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 <jonzing...@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 > > .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: > 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >
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