[FRIAM] stygmergy, CA's, and [biological] development

[Frank Wimberly]

I want to clarify what a dual space is.  I think it is much more specific
than Jon thinks it is.  A vector space is a linear  space which consists of
vectors for which addition and scalar multiplication are defined.  Scalars
are usually real numbers but may elements of other fields such a complex
numbers.

An important example of a (finite dimensional) vector space is the set of
n-tuples of real numbers.  A linear functional on a vector space is a
function defined on the set of vectors the into the set of scalars.  This
is a (0,1) tensor on the space.  The set of linear functionals is also a
vector space called the dual space.  The vectors of the original space
define linear functionals on the dual space as follows:  v(f) = f(v).

The only other dual space of which I am aware of is the dual topological
space which Google tells me is

In functional analysis and related areas of mathematics a dual topology is
a *locally convex topology on a dual* pair, two vector spaces with a
bilinear form defined on them, so that one vector space becomes the
continuous dual of the other space.

I am being very succinct and may not remember these definitions correctly.
Jon may be speaking "metaphorically" when he talks about pheromone trails
being duals of ants.

Anyway...

Frank

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Sun, Oct 24, 2021, 1:05 PM Jon Zingale <jonzingale at gmail.com> wrote:

> """
> The problem for me with this view is that I don't understand how seeing
> pheromone as 'organizing itself in space' is intuitively useful.
> """
>
> I suppose that even if I didn't find this view *useful*, which I do and
> will attempt to explain momentarily, I continue to find that it offers
> a theoretical completeness that I find aesthetically compelling. Much
> like magnetism and electricity can appear as distinct phenomena or as
> two aspects of an integrated whole, stigmergy points to a similar duality
> between agent and environment, another integrated whole. Lifting to such
> a perspective offers insights into a class of possible implementations,
> all preserving the underlying dynamics.
>
> For instance, when attempting to reason about the ant-pheromone system,
> I find it useful to view the ants as inefficient *raster-like* update
> to the state, but of course one could also choose a less brownian,
> less resource-limited or less discrete approach. For instance, I believe
> it makes the analysis more clear if we instead picture a continuum of
> ants acting on the space and begin with pheromone of very little effect.
> Then slowly turning up the potency, we begin to see the pheromone
> organize and as a side-effect (and as an epiphenomenon from my view)
> the ants follow suit. To view the ants as an implementation detail, for
> me, yields clarity into the problem, while the pheromone takes the role
> of first class citizens in the ABM.
>
> """
> Toward the end, you wrote, "I only meant to emphasize that stigmergy
> appears to me as a local concept." I'm not sure what that means.
> """
>
> By local, I mean local as it often manifests in mathematics, but I
> gather that you would prefer a different tack. Here I am referring to a
> pair of related concepts for me:
>
> 1. Excision of glider's from Conway's game.
>
> 2. Characterization of subjectivities (one's subjective experience, say)
> relative to objectivity.
>
> When one watches Conway's game unfold, it is challenging to maintain the
> view that gliders are not agents but simply a local patch of board state
> in the process of updating itself as a whole. The ease with which we
> perceive gliders as agents facilitated the discovery of "glider guns",
> and ultimately the construction of assemblages of "glider guns" to make
> logic gates. Further, there is a smallest possible toroidal board such
> that one can have a glider (and only a glider) walk along the surface
> forever. The relation of this smallest board to any other board state is
> (in my view) an analogy, an inclusion relation.
>
> Localization, here for me, is a way of bracketing the baby from the
> bathwater while continuing to acknowledge that the meaning of both is
> in relation to a whole. There have been a number of discussions on this
> forum (and quite a few papers by its participants) where work is done
> to emphasize the complications associated with non-trivial mereological
> systems, systems of parts with non-trivial structural constraints. Two
> recent papers that come to mind are:
>
> 1. https://www.biorxiv.org/content/10.1101/2021.02.09.430402v1.full
> 2. https://arxiv.org/pdf/1811.00420.pdf
>
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