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This is an exploration of partial or inhibited comprehension of insights that are said to be of the most profound significance. Unfortunately the language in which the insights are commonly expressed is one which I find essentially alienating. The formalization considered essential to articulation of those insights is precisely what inhibits my engagement -- despite a degree of intuitive sense of the potential meaning and its value for me.
To be clear, although I have studied mathematics through three years of university and although I have since written a set of papers on the potential significance of different branches of mathematics, the tantalizing set of insights (which are a continuing attractor) continues to be elusive.
A fundamental reason for my inhibited comprehension is that I am not satisfied by explication through formalization -- however much I respect such language, notably through decades of computer programming. My comprehension of the necessary formal operations, whether incomplete or adequate for a specific purpose, does not provide a psychological sense of completeness nor does it enhance my sense of what such completeness might be -- as I intuit that it might. From this perspective a "proof" is clearly formally adequate within mathematics as commonly understood but yet fails qualitatively to constitute the satisfier that seems possible.
None of this can be construed as a criticism of mathematics or of the explanatory power of mathematicians. It has much to do with my own intellectual inadequacy and the process of my mathematical education. Having attended some 10 schools in different countries it could be argued that this undermined a degree of continuity which might have brought the desired insights into focus on an appropriate foundation -- but then I would never have engaged in all the other activities for which I believe that mathematics has some as yet unrealized relevance.
The following is therefore an exploration of symmetry group theory as elegantly presented by Marcus du Sautoy (Finding Moonshine: a mathematician's journey through symmetry, 2008). This follows an earlier exploration of a related journey by Mark Ronan (Symmetry and the Monster: one of the greatest quests of mathematics, 2006) which I described -- according to my understanding -- in two complementary papers (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).
The focus here is necessarily on the challenge of the explanation of symmetry group theory to my comprehension of its implication for my understanding. Marcus du Sautoy introduces his own exploration with a very meaningful quote from Paul Valéry:
The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect
That theme could be understood as having been explored by the cognitive linguist George Lakoff and the psychologist Rafael E. Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).
The challenge would therefore seem to be the personal (necessarily subjective) comprehension of the relationship, or resonance, between that "inner structure" and what is offered objectively as a formal explication of that "profound symmetry".
There have been centuries of struggle by mathematicians of the highest ability to understand and give meaning to that profound symmetry. Given their many partial steps on the way (with their associated partial comprehension), it is appropriate to question whether a modern "explanation" is necessarily immediately meaningful -- whatever one's capacity or degree of application. This is clearly unfortunate if such symmetry is of such profound significance and implicated to such a degree in the inner structure of our intellect, whether individually or in terms of any understanding of collective intelligence and its application to the challenges of the times..
There is also the question of the profound significance now of the symmetry that only future generations of mathematicians will come to comprehend and explicate.
The explanation described above can be usefully reframed in terms of an experience over time implying a process of communication. This might be expressed as follows.
Legend:
It is appropriate to note that a mathematical "proof" -- confirming the existence of the Monster group -- which takes the form of 10,000 pages spread across 500 journals (as also indicated by Marcus du Sautoy) raises important issues regarding the nature of the connectivity constituting such objects, and their credibility and communicability with respect to human comprehension. This theme is discussed at greater length in the earlier comment on the Monster group (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007). Related factors might be implied by the cost (say $20) for anyone seeking to access each such copyrighted journal article over the web, and the extent to which such knowledge then remains essentially confidential to a particular community, to be affirmed as a credo by those unable to confirm the proof, especially if they lack the capacity to comprehend it !
With respect to the above table, given the assumptions regarding the fundamental relationship between symmetry and the inner structure of the intellect, it might be assumed that the theory of recapitulation -- expressed as ontogeny recapitulates phylogeny -- also applies to progressive comprehension of such symmetry. In evolutionary biology the theory holds that embryonal development of an individual organism (its ontogeny) follows the same path as the evolutionary history of its species (its phylogeny). The theory has been variously refuted although it continues to be held to offer interesting insights as a first approximation. It may be equally suggestive in the case of progressive comprehension and learning.
Giorgio T. Bagni, et al. (History and Epistemology in Mathematics Education, 2003) indicate that this possibility of mathematics education "had its apotheosis" in a famous book by Benchara Branford (1921). Fulvia Furinghetti and Luis Radford (Historical Conceptual Developments and the Teaching of Mathematics: from phylogenesis and ontogenesis theory to classroom practice, 2000) review current renewed interest in this approach from the following perspective:
That is, in considering history not only as a window from where to draw a better knowledge of the nature of mathematics but as a means to transform the teaching itself. The specificity of this pedagogical use of history is that it interweaves our knowledge of past conceptual developments with the design of classroom activities, the goal of which is to enhance the students' development of mathematical thinking.... the question is how to relate the development of students' mathematical thinking to historical conceptual developments. Psychological recapitulation, which transposes the law of biological recapitulation, claims that in their intellectual development our students naturally traverse more or less the same stages as mankind once did; it has been taken as a guarantee (sometimes implicitly) to ensure the link between both domains. In its different variants, however, psychological recapitulation has been subject to deep revision recently, in part because of the emergence of new conceptions about the role of culture in the way we come to know and think.
A learner (whether child or adult) has to struggle ab initio -- at whatever rate -- through various inappropriate understandings of what can be comprehended, whatever the stimulus for learning. In this sense the pattern of learning may be understood as involving processes of experimentation with forms and stages of order somewhat analogous to the pattern in the biological case. In this respect the authors note the conclusion of Jean Piaget:
We mustn't exaggerate the parallel between history and the individual development, but in broad outline there certainly are stages that are the same.
And with respect to mathematics, they note the argument of physicist Rolando Garcia (Bringuier, 1980):
In modern mathematics, at the level of algebraic geometry, of quantum mechanics, although it's a much higher level of abstraction, you find the same mechanisms in action -- the processes of the development of knowledge or the cognitive system are constructed according to the same kinds of evolutionary laws.
Garcia argued (Bringuier, 1980, p. 103) that:
And it means that one can explain the development of knowledge by starting with biology; in other words, it is the developing biological being that becomes a thinking being, even a scientist, capable of making systems that explain nature -- not the system that explains nature but some systems that explain part of nature.
Of the argument of Garcia, Piaget notes (Bringuier, 1980, p. 95-6):
...take the history of geometry, for example... you find what I call "common mechanisms". In geometry, the common mechanisms are these: in the first stage all the geometric spatial relations the child constructs are strictly intrafigural, just as for Euclid.... the second stage is interfigural. It is the Cartesian coordinates... The third step is the algebraization of geometry, starting with Klein and the Erlangen program; all geometries are reduced to displacement groups or transformation groups. Now that's a mechanims common to the history of sciernce and psychogenesis.... You see how the elementary laws of formation appear, from simple to complex... It can't be done any other way. If you began with structures and ended with a description of the elements, you'd be reversing an order that is, as I call it, "natural" because it's required, so to speak, by the very nature of things.
With respect to what was then termed genetic development, Piaget and Garcia subsequently collaborated in a book Psychogenesis and the Hisotry of Science (1989) to counteract overly simplistic psychological interpretations of recapitulation. The concept was a feature of Piaget's theory of genetic epistemology. As noted in the major collective study by John Fauvel and Jan A. Maanen (History in Mathematics Education: the ICMI Study, 2000), they argued that we should try to understand the problem of knowledge in terms of the intellectual instruments and mechanisms allowing its acquisition. Citing the challenges to this approach by Lev Vygotsky (1997) from a cultural perspective, the study concludes (p. 147):
The examples of Piaget and Garcia and of Vygotsky, uncover the complexity of the problem of the relationship between phylogenesis and ontogenesis and the importance of working towards a clear theoretical framework.
And, more specifically, with the reservations (pp 168-170):
Indeed any attempt to put in relation the history of mathematics and the teaching or learning of mathematics necessarily induces an epistemological questioning both of individual cognitive development and of the interpretations of the historical development of mathematics.... an epistemological reflection on the development of ideas in the history of mathematics can enrich didactical analysis by providing essential clues which may specify the nature of the knowledge to be taught, and explore different ways of access to that knowledge. Nevertheless what appears to have happened in history does not cover all the possibilities..
In this light an individual struggles with understandings and hypotheses -- that may indeed be considered inadequate by others if they could be adequately communicated -- perhaps discarding them quickly, perhaps retaining them inappropriately for an undue amount of time. This may also be true of a community introduced to mathematical concepts for the first time (such as after some civilizational disaster).
Whilst the table highlights the progressive increase in comprehension of symmetry over time, it also implies a corresponding ignorance of such symmetry by those who have not matched the insights acquired by others. In each case there is a sense of struggling with what is known (correctly or incorrectly) and what is not known -- and remains to be discovered (whether from others or by future generations). Ignorance may also be intimately related to information accessibility, overload and imagined priorities regarding what it is worth endeavouring to know -- notably as constrained by cost and confidentiality.
It is also presumably the case that some may have an understanding of profound symmetry -- in resonance with their own sense of the "inner structure" of their intellect -- without being able to communicate it through conventional mathematical formalism. This is clearly the case with music and architectural patterns, for example, which may offer the possibility of reinterpretation in mathematical form -- as discussed by Marcus du Sautoy. Appreciation of pattern may also be cultivated -- quite independently of mathematics -- as it is expressed through nature, as notably highlighted by Christopher Alexander (The Nature of Order: an essay on the art of building and the nature of the universe, 2003-2004). It is of course also the case that some, later acclaimed to be mathematical geniuses, may express their insights through formalisms foreign to conventional mathematics, as was the case with Srinivasa Ramanujan.
Given this struggle between knowledge and ignorance, life-long experience may be understood as being on the boundary between comprehension of profound symmetry (in resonance with inner structure of the intellect) and profound ignorance or misunderstanding of any such relationship -- namely on the boundary between order and chaos (cf Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005). The implication is that appropriate insight will be forthcoming when appropriate knowledge is acquired through learning. For mathematicians this is dependent on the "proof" of such a higher order of symmetry -- understood as the end of a kind of journey to that level of comprehension.
The linearity of such a progression -- especially if one has to live with one's own ignorance in failing to comprehend some intuited higher order of symmetry (possibly illusory) -- suggests that the understanding or realization of such "completion" may itself (always) be incomplete, as discussed from a different perspective (Happiness and Unhappiness through Naysign and Nescience: comprehending the essence of sustainability? 2008).
The progressive comprehension of higher orders of symmetry is intimately related -- in mathematics at least -- with a particular understanding of increasing levels of abstraction. A proper understanding of abstraction is considered to be a prerequisite for explaining mathematical concepts and for doing mathematics. As a concern of philosophy, however, abstraction has been a focus of continuing debate since Aristotle -- who introduced the notion to the west (cf Andrzej Maryniarczyk, Abstraction, Powszechna Encyklopedia Filozofii). Mehmet Fatih Ozmantar and John Monaghan (A Dialectical Approach to the Formation of Mathematical Abstractions. Mathematics Education Research Journal, 2007) offer a useful summary of current perspectives on the nature of abstraction.
In a helpfully extensive review of the challenges of abstraction from an educational perspective, Agnes Edling (Abstraction and Authority in Textbooks: the textual paths towards specialized language, 2006) makes the point that:
Abstraction can be seen as an important feature of many varieties of specialized language. The function of specialized language is not just to use language in a pretentious way which excludes the uninitiated. It is rather a necessary part of that knowledge. Academic contexts are constituted through abstraction in texts.
Abstraction in general can be viewed as an essential feature of specialized texts in a society where expertise and specialization are increasingly significant. The focus of the study is on the different levels of abstraction and the transitions between them in literary, social science and natural science texts. This requires clarification of the nature of abstraction, presented in this case in terms of the scales concrete-abstract and specific-general. The point is made that:
But the view of abstraction as enabling large meanings in a concise form is not the only understanding of abstraction. In some contexts, abstraction is regarded as a feature that complicates and hinders students' access to texts. In a pedagogical context, where researchers discuss ways of facilitating the encounter between reader and text, abstraction, as well as other features of specialized discourses, is sometimes seen as problematic.
In the case of objected-oriented programming -- so fundamental to modern computing -- Alex Mueller (What is abstraction? 2006) argues:
Abstraction, in my opinion, is one of the more complex concepts to understand in object-oriented programming. First, the understanding of abstraction requires a good cognizance of objects, classes, inheritance, encapsulation, and polymorphism. Second, the definition of abstraction is somewhat abstract, in that it is not specific. Third, the ability to apply abstraction when designing a system demands OO concept recognition, which takes time and practice.
The argument is echoed in a detailed, cross-disciplinary review of the literature relevant to computer programming by Jeff Kramer (Abstraction: the key to computing?), noting the constrained capacity of some students with respect to the handling of complexity, the production of elegant models and designs, and the applicability of various modelling notations and other subtle issues:
They tend to find distributed algorithms very difficult, do not appreciate the utility of modelling, find it difficult to identify what is important in a problem, and produce convoluted solutions that replicate the problem complexities. Why? What is it that makes the good students so able? What is lacking in the weaker ones? Is it some aspect of intelligence? I believe that the key lies in abstraction: the ability to perform abstract thinking and to exhibit abstraction skills.
Whilst this clearly has fundamental implications for the possibility of abstraction in policy-making, with respect to mathematics, Raymond Duval (A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, 2006) notes:
To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, are some of them specific to mathematical activity?
Victor Katz et al (The Role of Historical Analysis in Predicting and Interpreting Students' Difficulties in Mathematics, 2000, p. 153) note:
A final common student difficulty involves the transition to abstraction. As a typical example, many instances of what today are called groups were known in the first decades of the nineteenth century -- and some were known even earlier. Yet it was not until 1882 that the first complete formal definition of this abstract concept was given. Nevertheless, many current textbooks in abstract algebra begin by giving a formal definition of a group before the student has experienced many of these examples. It is not surprising that students have difficulties making the leap to abstraction; too little attention has been paid to the necessary steps that historically preceded this leap.
One of the continuing challenges is the nature of the "objects" that are "discovered" through the development of mathematical insight and the vexed issue of whether they "exist" independently of that discipline or its practitioners -- as presented, for example, by Øystein Linnebo (Epistemological Challenges to Mathematical Platonism, Philosophical Studies, 2006; The Nature of Mathematical Objects, 2008).
Do the skills that enable the discovery of those object, and others beyond ordinary understanding, reflect a view of symmetry by people with a particular mindset, with particular skills, and with a particular understanding of generality? To what extent is their appreciation of higher orders of symmetry an "acquired taste" -- dependent on special training of the "palette"?
To the extent that their beautiful symmetry can be understood by others, why are so many such mathematical objects apparently without relevance to the psycho-social domain -- other than as nice posters and art objects?
Is it to be assumed that the "objects" discovered will be described for all time through the notations favoured for comprehension at this time? Is the mathematical language through which they are described to be understood as a kind of cognitive "exoskeleton" -- with the possibility that artificial intelligence might come to be analogous to a powered human exoskeleton or even to a cyborg -- empowered to appreciate even higher orders of symmetry beyond human ken? Might this be an implication of the "scaffolding" explored by Mehmet Fatih Ozmantar (An investigation of the formation of mathematical abstractions through scaffolding, 2005)?
How many levels of abstraction might there be? To what extent is abstraction self-reflexively defined, possibly in the case of symmetry as:
Might there be other ways of comprehending such symmetry? In other words, are the objects discovered more intimately related to particular modes of understanding than is currently suspected? How could one demonstrate otherwise?
Curiously the studies of abstraction do not identify many degrees or levels of abstraction. This is strange in that:
It would appear that neither mathematics, nor computing, nor education has as yet clarified the shifts in understanding associated with shifts in levels of abstraction. Even the tantalizing title of the study by J. Mason (Mathematical Abstraction as a Result of a Delicate Shift of Attention. For the Learning of Mathematics, 1989) seems to imply only a single shift in attention -- rather than either a linear series of such shifts, or possibly a non-linear progressive comprehension of a system (as hypothesized in Authentic Grokking: Emergence of Homo conjugens, 2003).
And yet the history of mathematical discovery with regard to symmetry suggests that a succession of assumptions is made that successively challenges:
Despite the reservations regarding psychological recapitulation, that history of discovery might therefore be understood as itself mapping or encoding the successive levels of abstraction -- and the shifts in perspective necessary to appreciate the underlying and higher orders of symmetry to be discovered. In that sense, as argued above, the ontogeny of the student's comprehension may indeed recapitulate the phylogeny of the discipline of mathematics -- whether the beauty to be discovered "exists" or is a "taste" acquired through education of the suitably predisposed student's "palette".
The "levels of abstraction" have effectively been explored to a greater degree through the term "levels of reality" on which there is an extensive literature, notably the work of physicist Basarab Nicolescu (Transdisciplinarity and Complexity: Levels of Reality as Source of Indeterminacy, 2000; Levels of Reality and the Sacred, 2002) and John van Breda (Exploring Non-Reductionism and Levels of Reality: on the importance of the non-separability of discontinuity and continuity of the different levels of reality, 2008).
In the earlier exploration cited (Varieties of Rebirth: distinguishing ways of being "born again", 2004) clusters of contexts were identified and tentatively ordered in terms of increasing experiential implications for the individual -- a sense of some higher form of order. Two "paths" were distinguished in the following table (linking to that exploration) in order to relate the clusters:
Fig. 2: Varieties of cognitive "rebirth" | |
G. Experiential rebirth (operacy, flow, emdiment of mind, speaking with God, born-again, possession, psychedelic experience, embodiment in song, spiritual rebirth) | F. Cognitive perspective (metacognition, critical thinking, philosophy, aesthetic sensibility, orders of thinking, systematics, orders of abstraction, disciplines of action) |
E. Therapeutical rebirth (release from trauma, mentors, self-help, discipleship) | D. Developmental rebirth (education, perspective, initiation, cultural creativity, individuation) |
C. Psycho-behavioural rebirth (sin-to-virtue, changing patterns of consumption, conversion) | B. Socio-religious rebirth (birthright, destiny, reincarnation, socal status, ceremony, ritual, group affiliation, games, sports) |
A.
Cultural rebirth (renaissance, aesthetic birth, mytho-poesis) |
Given the bias here regarding the potential psycho-social relevance of higher orders of symmetry and the epistemological challenges to their comprehension, it is appropriate to note a classical metaphorical approach to distinguishing such phases of abstraction namely that of the traditional "10 ox-herding pictures" of Zen (D.T. Suzuki, The Ten Oxherding Pictures from The Manual of Zen Buddhism). With respect to psycho-social implications, these stages were compared to the challenge of the "7 blind men seeing the elephant" as discussed with respect to a critical current issue (Climate Change and the Elephant in the Living Room: in quest of an endangered species, 2008). Mathematicians in the early stages of the quest for the Monster group might have been compared to the blind men.
In the Zen case, however, a graded series of 10 stages in ability to relate "to the ox" is carefully articulated. The challenge of "seeing the elephant" might even be compared to the first such stage, with only an implication of the challenges of the later stages. These stages of abstraction were tentatively reframed in a commentary on the Integration of perceived problems (in the Encyclopedia of World Problems and Human Potential) under the headings:
There is a sense in which these stages are to be understood as a progression towards "seeing the elephant" as the "pattern that connects", as articulated by Gregory Bateson (Mind and Nature: a necessary unity, 1979):
The pattern which connects is a metapattern. It is a pattern of patterns. It is that metapattern which defines the vast generalization that, indeed, it is patterns which connect.
This concern with pattern can be related to the mathematical concern with symmetry and to the "pattern language" initially developed by Christopher Alexander (A Pattern Language, 1977) -- subsequently to be used in software engineering and, more generally, in computer science, as well as in interaction designs. Pedagogical patterns are used to document good practices in teaching (notably Learning patterns for the design and deployment of mathematical games). Joseph Bergin has articulated Fourteen Pedagogical Patterns (2007) for computer science course development which he recognizes might have application to other fields as well -- such as those identified by the Pedagogical Patterns Project.
A more empirical approach to conceptual patterns is reflected in an earlier study (Patterns of Conceptual Integration, 1984) and notably its documentation of Examples of Integrated, Multi-set Concept Schemes (1984). The study included an exercise in Distinguishing Levels of Declarations of Principles (1980). The latter endeavoured to detect the challenges to comprehension in formulating declarations with different 20 degrees of uncertainty -- and the consequent (constrained) potential for encompassing the diversity of perspectives. This followed a study on Representation, Comprehension and Communication of Sets: the role of number (1978).
It is most intriguing that such a fundamental domain of mathematics, symmetry group theory, should be (proudly) held by mathematicians to be irrelevant to the psycho-social challenges of the mundane world -- whilst being upheld as the most competent in identifying and describing the beauty to be found in the universe. This is especially irritating in a world challenged by a need for greater harmony when symmetry group theory is recognized -- through music -- as indeed being fundamental to human understanding of harmony in all its variety. As a musician himself, Marcus du Sautoy stresses this relationship. But why is it that the forms of "harmony" that are the focus of strategic initiatives -- "harmonization" as advocated within the European Union -- are so simplistic in terms of the insights into harmony offered by music and symmetry group theory? Do understandings of "harmony" and "integration" need to be "liberated" as explored elsewhere (Liberation of Integration, Universality and Concord -- through pattern, oscillation, harmony and embodiment, 1980)?
One expression of this frustration is indicated elsewhere with respect to the territorial challenges of the Middle East and other conflicts (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000). The account of Marcus du Sautoy is explicit in indicating his personal appreciation of the understanding of mathematics in the Jewish and Muslim cultures. But somehow those cultures have proven unable to understand the relevance of their insights into symmetry groups as they might relate to challenges of territorial apportionment -- considered mathematically "trivial" even though the simplistic "solutions" currently promoted give rise to bloody conflict.
Unfortunately mathematicians have primarily contributed their insights to the military applications associated with this lack of understanding and its reinforcement, rather than applying their insights into harmony to reconcile the differences of the parties -- on the assumption, for example, that this might require symmetry of a higher order. It would seem that the military applications are mathematically more interesting -- perhaps precisely because those regarding harmony are explored at too simplistic a level?
It is perhaps the most supreme form of irony that insights into the highest forms of symmetry -- fundamental to the structure of the universe -- emerge from a "group theory" of mathematics which is considered totally distinct from a "group theory" of the social sciences (as in Mary J. Fambrough, The Changing Epistemological Assumptions of Group Theory, The Journal of Applied Behavioral Science, 2006). It is the insights of the latter which are supposedly of relevance to the alleviation of the bloody lack of harmony between different perspectives in a global civilization. This confusion is only too evident in web searches. It matches that described with respect to the "correspondences" considered fundamental to the mathematical "moonshine" conjecture (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).
In addition to the "social science" focus implied by issues of mathematics education (cf International Commission on Mathematical Instruction; International Group for the Psychology of Mathematics Education), one exception to this lack of connectivity between the two forms of group theory is perhaps to be found in the recognition by the International Society for Group Theory in Cognitive Science that:
Group Theory has emerged as a powerful tool for analyzing cognitive structure. The number of cognitive disciplines using group theory is now enormous. The power of group theory lies in its ability to identify organization, and to express organization in terms of generative actions that structure a space.
A series of relevant studies has been produced by the president of the group, Michael Leyton (Shape as Memory: a geometric theory of architecture, 2006; A Generative Theory of Shape, 2001; Symmetry, Causality, Mind, 1992).
The key question for the following exploration of symmetry group theory is associated with the manner in which mundane "differences" are transformed through the process of abstraction into mathematical objects. Studies of symmetry acquire their power and interest beyond binary relationships, however implicit these may continue to be in the mathematical objects that subsequently emerge. Interest might be said to start with the triangle, as it does in Marcus du Sautoy's account. But a triangle of what?
Abstraction, and the practice of mathematics, requires that any such triangle be a triangle of "points" defining the differences between three related "somethings" -- whose nature is quickly to be forgotten as being irrelevant to the efficacy of subsequent discovery of symmetry. It is assumed that the essence defining the significance of those points is appropriately retained by the process of abstraction. This assumption may be fruitfully challenged as illustrated by the following cases:
Whilst mathematics may indeed offer a description of the sets of relationships obtained by abstraction (in the above table), the challenge in each case is the identification by individuals (or cultures) with what is described -- when they are implied by the description as in the psycho-social contexts of interest here. In this sense the "points" are points of identification -- which may well be psychologically highly charged. These may be clustered to form a set in which the relationships between the points may also be a focus of identification -- and of identity games (as typical of many social groups, including mathematicians). The distinctions between the points are in this sense one of complementarity essential to the operation of the set as a system -- with which larger framework an individual (as a "point") may identify.
In the right-hand portion of the table, the points find their identity challenged -- leading to competing sets ("them and us") which may well be reframed as distinct "points" in further description. In many situations the left and right-hand portions are to some degree interwoven as in competing ball games or (amicable) debate. This integration may however fail and lead to conflict that is not so contained.
The point to be emphasized is illustrated by the last row of the table. How are the "insights" of symmetry group theory to be understood in terms of a triangle, for example, where (rather than the conventional geometric or algebraic depiction) the "points" are associated cognitively with different "ways of knowing":
As noted by Marcus du Sautoy, as a triangle, any such triangle has six "symmetries". From what understanding are such symmetries then recognized? Even when the more general term, "understanding", is used (rather than the vision-biased "perspective"), does the use of "from" then inappropriately imply a metaphorical "distance", as well-argued by George Lakoff and Mark Johnson (Metaphors We Live By, 1980)?
Marcus du Sautoy makes the point that a:
...triangle's symmetry was captured by things I could do to it that would leave it looking the same...the number of ways that I could pick up the triangle and put it down so that it fitted back exactly inside its outline on the paper. Each of these moves... was a 'symmetry' of the triangle. So a symmetry was something active, not passive.... symmetry as an action that I could perform on the triangle to replace it inside its outline, rather than some innate property of the triangle itself.... one could think of the total symmetry of an object as all the moves that the mathematicians could make to trick you into thinking that he hadn't touched it at all.
In geometric (and algebraic) terms these "moves" are described as including:
Such symmetries, and others, may be combined such that for a given form (like a triangle) the set of combinations constitutes a "group" -- hence the term "symmetry group theory".
Clearly the question with respect to the example of "ways of knowing" is how to understand possible analogues to these "moves", especially when there is some form of psycho-social identification with "points" configured as a triangle, for example -- whether the identification is by a person, a group, a discipline, a "school of thought", a belief system, or otherwise. The challenge in these cases has much to do with the recognition of invariance despite a range of "moves" -- notably interesting in the case of ideological groups. How is the sameness of a behvioural pattern to be recognized underlying such "moves"?
The challenge to the conventional, and supposedly dominant, mode of understanding is evident in the range of such modalities as variously identified (Systems of Categories Distinguishing Cultural Biases, 1993). Of relevance here is how these variously understand "symmetry" and prefer to represent it in support of their own approach to psycho-social organization. The potential of such alternative perspectives has been elegantly argued, emphasizing Asian insights, by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999) and appropriately recognized, in traditional knowledge systems, by Darrell A. Posey (Cultural and Spiritual Values of Biodiversity, 1999).
Virtualization of identity through abstraction Kenneth Boulding. Ecodynamics; a new theory of societal evolution, 1978. |
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. |
A key to understanding what might be the possible "moves" associated with "psycho-social symmetry" might be the manner in which symmetry attracts -- and functions as an attractor. Marcus du Sautoy helpfully points to some of the attractiveness of symmetry, and the dependence on it in nature:
He makes the point that: "Huge swathes of mathematics, physics and chemistry can be explained in terms of the underlying symmetry of the structures under investigation" (p. 21). Of continuing concern is in what way the satisfactory nature of the explanation derives from the satisfaction with the symmetry "somehow present in the inner structure of our intellect".
As he indicates:
The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure....Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape.
But, as he asks: "symmetry is a slippery concept. What exactly is it?". This is the question that has motivated the work of many mathematicians in seeking to classify its forms and to discover its basis. However, before venturing down that trail, it is appropriate to reinforce the relevance of symmetry as an attractor in psycho-social organization in response to contemporary challenges. For example:
In each case there is the challenge of understanding the abstraction through which higher symmetry may need to be appreciated.
The exploration of such possibilities should clearly be informed by subtler insights from symmetry group theory and how they are communicated. This is especially the case where they might be enhanced by the development of an appropriate pattern language (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008). Of significance in this respect is the relevance of rendering evident the aesthetics of subtler forms of symmetry to increase the comprehensibility of such possibilities. The advent of the web, and the potential of the semantic web, suggests that symmetry might play a vital role in the design of new communication algorithms and protocols to enhance the possibility of emergence of psycho-social structures of greater subtlety and attractiveness. These might be said to have been curiously prefigured by the global role of music -- with presumably little recognition of the forms of symmetry that music promotes in psycho-social structures (cf Jacques Attali, Noise; the political economy of music, 1985).
The full flowering of symmetry group theory is recognized to have only been possible through shifting away from dependence on geometric depiction with its limitation to two and three dimensions. The key has been the use of abstract algebraic language that was unconstrained in this respect -- allowing thousands of dimensions to be handled. The point is made by Ladislav Kvasz (The History of Algebra and the Development of the Form of its Language, Philosophia Mathematica, 2006):
It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form (in the sense of Wittgenstein's Tractatus) of the symbolic language of algebra.
Of course this required that the cognitive implications be set aside, since the objects that emerge through this language are not intended to be comprehended in any conventional manner. For Marcus du Sautoy:
I can portray the geometry in four dimensions without having to concern myself at all with trying to visualize it... Although I can't see the hypercube, the mathematical language allows me to manipulate it and explore its symmetries. The numbers give me... a sixth sense -- the feeling that I really can see in four dimensions. (p. 27).
The question raised above is how such objects are to be expected to engage the attention of those implicated in any such description. Who is then able to "see" them and what conclusions are they then empowered to reach? More intriguing is the fact that although all the building blocks of symmetry have been successfully incorporated into a periodic table of symmetry (J. H. Conway, et al. Atlas of Finite Groups, 1985), there is still very little understanding of what can be built from those elements -- and this also applies in the psycho-social case. Or does it?
A related challenge to communication of insights into symmetry is associated with the assumption that it can be appropriately depicted on the flat surface typical of text -- or the presentation of any mathematical proof or tabular representation (as in an "atlas"). The plane as a surface of representation, in contrast with the torus, has become significant to the discussion of the relationship between form and medium in advanced theories of communication.
As discussed in greater detail elsewhere (Beyond the plane: form and medium in terms of the calculus of indications, 2006), this notably featured in the work of Niklas Luhmann (Die Gesellschaft der Gesellschaft, 1997) and discussed by Michael Schiltz (Form and Medium: a mathematical reconstruction, Image [&] Narrative, 6, 2003) in relation to the calculus of indications of George Spencer-Brown (Laws of Form, 1969/1994). Schiltz notes that form/medium is "the image for systemic connectivity and concatenation", as described by Humberto Maturana and Francesco Varela. Schiltz notes, that the notion of "space" is the key to reflexivity appropriate to any discussion of form and medium, citing Spencer-Brown as follows:
In all mathematics it becomes apparent, at some stage, that we have for some time been following a rule without being aware of it. This might be described as the use of a covert convention. [… Its] use can be considered as the presence of an arrangement in the absence of an agreement. For example, in the statement and theorem.... it is arranged (although not agreed) that we shall write on a plane surface. If we write on the surface of a torus the theorem is not true […] The fact that men have for centuries used a plane surface for writing means that, at this point in the text, both author and reader are ready to be conned into the assumption of a plane writing surface without question. But, like any other assumption, it is not unquestionable, and the fact that we can question it here means that we can question it elsewhere […]
Schiltz then comments (emphasis added) :
It was our choice to write in a plane surface that has made that distinctions indeed do cut off an inside from an outside, that 'differences do make a difference' (Gregory Bateson). Covert conventions at a level deeper than the level of form, preceding the level of form, have determined what the form would do. There lies a chance for developing a medium theory here. In this concrete case: the medium of the plane surface makes the difference. And in general: the topology of the medium makes the difference between distinctions making a difference and distinctions not making a difference. 'It is now evident that if a different surface is used, what is written on it, although identical in marking may be not identical in meaning"... Spencer-Brown has shown us that the 'medium is the message' (Marshall MacLuhan).....
Hence, we are writing in a space that connects the level of first-order (operand) and second-order (operator) observations. That space is a torus. If considered operationally, distinctions written on a torus can subvert their boundaries and re-enter the space they distinguish, turning up in their own form....
Such conceptualization diverts sharply from an intuitive understanding of a medium. As seen here, a medium is far from a Euclidean container. Rather is it introverted space, it is identical to the topology of the form, it is the form's 'deep structure'.
There is at least the possibility that shifting out of planar articulation of any "peace process" in the Middle East, for example, might clarify more coherent options.
Schiltz concludes:
If the medium of meaning is indeed the ultimate medium of psychic and social systems, i.e. if meaning is 'the medium of itself', then what is its 'form', the distinction through which it can be expressed? I perceive only one answer: the medium of meaning must be identical to the difference between form and medium, and the re-entry of that distinction into itself. Its consequent indecidability is the symbol of our dealing with the world. It expresses the fact that all our attempts to get a hold of the world are doomed to frustration.... Meaning as our phenomenology of this world can only be partial, as the difference between form/medium can only be actualized as a form. In mathematical terms: meaning is a lambda-domain occupied by communications that, by acting on themselves (= being a function of themselves), produce new communications in the same domain which can in turn act on themselves and further expand the domain. [cf Louis H. Kauffman, The Mathematics of Charles Sanders Peirce, Cybernetics and Human Knowing, 8, 1-2, 2001]
To what extent are the various approaches to sustainable development, and the search for alternative paradigms, to be considered as efforts to achieve new -- and more encompassing -- forms of closure?
Symmetry and group theory, as explored by mathematics, derive initially from visual observations -- namely using the sense of vision. This has led to the recognition of the symmetry associated with forms such as the triangle, the square, the pentagon, the hexagon, etc. This understanding of symmetry in 2 dimensions has been extended into 3 dimensions through combining such forms into the tetrahedron, the cube, the octahedron, the icoshedron and the dodecahedron, to name only the Platonic polyhedra. Through group theory forms of symmetry have been explored in 4 dimensions and many more.
Whilst such forms are basic to patterns appreciated for their aesthetic qualities, most notably in architecture, they seem to be irrelevant to qualities that are appreciated in terms of the sense of sound, taste, smell, and touch. The point to be made in what follows can however be introduced through an aspect of vision, namely colour, through which the aesthetics of patterns may be distinguished.
Clearly it is possible to position 3 colours at the vertices of a triangle, or 4 at the vertices of a square, etc. Well chosen these may be appreciated as complementary, offering an aesthetic effect. The question is whether selections of colours offering an aesthetic benefit through their distribution onto more complex symmetrical forms in 2 dimensions or 3.
Generalizing from this case, consider use of the same approach to select and distribute:
The examples given in each case are simple and rely to some degree on familiarity with the visual example by which they were introduced -- to give a sense of their complementarity through that symmetry. However, there is no reason why the argument should be limited to these simple cases. The qualitative descriptors used in the appreciation of wines (tastes) or perfumes (smells) are far more complex.
Group theory enables much more complex patterns of symmetry to be explored and distinguished by a suitasble notation system -- beyond the capacity for them to be visualized, for example. The questions to be explored are:
Following from the preoccupations of group theory, associated questions (of possibly quite different degrees of significance) might include:
These questions aside, to what degree can group theory contribute to qualitative experience of subtler aesthetic experiences, notably in a web environment? Would enabling such possibilities dispose people to an understanding of psycho-social possibilities of relevance to the challenges of the times?
As indicated, this started as an exploration of the succession of insights into symmetry group theory, as noted by Marcus du Sautoy, that enabled discovery of the Monster. His account and style seemed to lend itself to naming those insights as cognitive transitions of relevance to anyone tracing that journey -- but these proved to be elusive, as in other sources.
There are suggestive leads, such as various approaches to a theory of generalization, notably in mathematical pedagogy (V. V. Davydov, Types of Generalization in Instruction: logical and psychological problems in the structuring of school curricula, 1990; David H. Wolpert, The Mathematics of Generalization, 1995; David A. Plaisted, Abstraction using Generalization Functions, 1986). These do not seem to address the challenge of degrees of abstraction (as opposed to recognition of commonality in different types of content) -- as in the case of the challenged approach of Jean Piaget, Rolando Garcia, et al. (Toward a Logic of Meanings, 1991).
Although the focus on symmetry here has necessarily concentrated on mathematics, the possibility of marrying the formalism of group theory with that of distinguishing differences more generically (especially those arising from different and multi-sensorial modes of cognition) would seem to be a challenge for the future -- despite the focus of the International Society for Group Theory in Cognitive Science.
Similarly postponed is the delightful possibility that the riches of symmetry generated by the simplest polygons (of less than 6 "points") are intimately related both to the number of human senses (and their use in various combinations) and to the classic constraint in psycho-social praxis as suggested by George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956) -- and perhaps even to be reflected in the use and symbolism of the hand as the most basic instrument of mathematics and design (Handing Over: handy metaphors for the communication of intent, 2006). The conclusions of Miller's study have since been confirmed by John S. Nicolis and Ichiro Tsuda (Chaotic Dynamics of Information Processing: The 'magic number seven plus-minus two' revisited, Bulletin of Mathematical Biology, 1985).
Given the psycho-social preoccupation of this exploration, more striking in the account of Marcus du Sautoy was the appallingly shoddy dynamic that seemed to characterize the failure by mathematicians to recognize a (statistically significant?) number of discoveries in group theory -- and the associated treatment of their discoverers, or the dynamics between those who laid claim to such discoveries. This points to the inadequacy of symmetry group theory in abstracting its preoccupations out of the matrix within which it is engendered -- without having the slightest motivation to treat those characteristic psycho-social phenomena as factors in any more widely relevant understanding of "group theory".
The case has been made since the book by Marcus du Sautoy went to print -- through the example of a paper by Antony Garrett Lisi (An Exceptionally Simple Theory of Everything, arxiv:0711.0770) that has bypassed the formal peer-reviewed publication process. As noted by Wikipedia:
The theory "received accolades from a few physicists amid a flurry of media coverage," but also "widespread skepticism.". Scientific American reported in March 2008 that the theory was being "largely but not entirely ignored" by the mainstream physics community, with a few physicists picking up the work to develop it further. However, as of July 2008, the paper had nine citations from other arXiv preprints, and was the most downloaded preprint on the arXiv....
In Lisi's model, the base is a four-dimensional surface -- our spacetime -- and the fiber is the E8 Lie group, a complicated 248 dimensional shape, which some mathematicians consider to be the most beautiful shape in mathematics. In this theory, each of the 248 symmetries of E8 corresponds to a different elementary particle, which can interact according to the geometry of E8. As Lisi describes it: "The principal bundle connection and its curvature describe how the E8 manifold twists and turns over spacetime, reproducing all known ﬁelds and dynamics through pure geometry."
For a discipline that is defined as the epitome of generalization and abstraction, self-reflexivity of any kind would seem to have been designed out of the interaction between mathematicians -- or effectively to degrade in practice to the binary dynamic from which society suffers. This has been well identified by Edward de Bono (I Am Right and You Are Wrong, 1990) in his quest for "new thinking". The group dynamics of symmetry group theorizing clearly has an "elephant in its living room" -- whether or not the Monster group is a key to understanding it. How to explain the binary dynamics that "new thinking" is ignored as "wrong" until it becomes "fashionable" and therefore "right" -- for the while?
This highlights the question of why symmetry group theory is not used to address and reframe the different perspectives ("points" of view) of the schools of thought of mathematics in general and of group theory in particular -- or of the "mindfield" of opposing theories of mathematical education. Why indeed is there concern on the part of some mathematicians for the need for mathematical education and the communication of mathematical insights to a wider society judged to be increasingly disinterested? Within what cognitive framework does a mathematician such as Marcus du Sautoy reconcile his focus on symmetry group theory with his appointment to the Simonyi Professorship of the Public Understanding of Science. He succeeds Richard Dawkins, famed for his controversial (The God Delusion, 2006) and his repeated affirmation:
It is absolutely safe to say that if you meet somebody who claims not to believe in evolution, that person is ignorant, stupid or insane (or wicked, but I'd rather not consider that). (Ignorance is No Crime, 2006)
From symmetry group theory, with the advance of science, people are now called upon to attach credibility to the belief of an elite group of mathematicians in the existence of a "Monster" -- a "preposterous snowflake" that can only be "faithfully represented" in 196883 "dimensions" and contains more than 8x10^{53} "elements" offering 1050 symmetries. It might be suggested that there is a certain simple degree of symmetry between the expectation by Richard Dawkins of people with regard to the "God" delusion and that of Marcus du Sautoy with regard to the existence of such a "Monster" -- especially when failure to believe in the evidence for either may be judged only in binary terms.
There would seem to be a strong case for "decoding", into its simplest cognitive terms, the elusive references to the "delicate shift of attention" (Mason, 1989) associated with each significant breakthrough in symmetry group theory -- beyond such simple binary symmetry. To that end timelines such as the following are a valuable resource:
However it is tempting to see mathematics as having developed a valuable formal language that needs to be "confronted" with other approaches to generalization and abstraction, preferably with a significant degree of self-reflexive cognitive implication. The question is how to determine the set of approaches to understanding generalization that could fruitfully challenge each other. To ensure requisite variety, they might include:
In 2007, a team of mathematicians and computer scientists managed to provide a description of the Lie symmetry group E8 in 248-dimensions (as mentioned above) -- consisting of 60 gigabytes of data. The human genome contains less than a gigabyte of data (Marianne Freiberger, Solving Symmetry, Plus Maths, Jan-April 2007). This achievement is considered invaluable for future mathematicians and scientists. Whilst some physicisits, like Eugene P. Wigner (Symmetries and Reflections, 1970), have long predicted a convergence of psychology and physics, the question is whether objects like the E8 group or the Monster might then be the basis for psychological description only, or whether they will enhance the meaning to human consciousness of personal identity -- challenged in a larger context by other identities with radically "different perspectives".
More speculatively of course, given the suggestive title of the study by mathematician Ron Atkin (Multidimensional Man: can man live in three dimensional space?, 1981), the willingness to envisage "objects" that only exist in 248 dimensions raises the question of how much data is required to define "life" -- and under what conditions such "objects" might be considered "alive" in ways beyond human knowledge. Presumably any valid Theory of Everything will necessarily define life in all its forms -- at every level of reality.
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:
Fig. 4: Indication of relationship
between dimensions discussed relating to engagement with symmetry (tentative) |
This simplistic figure holds some dimensions of the more complex diagram that is the subject of Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation (2007) and of an animation in Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori (2006). |
The constraints imposed by comprehension have been fruitfully analyzed from a mathematical perspective by Ron Atkin. Combinatorial Connectivities in Social Systems; an application of simplicial complex structures to the study of large organizations, 1977; Multidimensional Man: Can man live in three dimensional space?, 1981). Of particular interest, in the light of the reference to a triangle above, is Atkin's use of a triangle with vertices of 3 distinct colours in order to explain the challenge of comprehending their combinations -- and the experience of the "geometry" of what is not fully comprehended.
The discoveries of mathematicians with respect to symmetry are truly awesome. It is however curious that, just as the implications of other ways of knowing symmetry are considered irrelevant, so are the implications of not being able to comprehend such discoveries. Whether it is a question of deficiency in IQ (perhaps to be measured as in terms of a "Symmetry Quotient") or an inappropriate bias towards another mode of intelligence (cf Antonio de Nicolas, Habits of Mind, 2000), the challenge is how to design into one's psycho-social "home" the highest level of symmetry that one can comprehend.
The home construction metaphor is useful in that it highlights the reality of the cognitive space in which most are obliged to dwell. It clarifies the inappropriateness of appeals to higher orders of symmetry that one is unable effectively to comprehend, especially in the language in which they are communicated. Just as with the shacks in which the impoverished may be obliged to live in some socio-economic contexts, many may be obliged to dwell in psycho-social shacks constructed with elements of a relatively low order of symmetry -- for lack of materials or for lack of know-how. Extolling their inhabitants to more sophisticated construction may be as unhelpful and inappropriate in the second case as it is in the first.
More interesting from a symmetry perspective is the sense -- as with Alexander's pattern language (specifically focused on building design) -- that there may be many ways to design psycho-social symmetry elements into one's cognitive home, in a manner consistent with one's philosophy or belief system. The repertoire of patterns of symmetry that mathematicians have painstakingly classified could be understood as a resource to that end. This emphasizes the possibility of building one's house on the basis of the symmetry one "understands" -- of the configuration of design elements that one then "stands under".
Most depressing is the sense in which conventional understandings of a limited range of patterns leads to psycho-social dwellings conforming to a limited range of stereotypes -- as with the suburbia of the most developed countries. Constraints on innovative construction that are evident in such settings -- building codes, health and safety, fashion, etc -- have their equivalent in psycho-social construction for individuals, groups or communities.
Perhaps even more problematic is a potential implication that higher symmetry is better, even if it cannot be comprehended. This is where the mathematical capacity to handle spaces of thousands of dimensions -- within which such dwellings potentially emerge -- is a form of illusion. It may be possible to dream of dwelling in such celestial spaces -- encouraged by science fiction -- but there is a real need to render comprehensible higher orders of symmetry that are characterized by accessibility. Computer augmented intelligence may indeed facilitate the process through sophisticated management of communication protocols that stabilize such psycho-social constructs as viable environments. Experimentation in successive generations of virtual worlds (Active Worlds, Second Life, Emerging Virtual Institutions, Metaverse, etc) may be extremely helpful in that respect. There would appear to be many possibilities for organizing websites in terms of higher orders of symmetry (Transforming Static Websites into Mobile "Wizdomes": enabling change through intertwining dynamic and configurative metaphors, 2007).
Negative Symmetry Capability (adapted from the Negative Capability of poet John Keats, with apologies) |
Capable of being in uncertainties, mysteries, doubts without any irritable reaching after fact, reason and higher orders of symmetry |
Christopher Alexander. The Nature of Order: an essay on the art of building and the nature of the universe. Center for Environmental Structure, 2003-2004 [summary]
Ron Atkin:
Giorgio T. Bagni, Fulvia Furinghetti and Filippo Spagnolo. History and epistemology in mathematics education. In: Cannizzato, L., Pesci A., Robutti, O. (Eds.), Italian research in Mathematics Education, 2000-2003 [text]
Gregory Bateson. Mind and Nature: a necessary unity, 1979
Benchara Branford. A Study of Mathematical Education. Oxford, Clarendon Press, 1908/1921 [review].
Jean Claude Bringuier. Conversations with Jean Piaget. University of Chicago Press, 1980
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, 1985. [summary]
Philip J. Davis and Reuben Hersh. The Mathematical Experience. Houghton Mifflin, 1981.
V. V. Davydov. Types of Generalization in Instruction: logical and psychological problems in the structuring of school curricula. Soviet Studies in Mathematics Education, 1990, Volume 2. [abstract]
Edward de Bono. I Am Right, You Are Wrong: New Renaissance: From Rock Logic to Water Logic. Penguin, 1990
Antonio T. de Nicolas. Habits of Mind: an introduction to the philosophy of education. iUniverse, 2000 [text]
F. Detienne. Assessing the Cognitive Consequences of the Object-oriented Approach: a survey of empirical research on object-oriented design by individuals and teams. Interacting with Computers, 9, 1997, pp. 47-72.
J. L. Dorier. Meta Level in the Teaching of Generalizing Concepts in Mathematics. Educational Studies in Mathematics, 29, 1995, pp. 175-197
Marcus du Sautoy. Finding Moonshine: a mathematician's journey through symmetry. Fourth Estate, 2008
Ed Dubinksy:
Raymond Duval. A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, Volume 61, Numbers 1-2, February 2006, pp. 103-131(29) [abstract]
Agnes Edling. Abstraction and Authority in Textbooks: the textual paths towards specialized language. Uppsala, Acta Universitatis Uppsaliensis, 2006 [text]
David Eppstein:
Mary J. Fambrough. The Changing Epistemological Assumptions of Group Theory. The Journal of Applied Behavioral Science, 42, 2006, 3, pp. 330-349 [text]
John Fauvel and Jan A. Maanen (Eds.). History in Mathematics Education: the ICMI Study. Springer, 2000) [text]
Jacob Feldman. Perceptual Grouping into Visual "objects": a detailed chronology. Journal of Vision, 2, 2003, 7 [abstract]
P. Frorer, Orrit Hazzan. and M. Manes. Revealing the Faces of Abstraction. International Journal of Computers for Mathematical Learning, 2, 1997, pp. 217-228
Fulvia Furinghetti and Luis Radford. Hisotrical Conceptual Developments and the Teaching of Mathematics: from phylogenesis and ontogenesis theory to classroom practice. In: John Fauvel and Jan A. Maanen (Eds.). History in Mathematics Education: the ICMI Study. Springer, 2000) [text]
Susantha Goonatilake. Toward a Global Science: mining civilizational knowledge. Indiana University Press, 1999
Orrit Hazzan:
Martin Hughes. Children and Number: difficulties in learning mathematics. Blackwell Publishing, 1986
Guy Inchbald. Polytopes, duality and precursors. 2005 [text]
Victor Katz, Jean-Luc Dorier, Otto Bekken and Anna Sierpinska. The Role of Historical Analysis in Predicting and Interpreting Students' Difficulties in Mathematics. In: John Fauvel and Jan A. Maanen (Eds.). History in Mathematics Education: the ICMI Study. Springer, 2000 [text]
Anthony W. Knapp. Group Representations and Harmonic Analysis from Euler to Langlands. Notices of the AMS, 3, 1996, 4 [text]
Jeff Kramer. Abstraction - the key to Computing? [text]
KTH Fysik:
Ladislav Kvasz. The History of Algebra and the Development of the Form of its Language. Philosophia Mathematica 2006 14(3):287-317 [abstract]
George Lakoff and Mark Johnson. Metaphors We Live By. University of Chicago Press, 2003
George Lakoff and Rafael E. Núñez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2000 [summary]
T. Y. Lam:
J. M. Lévy-Leblond. The pedagogical role and epistemological significance of group theory in quantum mechanics. La Rivista del Nuovo Cimento, 4, 1, January, 1974 pp. 99-143 [abstract]
Michael Leyton:
Margarita Limón and Lucia Mason. Reconsidering Conceptual Change: issues in theory and practice. Springer, 2002 [text]
Øystein Linnebo:
J. R. Martin:
Andrzej Maryniarczyk. Abstraction. Powszechna Encyklopedia Filozofii [text]
J. Mason. Mathematical Abstraction as a Result of a Delicate Shift of Attention. For the Learning of Mathematics, 9, 1989, pp. 2-8.
Alexander Masters. The Genius in My Basement: the biography of a happy man. Fourth Estate, 2011
Guerino Mazzola. Mathematical Music Theory: Status Quo 2000. 2001 [text]
Vann McGee. How We Learn Mathematical Language. Philosophical Review, 106, 1997, pp. 35- 68.
P. Mengal (Ed.). Histoire du concept de récapitulation. Paris, Masson, 1993
George Miller. The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information. Psychological Review, 63, 1956, pp. 81-97 [summary]
Basarab Nicolescu:
John S. Nicolis and Ichiro Tsuda. Chaotic Dynamics of Information Processing: the 'magic number seven plus-minus two' revisited. Bulletin of Mathematical Biology, 47, 1985, 3, pp. 343-365 [abstract]
J. J. O'Connor and E. F. Robertson. The Development of Group Theory. 1996 [text]
Magnus Österholm. Metacognition and Reading: criteria for comprehension of mathematics texts. In: Novotná, J., Moraová, H., Krátká, M. and Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, 2006, pp. 289-296. [text]
S. Ohlsson and E. Lehtinen. Abstraction and the Acquisition of Complex Ideas. International Journal of Educational Research, 27, 1997, pp. 37-48.
Mehmet Fatih Ozmantar. An investigation of the formation of mathematical abstractions through scaffolding. Unpublished Ph.D. thesis, University of Leeds, 2005.
Mehmet Fatih Ozmantar and John Monaghan. A Dialectical Approach to the Formation of Mathematical Abstractions. Mathematics Education Research Journal, 2007, 19, No. 2, pp. 89-112 [text]
N. Pennington, A. Y. Lee and B. Rehder. Cognitive Activities and Levels of Abstraction in Procedural and Object-oriented Design. Human-Computer Interaction, 10, 1995, pp. 171-226
Jean Piaget and Rolando Garcia. Psychogenesis and the History of Science. Columbia University Press, 1989
Jean Piaget, Rolando Garcia, Philip M. Davidson and Jack Easley. Toward a Logic of Meanings. Hillsdale NJ, Lawrence Erlbaum Associates, 1991 [contents]
David A. Plaisted. Abstraction using Generalization Functions. 8th International Conference on Automated Deduction, Volume 230/1986, pp. 365- 376 [abstract]
Darrell A. Posey (Editor). Cultural and Spiritual Values of Biodiversity: a complementary contribution to Global Biodiversity Assessment, Intermediate Technology, 1999 (for the United Nations Environment Programme)
Luis Radford, Paolo Boero and Carlos Vasco. Epistemological Assumptions Framing Interpretations of Students Understanding of Mathematics. In: John Fauvel and Jan A. Maanen (Eds.). History in Mathematics Education: the ICMI Study. Springer, 2000, pp. 162-167 [text]
Luis Radford. On psychology, historical epistemology and the teaching of mathematics: towards a socio-cultural history of mathematics. For the Learning of Mathematics, 17, 1997, 1, pp. 26-33
Mark Ronan. Symmetry and the Monster: one of the greatest quests of mathematics. Oxford University Press, 2006 [review]
Anna Sfard:
A. Sierpinska. Understanding in Mathematics. Falmer Press, 1994.
N. Stehliková. Emergence of Mathematical Knowledge Structures: introspection. In: N. A. Pateman, B. J. Dougherty, and J. T. Zilliox (Eds.), Proceedings of the 27th International Conference for the Psychology of Mathematics Education, 4, 2003, pp. 251-258
Radegundis Stolze. Levels of Abstraction in Specialist Concepts as a Translation Problem. In: C. Zelinsky-Wibbelt (Ed.) Text, Context, Concepts, Berlin, Mouton de Gruyter, 2003
John van Breda. Exploring Non-Reductionism and Levels of Reality: on the importance of the non-separability of discontinuity and continuity of the different levels of reality (Paper for the conference of the Metanexus Institute Subject, Self, and Soul: Transdisciplinary Approaches to Personhood, Madrid, 2008). The Global Spiral, 2008 [text]
C. E. Vasco. History of Mathematics as a Tool for Teaching Mathematics for Understanding. In: D. N. Perkins, et al. Software Goes to School: teaching for understanding with new technologies. Oxford University Press, 1995, pp 56-69
Lev S Vygotsky. The History of the Development of Higher Mental Functions. Plenum Press, 1997
Eugene P. Wigner. Symmetries and Reflections: Scientific Essays. MIT Press, 1970
David H. Wolpert. The Mathematics of Generalization. Addison Wesley Longman, 1995