Sunday, 23 October 2011

Learning and a Calculus of Anticipation

The outward manifestation of 'learning' is a change in patterns of communication, or 'positioning': the pattern of communications that exist between people who learn changes as utterances become different, and their social context consequently also changes.

The inward manifestation of 'learning' which makes possible the change in positioning, is a change in an individual's inner 'story', or what Harre calls a 'storyline'. But what is that individual 'story'... I think it is fundamentally anticipatory in nature: the change in storyline is equivalent to a change in the way individuals 'anticipate' things.

There are three questions here:

1. Can we deduce the ways in which anticipation changes through looking at the outward manifestations of learning (this may be what Leydesdorff does with his communications analysis)?
2. Can we model the way anticipation might work such that it produces emerging patterns of communication consistent with changing positions?
3. What is the nature of time in anticipation, and in the way in which shifting patterns of communication might reveal anticipation?

In thinking about anticipation itself, I'm interested in the maths of it. In particular I want to know if a mathematics of anticipation can exclude time a a variable.

It could be argued that anticipation of a sort is the feature of differential calculus, for to know the gradient of a curve at a particular point is to anticipate where it is likely to go next. But in my thinking about anticipation, it isn't so much about looking at the gradient at a particular point, but looking at the  symmetry of a particular point in relation to another point and defining a hierarchy of possible actions from which to move next. At the very least, there are always two possibilities.... that's like two possible gradients at each point... which in a Cartesian system isn't possible.

An alternative idea is to see anticipation as a sort of 'fractal'. The images below suggest how this might work.
First of all, an event sets up a symmetry as a fractal (here using a Mandelbrot set shown below): Within the local symmetry, there are things which can be immediately anticipated because they are within the symmetry.

But there is a larger symmetry that the immediate event symmetry is contained by. There are things which may be less immediately anticipated, but which are nevertheless consistent with the symmetry. The effect of these things happening is to shift the context of what might be forseen.
But as events become focused on one level of a symmetry, so the greater dimensions of the symmetry that contains that symmetry become more possible and anticipation moves out to the greater symmetry - maybe as way of increasing the variety of possible events...

So we zoom out again...
But the symmetries here are 'ideal'.. events do not conform to expectations. They transform expectations, and consequently transform the symmetries. In this way, anticipation is plastic, continually moulding expectations (any musician could tell you this!).. I imagine it might be like stretching these pictures on a sheet of rubber.

So what about a mathematics of anticipation? Partly this has to do with the calculus of fractals, but also I suspect something needs to be done in terms of the 'deformations' of symmetry. (That deformation has some sort of hysteresis, for example). But the fractal metaphor might show a way to think of anticipation and symmetry without time...

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