<!DOCTYPE html>
<html>
<head>
<meta http-equiv="Content-Type" content="text/xhtml; charset=utf-8">
</head>
<body>
<div style="font-family:sans-serif"><div style="white-space:normal"><blockquote style="border-left:2px solid #777; color:#777; margin:0 0 5px; padding-left:5px"><p dir="auto">On Sun, Oct 24, 2021 at 5:00 PM Frank Wimberly <wimberly3@gmail.com><br>
wrote:<br>
I want to clarify what a dual space is.</p>
</blockquote><br><p dir="auto"> The domain of dual.space can be mapped to any other domain :-)</p>
<p dir="auto">_______________________________________________________________________<br>
Stephen.Guerin@Simtable.com <stephen.guerin@simtable.com><br>
CEO, Simtable <a href="http://www.simtable.com" style="color:#3983C4">http://www.simtable.com</a><br>
1600 Lena St #D1, Santa Fe, NM 87505<br>
office: (505)995-0206 mobile: (505)577-5828<br>
twitter: @simtable<br>
z <<a href="http://zoom.com/j/5055775828" style="color:#3983C4">http://zoom.com/j/5055775828</a>>oom.simtable.com</p>
<br><p dir="auto">On Sun, Oct 24, 2021 at 5:00 PM Frank Wimberly <wimberly3@gmail.com> wrote:</p>
<blockquote style="border-left:2px solid #777; color:#777; margin:0 0 5px; padding-left:5px"><p dir="auto">I want to clarify what a dual space is. I think it is much more specific<br>
than Jon thinks it is. A vector space is a linear space which consists of<br>
vectors for which addition and scalar multiplication are defined. Scalars<br>
are usually real numbers but may elements of other fields such a complex<br>
numbers.<br>
<br>
An important example of a (finite dimensional) vector space is the set of<br>
n-tuples of real numbers. A linear functional on a vector space is a<br>
function defined on the set of vectors the into the set of scalars. This<br>
is a (0,1) tensor on the space. The set of linear functionals is also a<br>
vector space called the dual space. The vectors of the original space<br>
define linear functionals on the dual space as follows: v(f) = f(v).<br>
<br>
The only other dual space of which I am aware of is the dual topological<br>
space which Google tells me is<br>
<br>
In functional analysis and related areas of mathematics a dual topology is<br>
a *locally convex topology on a dual* pair, two vector spaces with a<br>
bilinear form defined on them, so that one vector space becomes the<br>
continuous dual of the other space.<br>
<br>
I am being very succinct and may not remember these definitions<br>
correctly. Jon may be speaking "metaphorically" when he talks about<br>
pheromone trails being duals of ants.<br>
<br>
Anyway...<br>
<br>
Frank<br>
<br>
---<br>
Frank C. Wimberly<br>
140 Calle Ojo Feliz,<br>
Santa Fe, NM 87505<br>
<br>
505 670-9918<br>
Santa Fe, NM<br>
<br>
On Sun, Oct 24, 2021, 1:05 PM Jon Zingale <jonzingale@gmail.com> wrote:<br>
</p>
<blockquote style="border-left:2px solid #777; color:#999; margin:0 0 5px; padding-left:5px; border-left-color:#999"><p dir="auto">"""<br>
The problem for me with this view is that I don't understand how seeing<br>
pheromone as 'organizing itself in space' is intuitively useful.<br>
"""<br>
<br>
I suppose that even if I didn't find this view *useful*, which I do and<br>
will attempt to explain momentarily, I continue to find that it offers<br>
a theoretical completeness that I find aesthetically compelling. Much<br>
like magnetism and electricity can appear as distinct phenomena or as<br>
two aspects of an integrated whole, stigmergy points to a similar duality<br>
between agent and environment, another integrated whole. Lifting to such<br>
a perspective offers insights into a class of possible implementations,<br>
all preserving the underlying dynamics.<br>
<br>
For instance, when attempting to reason about the ant-pheromone system,<br>
I find it useful to view the ants as inefficient *raster-like* update<br>
to the state, but of course one could also choose a less brownian,<br>
less resource-limited or less discrete approach. For instance, I believe<br>
it makes the analysis more clear if we instead picture a continuum of<br>
ants acting on the space and begin with pheromone of very little effect.<br>
Then slowly turning up the potency, we begin to see the pheromone<br>
organize and as a side-effect (and as an epiphenomenon from my view)<br>
the ants follow suit. To view the ants as an implementation detail, for<br>
me, yields clarity into the problem, while the pheromone takes the role<br>
of first class citizens in the ABM.<br>
<br>
"""<br>
Toward the end, you wrote, "I only meant to emphasize that stigmergy<br>
appears to me as a local concept." I'm not sure what that means.<br>
"""<br>
<br>
By local, I mean local as it often manifests in mathematics, but I<br>
gather that you would prefer a different tack. Here I am referring to a<br>
pair of related concepts for me:<br>
<br>
1. Excision of glider's from Conway's game.<br>
<br>
2. Characterization of subjectivities (one's subjective experience, say)<br>
relative to objectivity.<br>
<br>
When one watches Conway's game unfold, it is challenging to maintain the<br>
view that gliders are not agents but simply a local patch of board state<br>
in the process of updating itself as a whole. The ease with which we<br>
perceive gliders as agents facilitated the discovery of "glider guns",<br>
and ultimately the construction of assemblages of "glider guns" to make<br>
logic gates. Further, there is a smallest possible toroidal board such<br>
that one can have a glider (and only a glider) walk along the surface<br>
forever. The relation of this smallest board to any other board state is<br>
(in my view) an analogy, an inclusion relation.<br>
<br>
Localization, here for me, is a way of bracketing the baby from the<br>
bathwater while continuing to acknowledge that the meaning of both is<br>
in relation to a whole. There have been a number of discussions on this<br>
forum (and quite a few papers by its participants) where work is done<br>
to emphasize the complications associated with non-trivial mereological<br>
systems, systems of parts with non-trivial structural constraints. Two<br>
recent papers that come to mind are:<br>
<br>
1. <a href="https://www.biorxiv.org/content/10.1101/2021.02.09.430402v1.full" style="color:#999">https://www.biorxiv.org/content/10.1101/2021.02.09.430402v1.full</a><br>
2. <a href="https://arxiv.org/pdf/1811.00420.pdf" style="color:#999">https://arxiv.org/pdf/1811.00420.pdf</a><br>
<br>
.-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - .<br>
FRIAM Applied Complexity Group listserv<br>
Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam<br>
un/subscribe <a href="http://redfish.com/mailman/listinfo/friam_redfish.com" style="color:#999">http://redfish.com/mailman/listinfo/friam_redfish.com</a><br>
FRIAM-COMIC <a href="http://friam-comic.blogspot.com/" style="color:#999">http://friam-comic.blogspot.com/</a><br>
archives:<br>
5/2017 thru present <a href="https://redfish.com/pipermail/friam_redfish.com/" style="color:#999">https://redfish.com/pipermail/friam_redfish.com/</a><br>
1/2003 thru 6/2021 <a href="http://friam.383.s1.nabble.com/" style="color:#999">http://friam.383.s1.nabble.com/</a><br>
</p>
</blockquote><p dir="auto">.-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - .<br>
FRIAM Applied Complexity Group listserv<br>
Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam<br>
un/subscribe <a href="http://redfish.com/mailman/listinfo/friam_redfish.com" style="color:#777">http://redfish.com/mailman/listinfo/friam_redfish.com</a><br>
FRIAM-COMIC <a href="http://friam-comic.blogspot.com/" style="color:#777">http://friam-comic.blogspot.com/</a><br>
archives:<br>
5/2017 thru present <a href="https://redfish.com/pipermail/friam_redfish.com/" style="color:#777">https://redfish.com/pipermail/friam_redfish.com/</a><br>
1/2003 thru 6/2021 <a href="http://friam.383.s1.nabble.com/" style="color:#777">http://friam.383.s1.nabble.com/</a><br>
</p>
</blockquote></div>
</div>
</body>
</html>