Mathematical Learning: A Journey into the Deep Subconscious?
I am not a mathematician, which is not to say that mathematics does not intrigue me, because it does. Women may intrigue a man without there having to be any correlation between deepening intrigue and ‘deepening understanding’. The deeper one goes in, the more confounded one can become. What you thought you already knew slips away from you in your attempt to grasp it more firmly (isn’t this what has happened in particle physics?).. The explicit being and interactions of numerical arithmetic dissolves into a phantom-filled fog of tacit relationship in algebra, and sharply-etched delineations of geometric form melt like Salvadore Dali clocks as one lets go of the rigidity ‘of being’ and moves to the relational invariants of ‘becoming’ in topology (how would you create a representation of what was invariant in your own human body going from infancy through to old age? The orifices are distinctive invariants, and the animated graphic above is the way that some ancient thinkers pictured the ‘continuous becoming’ of a human. The flesh is secondary, like a ‘biofilm’ that gather in the energy flow, as with the tubular shells of tubeworms that form in the flow of clouds of bacteria precipitating in the hydro-thermal ambiance of volcanic vents deep in the ocean.
Before getting lost in the maze, the point of this post is to open up a place for discussion on ‘symmetry’ and how it relates to ‘understanding’.
I will kick it off from my own perspective which does seem in common with the perspectives of others (a ‘growing minority’?), as indicated in the following ‘formulation’ by Kirby Urner that seeks to capture what I would call the sourcing influence of the conjugate-habitat-inhabitant relational dynamic. What I mean by this is that we normally think of our form, behaviour and organization as originating from within (intrinsically influenced) but our life experience seems to be informing us that our form, behaviour and organization often originates from outside of us (extrinsic influence), e.g. the spatial dynamics we are situationally included in appear to orchestrate our individual and collective behaviour (e.g. the emergent opening of a hole in a clogged flow of traffic, orchestrates our simultaneous movements into the opening ‘hole’)
Kirby’s formulation is;
“The mantra of the extrinsic[-intrinsic] modeler might be “my community needs me, my community creates me”
For me, the term ‘community’ would imply ‘nature’ or ‘the world’ so that we might equally say; “the world needs me, the world creates me” (when the community is all-inclusive, there is no need to use the possessive ‘my’). Kirby’s formulation captures the understanding quite well, and ties to Emerson’s ‘transcendentalism’ where as Emerson says ‘The genius of nature not only inhabits the organism but creates it.” ‘The genius of nature’ in this case, corresponding to ‘the mantra’ in Kirby’s above formulation.
Before leaving this topic opening for discussion, I can recount one of the symmetry shifts I encountered which made a big impression on me and on most people who were close to it, which gives some everyday practical meaning to this notion of the ‘conjugate-habitat-inhabitant-relation.’
As part of an assignment to assist managers in the process of bringing information technology into their operations (things were in a costly, tower-of-babel disarray), a study was hatched (organized by myself) to first identify ‘exceptionally performing teams’ and see how information and knowledge flow was contributing to their exceptional performance (judged exceptional by host community, management, customers, suppliers, service providers, families), suggesting the way in which technology was being brought in to contribute to these results. All of the three teams studied had undergone a transformation of symmetry of the same basic ‘topology’.
For example, the largest team studied had just over 150 members, administrators, engineers, geoscientists, skilled trades and labourers. The team’s operations were losing money and senior management had tried everything they knew how to get the team’s operation back in the black, and had sent in their best managers to cut the overheads so that they would once again fall below revenues in a severe price-depressed business environment. Management announced to the team that it was going to sell off the assets that they had been working on and liquidate the operation (laying off pretty well everyone), unless the team could find a way to put the operation back in the black in a year.
The team’s organization had been the classic top-down, vision, mission, goals and objectives oriented structure that had been tuned to run as a finely oiled machine with an efficiency that was not exceeded by any of their peer teams (the problem was the difficult asset base and the regionally depressed pricing environment). It was a very curious scenario in that their senior management had simply giving up on them leaving a ‘hole’ at the top. They knew that they had already tried the best techniques of the best top-down management approaches piloted by the best managers available anywhere, and that had greatly reduced their overheads and increased their operating efficiencies but it hadn’t been enough to put them back into the black.
They all got under one roof, professionals and labourers and local managers in an aircraft hanger to see if everyone wanted to have a stab at another attempt. Almost all of them liked living where they did and their families were very happy there and if they were laid off they would definitely have to move to find new jobs, so they decided to try something very different (an idea that cropped up in their brainstorming sessions); i.e. to leave the hole in the top, and to let their operational form, behaviour and organization be orchestrated by their customers, suppliers, service providers and host community. Instead of producing into a faceless commodity market, they went out in buses to meet with the next-in-the-food-chain customers who received their produce and they asked how they could modify their ‘commodity’ product upstream in their operations so that it would move more smoothly into their customers’ operations (e.g. perform quality control that was easily accomplished upstream so that expensive QC would not have to be done at the receiving end etc.). They went to visit with their service providers and developed an arrangement whereby, instead of services (equipment maintenance etc.) being on an on-demand basis, the service providers would work it in when it was convenient for both; e.g. when they already had experts and maintenance equipment in the area etc.
As for their operations, they developed a network-based ‘vital signs’ real-time database that all team-members had access to, and when operations went through pulses where things were working really well, they invited any member who thought they knew why and could sustain that quality of operation to form their own teams and work with the full team to revise procedures to sustain the bursts of ‘exceptionally good health’. Likewise, when things were not going well, those (anyone in the full team) who thought they had input on why things were not going well were self-empowered to form subteams to work with the full team and revise procedures so as to attenuate or inhibit the ‘falling into bad health’.
The team achieved the unbelievable, succeeding in cutting the already stripped down to the barebones overheads in half in the following year and returned the operation to the black. The team was no longer ‘the team’; i.e. it was no longer the ‘local system with its own locally originating, internal-intellect and purpose-directed behaviour’ as it had been, but was allowing its identity to ‘blur’ by way of its conjugate relation with the dynamics of the other industries and communities it was situationally included in. One could not really say that the efficiencies gained were produced by the team since the efficiencies were coming from the conjugate community-team relation, much in the manner that wildgeese let their behaviours be orchestrated by the dynamics of the space they are included; i.e. allow their organization to be shaped by the co-cultivating of resonances in their conjugate habitat-inhabitant relation.
The local ‘nominal’ manager of the team was a man whose seasoned experience and natural wisdom transcended his ego and need to ‘be in control’. When his remote senior management called, he told them exactly what the team was doing, just as if it were him that was acting as the central authority issuing commands that would cascade down the control hierarchy. But there was no longer any control hierarchy. Everyone was responding to the dynamics of the community that their operations were included in, and to the real-time ‘vital signs’ that everyone had access to (over networked PCs that the team had ‘rigged’). The top local manager had transformed from the central direction-giver to the voice of the ‘collective consciousness’ of the team.
The ‘symmetry’ as I saw it, had undergone a shift in supra-system – system relation from that of Russian dolls (self-similar forms nesting inclusionally BUT DISCRETELY within one another) to self-similarity as in flow, that of ‘whorls-within-whorls’ where the suprasystem-system relation involves the continual ‘coniunctio’ of extrinsic with intrinsic flow, as in the relation between storm-cells and the flow of the atmosphere. as with multiple storm-cells in a common flow, the operations of the team-cell, customer-cells, service-provider-cells, supplier-cells were all simultaneously mutually influencing one another in a Mach’s principle manner (letting the common spatial dynamic become the intermediating orchestrator).
The team members had deliberately abandoned their prior ‘job-titles’ that implied hierarchy and had explicitly decided to put all of their ‘business cards’ away. The union laborer had become a full member of the new team which had no ‘class divisions’ within it, other than those implied by pay, which did not appear to be a show-stopper, as they were all ‘realists’ about the world we/they are all currently living in. So, one might say that something had also transformed in this suspending of the ego (the view of ‘self’).
What comes to mind is Nietzsche’s suggestion that science (e.g. the science of management as in the traditional top-down team structure) is ‘anthropomorphism’; i.e. our model of ‘self’, as a ‘local system with it’s own locally originating, intellect-and-intention-directed behaviour’ is something we infuse into our sciences (the biological model of a ‘cell’, an ‘organism’, and the sociological model of an ‘organization/team’). What this team had done can be thought of as transforming the model of ‘self’, superseding the ego-first model with the ‘self’ instead seen as a ‘conjugate habitat-inhabitant relation’ in the manner of the storm-cell in the flow of the atmosphere.
In terms of ‘mathematics’ as was discussed above, this was a move from “the sharply-etched delineations of geometric form [Russian dolls] melting like Salvadore Dali’s clocks as one lets go of the rigidity ‘of being’ and moves to the relational invariants of ‘becoming’ in topology (how would you create a representation what was invariant in your own [team] going from infancy through to old age?
As F. David Peat says, in ‘Mathematics and the Language of Nature’;
“The history of geometry demonstrates the discovery of deeper and more general levels, Euclidian geometry gives way to non-Euclidian, beneath geometry is topology, and topology itself is founded on even more general and beautiful mathematics. The longer a particular topic has been studied, the deeper mathematicians are able to move towards its foundations.
But Piaget, pointed out, this historical evolution is a direct reversal of the actual development of concepts of space in the infant. To the young child, the distinction between intersecting and non-intersecting figures is more immediate than between, say, a triangle, square and circle. To the infant’s developing mind, topology comes before geometry. In general, deeper and more fundamental logical operations are developed earlier than more specific rules and applications. The history of mathematics, which is generally taken as a process of moving towards deeper and more general levels of thought, could also be thought of as a process of excavation which attempts to uncover the earliest operations of thought in infancy. According to this argument, the very first operations exist at a pre-conscious level so that the more fundamental a logical operation happens to be, the earlier it was developed by the infant and the deeper it has become buried in the mind. Again, this suggests a reason why mathematics is so unreasonably effective, for the deeper it goes the more it becomes a formal expression of the ways in which with interact with, and learn about, the world.
But, it could be objected, if the history of mathematics and, to some extent, of theoretical physics, is simply that of uncovering, and formalizing, what we already know then how is it possible to create new ideas, like Einstein’s relativity, that totally lie outside our experience? The point is, however, that this equality or interdependence of space and time was already present in all the world’s language. Rather than coming to the revelation that time and space must be unified then have never really been linguistically separated! According to this general idea, what may appear to be novel in physics and mathematics is essentially the explicit unfolding of something that is already implicit within the structuring of human thought–of course physics itself also makes use of empirical observations and predictions. For this reason, the intelligent use of mathematics as a language for physics will necessarily make sense.”
While this makes a lot of sense to me, my impression is that Peat errs when he says;
“But, it could be objected, if the history of mathematics and, to some extent, of theoretical physics, is simply that of uncovering, and formalizing, what we already know then how is it possible to create new ideas, like Einstein’s relativity, that totally lie outside our experience?“
I don’t think that relativity “lies totally outside our experience”, my sense is that ‘we experience relativity’ (conjugate habitat-inhabitant relation), as in Ernst Mach’s principle of space-matter relativity; “The dynamics of habitat are conditioning the dynamics of the inhabitants at the same time as the dynamics of the inhabitants are conditioning the dynamics of habitat”. (e.g. as with storm-cells in the flow of the atmosphere, so it is with cells in the flow of the organism and so with all).
So what are the questions to ask and discuss here?
These will probably differ depending upon one’s preferred language game; i.e. mathematics, physics, sociology, psychiatry, pedagogy etc.
As a possible discussion topic, I will say this. It is my impression that our experience tells us that the more realistic model of the self puts our conjugate habitat-inhabitant relation in precedence over our Aristotelian standard/default; i.e. the ‘self’ as a; ‘local system with its own locally originating intellect-and-intention-directed behaviour’. Consistent with Piaget’s and Peat’s suggestion, the mental mathematics of the ‘conjugate habitat-inhabitant-relation’ type embraced by the exceptionally performing team described above represents a mathematical progression that takes us deeper into our own subconscious, our natural habitat-inhabitant beyond-good-and-evil relationship experienced in our infancy or embryo phase.
Any thoughts, comments, critiques or etc. on this or alternative hypotheses are welcomed.
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