Thanks, I’ll take a look!

]]>Life’s Irreducible Structure

Live mechanisms and information in DNA are boundary conditions with a sequence of boundaries above them

Science, New Series, 160 (3834)

June 21, 1968, 1308-1312.

This article presents some ideas that are relevant to

your topic.

Thank you for providing your coordinates Ben! I started a thread on the forum for us and anyone else who is interested to continue the discussion. I would continue here, but I prefer the convenient syntax over on the forum.

]]>Since you’re both already on the Azimuth Forum, I encourage you to talk about your ideas there! Or here – that’s fine too. Anyway, I’d like to listen in.

]]>Hey Cameron,

My name is Ben Sprott and I have included my home page with this post. I am posting a few ideas at nforum. Is there a bio or complexity forum you like, as I am iterested in seeing what other bio people might be doing and we could start a thread there.

]]>Thank you for sharing your thoughts here Bem. I don’t mean to be rude, but if you’d tell me who you are I’d like very much to continue this conversation perhaps outside of this comment thread. “Bem” doesn’t give me much to go on. You should be able to find my e-mail address if you click my name or if you are involved in the Azimuth forum we could discuss it there.

It might not matter, but I’m not sure how you would define an interface. It seems to me that in the example you give you’re already looking at an interface that lies on top of a lot of other interfaces. That could just be due to something idiosyncratic in the manner that I conceptualize interfaces. If we dig down to what is physically being used to represent information in the device you have conceived, then I think we have to build a lot of interfaces on interfaces to get up to the point of knobs, buttons, and a readout.

In any case, I think there are at least a few different ways to imagine representing functional hierarchies using categories. You could think of hierarchical levels as being limits or colimits within a particular category. This seems to be the approach of Ehresmann (Andrée not Charles). Another option I can imagine is to think about how higher order morphisms (e.g. functors or natural transformations) could emerge from lower order ones. The latter may involve defining a process operating on a category. Even if you start with something like a monoidal category where you can internally represent the interplay between serial and parallel processes it seems to also be necessary to define a meta-process that describes how those processes themselves evolve in time. If we define a meta-process a priori, however, then we cannot claim that it emerged from an implicit representation embedded in lower level processes. To me this sounds impossible on the surface, but without it we don’t have a very good intuitive match to our current understanding of biological evolution.

]]>I realize I am heaviiy necrotizing this post, but I wanted to share a recent thought. Since having read this post, I have been thinking about it in terms of categories as interfaces. You know, the thing about interfaces is that you can always cover an interface with another interface. Here is a thought about the simplest of all interfaces:

An apparatus has knobs, buttons and a readout. Those things comprise an interface. Furthermore, this interface is itself a category in the sense that turning a knob is a structure preserving map of the interface. (Very technically, we start with a monoidal category and produce the category of internal comonoids and then, humorously, categories themselves are monads in this category…eek!).

Turning the knob on your apparatus is a really complicated morphism. We generally liken it to a point in a 1-d manifold and also a field i.e. the real numbers. That’s way too much structure. Furthermore, there was a time in the history of our species when we had no apparatus at all. My feeling is that the apparatus has evolved over time and increased in complexity. At first, however, the apparatus was really simple.

How simple? How about a category with just one morphism. How is that for simple?!

So how do we interpret the category with just one morphism? Well, in the category of categories, the one morphism category is useful because any functor to it is a tool for saying “Hey! I have a transformation! Eh! There’s a transformation here!”. If we interpret the transformation as an event in our universe like the decoupling of EM radiation from the early cosmic soup, then our little one-morphism category is not very expressive. It only tells us that something happened. It does not tell us what happened.

Anton Zeilinger gave a talk once in which he exressed resect for the succinctness of a finite physics based on detector clicks. Simply put, a click is an event. But a click is not quite an event in a one morphism category. A click is more like a noise that is at one time there and then not there. It is more like a category with two morphisms: “Noise on” and “Noise off”. These are transformations of the same object and they are also inverses of one another.

So, what about our little one-morphism category? He seems somewhat more fundamental to the physics of causality. This category is like a pregnancy test. If the stick changes colour, then something happened. Otherwise, no baby. If the test changes colour, you throw it in the garbage and consider what to tell your husband.

Now that you know what the one morphism category is used for, you can start to think of richer structure….more morphisms!

For those of you who are really keen, you might consider the role of those categories that have no finite set of axioms, otherwise called locally finitely presentable.

]]>…but maybe then just call for an explicit answer to the question : “what would most critically distinguish armies from cases of hierarchical organization most relevant to Cameron Smith’s title ?”

]]>Another comment that may be too specific: again the key to numerically solving the exact “constant hierarchical block structured matrix” equation that Cameron posted is evaluating that matrix multiplying such a matrix by a vector, or evaluating a finite vector under a finite linear mapping. Now obviously a “co-ordinate-based” way of looking at this is that each component of the input vector results in a linear combination of the output basis vectors.

So if we apply a 2-D discrete wavelet transform to the matrix, in this particular case the 2-D discrete Haar transform, “all” we’re doing is chosing a different (but common) basis for both the input space and the output spaces (which in a diff. eqn. happen to actually be the same). But in this “in-out” basis combination such a matrix will be very sparse (most of the entries will be 0). So if one knows anything useful about inferring phenomena in a set of sparse differential equations this can be transferred to this “constant hierarchical block structured matrix” system.

This doesn’t directly address information flow up and down “detail scales”, but I bet people in the wavelet community have thought about it! (Ironically I didn’t initially think of this because with wavelets one often thinks about the boundaries between different regions as being important, whilst the description was in terms of the constantness being important apart from at some block boundaries.)

]]>You should take a look at this paper by McInerney and Farmer from Santa Fe.

“The Role of Design Complexity in Technology

Improvement”

They are basically extending the ideas of Muth and Simon in interesting ways. I am trying to use these in my real job, because it is all about how to to a job faster and more efficiently. The basic tool is the Design Structure Matrix, which is the same thing as the rooms example.

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