on the need for efficient, new techniques - based on sophisticated formal mathematics and software technology - for deriving benefit from the enormous knowledge available in current mathematical sources and for organizing that knowledge in a new way. On the other side, due its very nature, the realm of mathematical information would seem to be the best candidate for testing innovative theoretical and technological solutions for content-based systems, interoperability, management of machine understandable information, and the Semantic Web.
The challenge is also recognized, and variously addressed, as for example:
With respect to the argument here, however, there seems to be little concerted effort to focus on the order implied by the field of mathematical knowledge as a whole, as previously considered (Is the House of Mathematics in Order? -- are there vital insights from its design, 2000). One such attempt by Dave Rusin (The Mathematical Atlas: a gateway to modern mathematics, 2000) does indeed offer a modest map providing links into the Mathematics Subject Classification (MSC), discussed below. This offers a visual index to the subfields of mathematics. The question is whether more is possible -- benefitting from the sophisticated understandings of mathematics.
Despite such recent progress, and in the light of Corfield's earlier conclusion, a potentially interesting question is why there is seemingly no mind map -- even a crude one -- showing the entailment of the different mathematical approaches, and notably to any Periodic Table (given its potentially focal status). In how many innovative ways might such a map be represented and visualized (as previously illustrated)? For example, Guillermo Restrepo and Leonardo Pachón (Mathematical Aspects of the Periodic Law. Foundations of Chemistry, 2007) indicate:
Mathematics employed to study the periodic system includes number theory, information theory, order theory, set theory and topology. Each theory used shows that it is possible to provide the Periodic Law with a mathematical structure. We also show that it is possible to study the chemical elements taking advantage of their phenomenological properties, and that it is not always necessary to reduce the concept of chemical elements to the quantum atomic concept to be able to find interpretations for the Periodic Law. Finally, a connection is noted between the lengths of the periods of the Periodic Law and the philosophical Pythagorean doctrine.
Albert Khazan (Upper Limit in Mendeleev's Periodic Table: Element No.155. Svenska Fysikarkivet, 2009) focuses on a hyperbolic law whereby the content of an element in different chemical compounds can be described by the equation of an equilateral hyperbola.
Alain Connes (Noncommutative Geometry, 1994) argues with respect to the successful classification of the elements in the periodic table:
The theoretical explanation of this classification, by Schrödinger's equation and Pauli's exclusion principle, is an equivalent success of physics in the 20th century, and, more precisely, of quantum mechanics. One can look at this theory from very diverse points of view. With Planck, it has its origins in thermodynamics and manifests itself in the discretization of the energy levels of oscillators. With Bohr, it is the discretization of angular momentum. For de Broglie and Schrödinger, it is the wave nature of matter. These diverse points of view are all corollaries of that of Heisenberg: physical quantities are governed by noncommutative algebra.
A. John Coleman (Groups and Physics: Dogmatic Opinions of a Senior Citizen, Notices of the AMS, 1997) notes that the hamiltonian of an atom is invariant with respect to the rotation group in 3-space. A knowledge of the possible dimensions of irreducible representations of SO(3), together with the implications of the Pauli Principle, leads again to an explanation of the periodic table of the elements. He stresses that: It was variations on these themes involving quite delicate mathematics which filled the early papers and books on group theory and quantum mechanics...
The comprehensive review compiled by Denis H. Rouvray and R. Bruce King (The Mathematics of the Periodic Table, 2005) includes twelve presentations highlighting often-neglected mathematical features of the Periodic Table and several closely related topics. It considers predictions of what the ultimate size of the Periodic Table will be (D. H. Rouvray, The Ultimate Size of the Periodic Table) and examines the nature of its periodicity (N. N. Khramov et al., Statistical Modeling of Chemical Periodicity and Prediction of the Properties of the Superheavy Elements; P. G. Mezey, Syncopated Periodicity of Atoms in Molecules ). The nature of such a table is next considered in dimensions other than two (H. Hosoya et al., n-Dimensional Periodic Tables of the Elements) . The natural clustering of the elements into groups is then considered by three different but complementary routes:
Following a detailed investigation of the mathematical basis for the periodicity seen in atomic and molecular spectroscopy (K. Balasubramanian, Mathematical Basis of Periodicity in Atomic and Molecular Spectroscopy), three separate presentations delve into many different aspects of the group-theoretical structure of the Periodic Table:
These features of a possible entailment map are necessarily narrowly focused on the Periodic Table and do not appear to take account of:
Both points are of course relevant to the psychosocial implications of such periodicity as considered here. It is naturally tempting to consider that a more generic Periodic Table -- or one of psychosocial relevance -- might be elegantly self-reflexive in that the entailment map might be fruitfully structured as a Periodic Table -- of ways of knowing and understanding a Periodic Table.
The question of the respect in which "mathematics" is independent of the human mind is an old one. It is a question which many mathematicians consider to be irrelevant to their interests. For others the order discovered by mathematics is simply an exemplification of some understanding of the divine order within which human cognition has emerged.
For example, Reuben Hersh (What is Mathematics, Really? 1997) notes that most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. He reveals mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. This follows from his earlier collaborative investigation (Philip J. Davis and Reuben Hersh, The Mathematical Experience, 1981).
More recently Hersh has collected disparate essays on understandings of mathematics (18 Unconventional Essays on the Nature of Mathematics, 2005) that are tackling, from various points of view, the problem of giving an accounting of the nature, purpose, and justification of real mathematical practice -- mathematics as actually done by real live mathematicians. His concern is with the the nature of the objects being studied and what determines the directions and styles in which mathematics progresses (or, perhaps, degenerates).
The focus of several more recent studies (some cited by Hersh) indicates the challenge for the philosophy of mathematics at the crossroads of two schools of thought. On one side are the old school mathematicians who see mathematics as a foundation of science. On the other side is a small but growing group of scholars made up of cognitive psychologists, linguists, and neural biologists (and some mathematicians as well) who see mathematics as a function of the brain (John D. Barrow, Pi in the Sky: Counting, Thinking, and Being, 1994; Brian Butterworth, What Counts: how every brain is hardwired for math, 1999; Keith Devlin, The Math Gene: how mathematical thinking evolved and why numbers are like gossip, 2001; George Lakoff and Rafael Núñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).
These new neurobiological / linguistic / cognitive theories promise help in understanding how mathematics is learnt and comprehended. (Ironically the provocative list of "co-" terms above highlights a sense in which mathematics is "born" from the matrix of a "space" that they contribute to defining, as the problematic history of the treatment of mathematical innovators by their peers shows only too clearly)
In particular George Lakoff and Rafael Núñez advocate a cognitive idea analysis of mathematics in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to those ideas. Idea analysis is distinct from mathematics and cannot be performed by mathematicians unless they are trained in cognitive science. They are mainly concerned with proposing and establishing an alternative view of mathematics, one grounding the field in the realities of human biology and experience.
This emphasis accords with the interest in the cognitive status of a Periodic Table as a metaphor of a Periodic Pattern of Human Life. However the emphasis here is on the manner in which such a periodic table incorporates or embodies the challenges of learning. This implies a degree of self-reflexivity. There is an implication of one in the other.
As further indication of the array of methodologies applied to a "periodic table", John Baez (Computation and the Periodic Table, 2008). points out that:
By now there is an extensive network of interlocking analogies between physics, topology, logic and computer science, which can be seen most easily by comparing the roles that symmetric monoidal closed categories play in each subject. However, symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized "periodic table" of k-tuply monoidal n-categories. This raises the question of how these analogies extend.
The work of John Baez and James Dolan introduced the periodic table of mathematics in 1995. This identifies k-tuply monoidal n-categories and is said to mirror the table of homotopy groups of the spheres. These describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces (ignoring their precise geometry). Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
Care seems to be necessary in determing whether such explorations do indeed treat the Periodic Table of Elements as a specific instance -- even an extremely trivial instance -- of far more generic considerations, or whether such excitingly integrative possibilities are themselves an instance of a total disconnection from the reality which people are called upon to comprehend and with which they are expected to deal. There does not seem to be any extant explicit bridge between the two understandings of "periodic table". In particular, given the cognitive argument above, it is unclear how such abstractions are a product of human cognition and constrained by it -- and how this might be reflected in any periodic table of ways of knowing explicated or exemplified by mathematical categories and specializations.
More provocatively, to the extent that such abstraction approaches (if only asymptotically) the highest degrees of generality, it might be asked to what degree such wholiness is consistent with the omniscience of a Theory of Everything. Has divinity abandoned humanity?
Assuming a generic understanding of a periodic table (inclusive of the particular case, even if only potentially so), associated with that framing is the development of categorification, an interest which Baez shares with Corfield. They are key figures, with physicist Urs Schreiber, in the instigation of The n-Category Café, a physics / mathematics / philosophy blog. Categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues. When done successfully, it replaces sets by categories, functions by functors, and equations by natural isomorphisms of functors satisfying additional properties. It might be described as the process of identitying isomorphisms with more abstract (and typically generic) formulations. (John Baez and James Dolan, Categorification, 1998; David Corfield, Categorification as a Heuristic Device, 2005). Seemingly the focus on higher order categories has shifted away from any periodic table of categories or of mathematics (Eugenia Cheng and Aaron Lauda, Higher-Dimensional Categories: an illustrated guide book, 2004).
In its pursuit of higher order categories, categorification needs of course to be seen against decades of research, notably by the documentation sciences, into classification theory in all its flavours (library classification, taxonomic classification, scientific classification of organisms, classification in machine learning, statistical classification, document classification, classification theorems in mathematics). This is of course intimately related to categorization as the process by which ideas and objects are recognized, differentiated and understood, through grouping them into categories. Both are a proccupation of knowledge organization, notably as framed by the International Society for Knowledge Organization and its journal Knowledge Organization: International Journal devoted to Concept Theory, Classification, Indexing and Knowledge Representation 1973-), recognizing that there exist different historical and theoretical approaches to, and theories about, organizing kowledge, which are related to different views of knowledge, cognition, language, and social organization.
Within this framework, the articulation of John Baez (The Dimensional Ladder, 2005) regarding the "Periodic Table" (of n-categories) in the light of a "Ladder of n-Categories" is of particular interest, notably with respect to the relationships to physics (see also John Baez, Categories, Quantization, and Much More, 2006). .
The appropriateness of the "ladder" metaphor has of course been challenged from some perspectives, notably by feminists. The same might be said of the "table" metaphor, as discussed separately (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006). For example, Alison Bailey (Locating Traitorous Identities: toward a view of privilege-cognizant white character, In: Deborah Orr, et al. Feminist Politics, 2007) uses language that could appropriately inform a richer understanding of a Periodic Table of Life:
Not a climb up the ladder, but a discarding of the hierarchies and rigidites implicit in the ladder metaphor. It involves recognition of the plasticity and the thickness of identities, a new understanding of the ways in which identities interlock, being freed from old limitations, and the emergence of new possibilities.... When we pay attention only to statistical ladders, we tend to substitute two-dimensional markers for the multidimensional situations whose changes need to be evaluated. We fail to see the ways in which the ladder conceals the composition of the masses struggling at the bottom. The trick is to see "rising" more multidimensionally: not as progress up defined ladders, but as the yeast that allows the dough to spring back against the hands that knead it -- the pressure that expands. It empowers through change in the structure of our identities and the possibilities inherent in the categories that locate us; change in the categories that we locate; change in our relations to one another. (p. 181) .
Categorification might then be understood as a move towards a meta-process through which disparate schemas may be organized at a higher level of abstraction -- posing the question of whether their advocates subscribe to their being subsumed in this way.
Although seemingly (and perhaps necessarily) abstruse, categorification is currently a focus for interrelating disparate concerns as indicated by the preoccupations of a workshop on Categorification and Geometrisation from Representation Theory (Glasgow, 2009). This event is premised on the recognition that:
For a long time the idea of categorification has been in the background of many ideas in algebraic Lie theory and its connections to geometry. Several hard questions in Lie theory have been solved by translation (often via geometry) into combinatorics. For example, irreducible modules are labelled by combinatorial data and multiplicity formulas can be computed via combinatorially defined polynomials. On the other hand, topological questions are sometimes transferred into combinatorics in order to produce a clean answer: combinatorially defined knot invariants via polynomials; changing of coordinate systems via mutation rules; etc. It is becoming increasingly clear that the connecting principle of many such results in both Lie theory and topology is the idea of categorification. The notion of "ecategorification" goes back to Crane and Frenkel, motivated by mathematical physics, and in particular by the hope to construct higher dimensional topological quantum field theories.
Current concerns are to clarify the notion of categorification and its appearance in three different areas of mathematics: algebraic geometry, symplectic geometry and representation theory.
The further development of a generalized periodic table of n-categories is currently an active concern in relation to homology and cohomology, notably as a generalization of the Witt group. This is an abelian group whose elements are represented by symmetric bilinear forms over the field. Two symmetric bilinear forms are equivalent if one can be obtained from the other by adding zero or more copies of a hyperbolic plane (the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector). The Witt group is considered to be the simplest case of the cohomology of the periodic table of n-categories as recently discussed (Noah Snyder, The Witt group, or the cohomology of the periodic table of n-categories 30 March 2009). Homology in mathematics is a procedure to associate a sequence of abelian groups or modules with a given mathematical object. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Both might best be understood as methods for the detection of degrees of isomorphism.
Such developments are of course a delightful exploration of abstractions -- to the extent that anyone (especially this writer) can rise to the challenge of comprehension they represent. It is to be hoped that they give rise to greater insight into how the dramatic differences in society can be related, if this remains of relevance. Unfortunately it remains unclear as to whether such developments are capable of organizing the representation of the field of mathematics itself in more meaningful and accessible ways -- the challenge to which Corfield pointed (above) -- especially if there are "competing" views on how this is to be achieved and whether the formulations of such disparate views can themselves be integrated into an overarching theory, through some understanding of "complementarity".
Essentially it would appear that there is a capacity to discover and organize categories more coherently at higher orders of abstraction. The Periodic Table, as commonly known, is then potentially to be understood as a relatively simple instance of this. The question is whether it is the most readily comprehensible pattern which may be indicative of the nature of a possible Periodic Pattern of Human Life -- or whether such a pattern only emerges at yet higher orders of abstraction, beyond average capacity to comprehend it in any useful way.
Whether the Periodic Table can fruitfully be considered as a metaphor of human life, Kenneth Boulding, as cofounder of general systems theory, offers the following insight relating to such use of metaphor in providing an integrative understanding of human life:
Our consciousness of the unity of self in the middle of a vast complexity of images or material structures is at least a suitable metaphor for the unity of group, organization, department, discipline or science. If personification is a metaphor, let us not despise metaphors -- we might be one ourselves. (Ecodynamics; a new theory of societal evolution, 1978)
If "cohomology" is the key to the way forward (or "upward"), then it is appropriate to point out that the provocative checklist of "co-" terms (above) includes terms that bear on the self-reflexive comprehensiveness of any such pattern. Most of the relevant literature is subject to "copyright" and there is a high degree of (deniable) "competition" between mathematicians seeking to affirm their identities through claims on new conceptual territory -- replicating, without addressing, challenges faced by society, as discussed elsewhere (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000; Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). It might be asked whether there is any greater irony than efforts to copyright the ultimate periodic table of categories.
In the spirit of Gregory Bateson (Angels Fear: towards an epistemology of the sacred, 1988), this theme is explored (in Annex 1) as an exercise based on assumptions detailed there in relation to:
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. The "elements" of any such table are then to be understood as generative metaphors "through", or "out of", which ways of knowing are framed.
With regard to Corfield's concern with "ignorance" of what mathematics may have discovered, his focus on the algebra of symmetry offers a useful example. Separately the challenge of comprehending its psychological implications has been discussed, necessarily naively even though that is consistent with his "obstacle" (Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008). Therein a number of questions were posed (Possible questions for future symmetry-related exploration, 2008) as follows:
Regarding the organization of symmetry-related insight:
Regarding symmetry object detection as a result of abstraction:
Regarding psychological implication in symmetry objects of the mode of knowing associated with abstraction:
Regarding categories distinguished by psycho-social systems:
It was with these questions in mind that Figs. 1 and 2 were produced of which Fig. 3 is effectively a variant (Indication of relationship between dimensions discussed relating to engagement with symmetry).
Given the potentially fundamental importance of any Rosetta stone relating different modes of knowledge, the question would appear to be how to write about such matters meaningfully. How to write about what one does not understand -- especially when the longer the explanation, the greater the alienation of the reader.
Relevant to such questions is the view of Eugenia Cheng (n-categories with duals and TQFT, 2007) in discusing the work of John Baez. She stresses her bias in emphasizing the following viewpoints:
It is in this sense that a succinct periodic table offers mnemonic clues through its patterning. However it would appear that mathematics has its own challenges in this respect. Given the seeming lack of any overarching order to mathematics -- perhaps to be understood as a work in progress -- there is the intriguing possibility that mathematics itserlf might be "organized" as a periodic table (Is the House of Mathematics in Order? -- are there vital insights from its design, 2000). .
It is interesting that much benefit has been derived from the recognition of set of ways through which social organization might be represented and understood, notably as articulated by Gareth Morgan (Images of Organization, 1986; Imaginization: New Mindsets for Seeing, Organizing, and Managing, 1997 ). Curiously he first distinguishes eight "images": Machine, Organism, Brain, Culture, Political System, Psychic Prison, Flux and Transformation, and Instrument of Domination, and then six "models". Both potentially a helpful identification of the "groups" basic to the ordering of a form of periodic table? Is such "imagining", and its emergence, a key to understanding the nature of any periodic table of human life and knowing?
This theme is briefly discussed in Annex 2. As noted above, an extensive summary has recently been produced (D. H. Rouvray, et al., The Mathematics of the Periodic Table, 2005). The examples relating to the Periodic Table in that Annex are those of Jean-Claude Perez, Jozsef Garai and R. Buckminster Fuller. Their relevance here is that:
Also considered there as "comprehensive" frameworks are the development by Perez of his periodic table equation (into an Equation of Life), the Mandelbrot fractal, the Gaussian copula, and the set of higher order symmetry groups.
The question implied by each is how might any such framework affect cognition, especially if there is any implication that "coherence" calls for a cognitive relationship between them -- namely some kind of Rosetta stone, with the integrative comprehension that this implies. Who might be concerned that there is seemingly no connection between the periodic table of categories, and of mathematics (John Baez, et al.) and the preoccupations of Rouvray et al.? Given the ambitious undertaking of R. Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975), and its early indication of an approach to the organization of the periodic table now considered viable (see Annex 2), what questions does this raise with regard to his other explorations of more cognitive significance?
Given the challenge of complexity, potentially relevant is human dependence on the neglected nature of approximations as discussed by Valentin N. Ostrovsky (Towards a Philosophy of Approximations in the "Exact" Sciences. HYLE--International Journal for Philosophy of Chemistry, 2005). As one of the few mathematicians discussing epistemological issues, he demonstrates that
... approximations are in fact in the core of some recent discussions in the philosophy of chemistry: on the shape of molecules, the Born-Oppenheimer approximation, the role of orbitals, and the physical explanation of the Periodic Table of Elements. The ontological and epistemological significance of approximations in the exact sciences is analyzed. The crucial role of approximations in generating qualitative images and comprehensible models is emphasized. A complementarity relation between numerically "exact" theories and explanatory approximate approaches is claimed.
More generally such comprehensive formulations point to the ultimate cognitive challenge of a Theory of Everything and how people would be expected to relate to it were it to emerge from ongoing research in fundamental physics (cf Paul Halper, The Great Beyond: higher dimensions, parallel universes and the extraordinary search for a Theory of "Everything, 2004). The potential challenge is partially highlighted by the recent controversy regarding the publication of a An Exceptionally Simple Theory of Everything proposing a basis for a unified field theory, named E8 Theory, which attempts to describe all known fundamental interactions in physics, and to stand as a possible theory of everything (A. G. Lisi. An Exceptionally Simple Theory of Everything, 2007). A more recent example is that of George James Ducas (Trans-Dimensional Unified Field Theory Physics Theory: a theory that advances the unification of relativity with quantum mechanics and string theory 1974-2009) which
... explains the matrix and periodic table of a multidimensional universe.... Our universe is a multidimensional universe where processes and procedures involving natural physics relate and exist simultaneously in multiple dimensions. Natural occurrences are multidimensional. Historically we identify our existence within three dimensions or vectors of space. However, the matrix of space needs to be redefined as a periodic table of "components" or "vectors" which build up space-time and relate all physics within the relationship of space "component vectors" and "component matrixes".
In the following comments it is important to recognize that:
Any insights from mathematics included here are based on statements in the compilation by Denis H. Rouvray et al. (The Mathematics of the Periodic Table, 2005). Their selection (page numbers are given below where relevant) and interpretation are in themselves a reflection of extremely partial competence in these highly specialized matters (on the part of this writer). This exercise is in effect a provocation to those with the necessary competence to "re-read" such a compilation -- possibly substituting references to "atom" by references to more complex understandings of human "identity", for example (cf "Re-reading" patterns of concepts, 1995; .Principles of Re-reading and Rapplication, 2001).
The possibility of ordering global understandings of human living and knowing in terms of a periodic pattern calls for consideration and exploration of the following, which may well already have traces of periodic features. Possibilities include:
Andrew Aberdein. Mathematics and Argumentation. Foundations of Science, 14, 1-2, March 2009 [text]
Tanya Araujo and Francisco Louca. Bargaining Clouds, or Mathematics as a Metaphoric Exploration of the Unexpected. ISEG (School of Economics and Management) Technical University of Lisbon, 2008 [text]
Vladimir I. Arnol'd:
Andréa Asperti, Grzegorz Bancerek and Andrzej Trybulec. Mathematical knowledge management: third international conference, MKM 2004. Springer, 2004
John Baez. Categories, Quantization, and Much More. 2006 [text]
John Baez and James Dolan:
John D. Barrow. Pi in the Sky: Counting, Thinking, and Being. Back Bay Books, 1992 [review]
Toby Bartels. k-tuply monoidal n-category. 2009 [text]
Gregory Bateson and Mary Catherine Bateson. Angels Fear: towards an epistemology of the sacred. University of Chicago Press, 1988
Brian Butterworth. What Counts: how every brain is hardwired for math. Free Press, 1999
William Byers. How Mathematicians Think: using ambiguity, contradiction, and paradox to create mathematics. Princeton University Press, 2007 [extracts]
Eugenia Cheng and Nick Gurski. The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories. 2007 [text].
Eugenia Cheng and Aaron Lauda. Higher-Dimensional Categories: an illustrated guide book. University of Cambridge, 2004 [text]
Eugenia Cheng. Higher-Dimensional Category Theory: the architecture of mathematics. 2000 [text]
A. John Coleman. Groups and Physics: Dogmatic Opinions of a Senior Citizen. Notices of the AMS, 44, 1997, 1 [text]
N. D. Cook. The Attenuation of the Periodic Table. Modren Physics Letters A, Vol. 5, no. 17, 1990, PP. 1321-8 [text]
James Harold Davenport and Bruno Buchberger (Eds.). Mathematical knowledge management: second international conference (MKM 2003, Bertinoro, Italy, 2003). Springer, 2003 [extracts]
Philip J. Davis and Reuben Hersh. The Mathematical Experience. Houghton Mifflin, 1981.
Keith Devlin. The Math Gene: how mathematical thinking evolved and why numbers are like gossip.Basic Books , 2001
Antonella De Robbio and Dario Maguolo. Mathematics Subject Classification and related schemes in the OAI framework. Open Archives Initiative, 2002 [PPT]
Derek Dillon. Notes for a Theory of the Meta-loom. The Chiang Mai Papers [text]
F. Furst, M. Leclére and F. Trichet. Ontological Engineering and Mathematical Knowledge Management: a formalization of projective geometry. Annals of Mathematics and Artificial Intelligence, 38, 1-3, May 2003, pp. 65-89 [abstract]
Jozsef Garai. Mathematical Formulas Describing the Sequences of the Periodic Table.International Journal of Quantum Chemistry, 108, 2008, pp. 667-670 [text]
John J. Grebe. A Periodic Table for Fundamental Particles. Annals of the New York Academy of Sciences, Vol. 76, 1958, p. 3
J. Hadamard. The Psychology of Invention in the Mathematical Field. Princeton University Press, 1945
Michiel Hazewinkel. Mathematical Knowledge Management is needed. (Keynote speech, MKM meeting, Edinburg, 2003) [text]
Laurence Hecht. The Geometric Basis for the Periodicity of the Elements. 21st Century, May-June 1988, p.18
Mark R. Leach. The Periodic Table: What is it Showing? Periodic Tables and The Philosophy of Science. Chemogenesis, 1999-2009 [text]
I. d C. Marques. Mathematical metaphors and politics of presence/absence" Environment and Planning. D: Society and Space 22(1), 2004, pp.71-81 [abstract]
Thomas Mormann. Mathematical Metaphors in Natorp's Neo-Kantian Epistemology and Philosophy of Science. In: Activity and Sign: grounding mathematics education. Springer, 2005, pp. 229-239 [abstract]
Albert Khazan. Upper Limit in Mendeleev's Periodic Table Element No.155. Svenska Fysikarkivet,. 2009 [text]
M. R. Kibler. A Group-Theoretical Approach to the Periodic Table of Chemical Elements: Old and New Developments. 2005 [text]
George Lakoff and Rafael Nunez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2000 [summary]
George Lakoff. Women, Fire and Dangerous Things: what categories reveal about the mind. University of Chicago Press, 1987
A. G. Lisi. An Exceptionally Simple Theory of Everything. 2007 [text]
Yuri Ivanovich Manin. Mathematics as Metaphor: selected essays of Yuri I. Manin. AMS Bookstore, 2007 [extracts]
Deborah Orr, et al. Feminist Politics. Rowman and Littlefield, 2007 [extracts]
Valentin N. Ostrovsky. Towards a Philosophy of Approximations in the "Exact" Sciences. HYLE--International Journal for Philosophy of Chemistry, Vol. 11, No.2 2005, pp. 101-126.[text]
Jim Pitman, Brian Conrey and Gary King. Bibliographic Knowledge Network (Proposal submitted to the NSF Cyber-enabled Discovery and Innovation Program), 2008.[text]
Guillermo Restrepo and Leonardo A. Pachon. Mathematical Aspects of the Periodic Law. Foundations of Chemistry, Volume 9, Number 2, July 2007, pp. 189-214. [abstract]
Denis H. Rouvray and R. Bruce King (Eds.):
Eric R. Scerri. The Periodic Table: its story and its significance. Oxford University Press, 2006
George Spencer-Brown. Laws of Form. Allen and Unwin, 1969 [summary]
David Tall. Understanding the Processes of Advanced Mathematical Thinking. International Congress of Mathematicians, Zurich, August, 1994.[text]
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