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Variety of relationships between topos and topology

Psychodynamics of collective engagement with polyhedral value configurations (Part #7)

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The subtle relationships between topos, in its different senses, and topology merit careful exploration in the light of:

  • Memory: the classical importance attached to topos as mnemonic aids (known as the method of loci) as extensively explored by Frances Yates (The Art of Memory, 1966) regarding the classical Ars Memorativa. This used techniques such as the association of emotionally striking memory images within visualized locations (topos), the chaining or association of groups of images, the association of images with schematic graphics or notae, and the association of text with images. For Yates:
    There can be no doubt that these topoi used by persons with a trained memory must be mnemonic loci, and it is indeed probable that the very word 'topics' as used in dialectics arose through the places of mnemonics. Topics are the 'things' or subject matter of dialectic which came to be known as topoi through the places in which they were stored.
  • Learning: there are a number of interfaces between topology and learning, including applications of pattern recognition to artificial intelligence, notably using neural networks. Of particular relevance is the way in which a set of concepts can be represented formally as a topology, notably as a mind map of relationships between topoi. Learning can then be related to the capacity to internalize that network progressively. Communication of content to be learnt can be described in terms of the completeness of the topology successively enunciated, heard, comprehended and remembered. The challenge to learning and comprehension have been analyzed by Ron Atkin in terms of simplicial complexes, namely topological spaces of a particular kind (Combinatorial Connectivities in Social Systems; an application of simplicial complex structures to the study of large organizations, 1977) as summarized elsewhere (Social organization determined by incommunicability of insights).

  • Design: the design considerations of aesthetic environments intended to support more integrative insights, whether to create "an effect", as appropriate to a "sacred space" and/or for mnemonic purposes as in memory theatres, palaces and cathedrals (Jack Dann, The Memory Cathedral: A Secret History of Leonardo da Vinci, 1995). Such appropriately designed configurations have of course been of significance to "magical" settings, with their modern echoes in ritual and placement at negotiating tables.

  • Poiesis: this term, the etymological root of "poetry", was first a verb, an action that transforms and continues the world. Plato described how mortals strive for immortality in relation through poiesis a form of making/creating, a movement beyond the temporal cycle of birth and decay. It was described by Heidegger as a "bringing forth". The structure of poetry and music has been analyzed with the perspective of topology to understand how the relationships amongst their parts and themes (appropriately understood as topoi) engender aesthetic effects -- central to engagement and an integrated sense of the whole.

    As described by Daniel Belgrad (The Culture of Spontaneity, 1999), the poet Charles Olson adopted topology as an artistic principle. He used topos, in its Greek sense of a place of emotional significance, to encompass topology in both its geographical sense and in its mathematical sense. Topology is then understood as a model for the way a self or a society changes over time as articulated in his Maximus Poems (1983). Of relevance here is the sense in which poetry communicates a larger sense, through the pattern of associations (modelled by topology) between the valued topoi.

    It is of course significant that poiesis is now the root of autopoiesis, namely the self-organization of the complexity sciences (Humberto Maturana and Francisco Varela, Autopoiesis and Cognition: the Realization of the Living, 1973/1980). Poiesis is specifically related elsewhere to the articulation and engagement with strategy (Poetry-making and Policy-making: arranging a marriage between Beauty and the Beast, 1993; Ensuring Strategic Resilience through Haiku Patterns: reframing the scope of the "martial arts" in response to strategic threats, 2006).

  • Paradox: the role accorded to topology by Rosen (through such structures as the Mobius strip and the Klein bottle) to order understanding of the paradoxical complexity of cognitive relationship to the environment. This has also been explored from a different perspective (Psychosocial Work Cycle: beyond the plane of Möbius, 2007). Rosen's focus should also be seen within the context of the fundamental significance of topology as a branch of mathematics and the associated concern with topos and the topos theory of sets of categories in a topological space.

  • Self-reflexivity: the implication for articulation of more appropriate understanding through "writing" text on more complex topological surfaces as admirably articulated 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.

    This role of the torus has been further discussed elsewhere (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006). It raises the question as to whether checklists of values could be more appropriately and engagingly presented on such surfaces.

  • Biological ecosystems: With respect to the living environment, Rainer E. Zimmermann (Topological Aspects of Biosemiotics, tripleC, 2007) argues that:

    According to recent work, there is a close relationship between the conceptualization of biological life and mathematical conceptualization such that both of them co-depend on each other when discussing preliminary conditions for properties of biosystems. More precisely, such properties can be realized only, if the space of orbits of members of some topological space X by the set of functions governing the interactions of these members is compact and complete. This result has important consequences for the maximization of complementarity in habitat occupation as well as for the reciprocal contributions of sub(eco)systems with respect to their structural mutualism.

    Zimmerman shows what this more technical result means in philosophical terms with a view to the biosemiotic consequences. He shows that topology as formal nucleus of spatial modeling is more than relevant for the understanding of representing and co-creating the world as it is cognitively perceived and communicated in its design.

Aside from reframing the human relationship to the environment, these considerations point to other ways of framing the significance variously associated by the different Abrahamic religions with the central symbolic importance of Jerusalem as the epitome of one form of psychoactive environment --- where every wall may "speak" and engage memory in a challenge to "re-member". As such, Jerusalem is a memory-charged topological configuration -- with all the challenges that clearly implies.

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