Friday, May 26, 2006

The secret life of space


In his series of books called the Nature of Order, Christopher Alexander makes an argument that life is an objectively measurable property of certain configurations of physical space. In his first book, The Phenomenon of Life, he makes the claim to have mathematically formalized this idea. While i think his formalization falls far short of his claim, i believe that his basic idea has legs, so to speak, and can be given a formalization. Below i sketch out a narrative of the development of some ideas that i believe can achieve the formalization -- and attendant framework for doing calculation and prediction -- of Alexander's notions. Essentially, the idea is to find a computational framework for expressing dynamics so general that it is capable of internalizing both the physics of (quantum) gravity and the formal theories of agency and then to show that the proposals for the physics of gravity -- when viewed computationally -- cause the very fabric of spacetime to be made of computational agents of such complexity that they are capable of the things we typically associate with living agents -- even more with reasoning agents. Spacetime itself is not just alive, it is intelligent.

My working hypothesis is that the process algebras constitute such a computational framework and that the bulk of the work is simply to exhibit the process algebraic representations of (quantum) gravity and various characterizations of agency already in the literature. We know that this must be possible, in principle, for any computationally realizable theory of (quantum) gravity and notions of agency. Why? because the process algebras are computationally complete. Thus, if these theories are to be realized as programs they must be realizable as programs in the process algebra setting. Of course, the same could be said of Turing machines (TMs) or the lambda calculus (LC) or cellular automata (CAs). The reasons for choosing the process algebras are these:

  1. The process algebras give a compositional account of computation (neither TMs nor CAs do -- how do you build a TM/CA out of little TMs/CAs at the level of the theory of TMs/CAs?)
  2. With a compositional account of autonomous and concurrent computation (neither TMs nor LC nor CAs support this)
    1. Note that taken together these two notions (and especially the latter) give rise to a natural interpretation of agency as (autonomous) computation
  3. The process algebras have a well-developed family of logics (the so-called Hennessy-Milner logics) dual to the algebras giving an account of properties of computations and hence collections of computations (correlated by exhibiting such property)
In the discussion below i pursue only the loop approach to quantum gravity. i have serious philosophical and computational concerns about string theory. i will detail these in another entry.

Space as dynamics

This winter i developed an interpretation of spin networks as processes. This builds from my interpretation of knots as processes. The essential idea is that the network can be represented as a process in an algebra built out of names that are elements of compact Lie groups that are associated with the guage-invariance of the particular field being quantized. i am in the process of working with a stellar group of researchers to tighten up and publish the results on knots as processes. From there the spin network results are a very natural extension. However, i have found an even more pleasant presentation of the results in which they are factored into some results about the representations of graphs as processes and then specialized to knots and spin networks. This allows the bulk of the technical results to be qa'd by the process algebra community -- who are arguably the best folks to spot bugs in these results.

Of general interest in these researches is the notion that nothing is static. Or, rather that what we think of as stasis is actually a property of dynamic systems and that all systems in actual world are dynamic systems. If they seem static it is because they exhibit a certain property -- a kind of recursive structure in the description of the computation they realize -- that we recognize as stasis. i will post more on this later.

Epistemology and action

In connection with the last idea, the core intuition here is that to know is to exhibit the ability to do. Even knowing is a kind of dynamics. Last year, in the spring, i began to pursue more seriously some thoughts i'd had regarding how this intuition could be formalized in the relationship between epistemic logics (ELs) and Hennessy-Milner logics (HMLs). The basic intuition was that HMLs were a reasonable candidate for ELs -- in fact better than the home-grown ELs proposed by the community -- because

  • the notion of agent is explicit and compositionally characterized in the process algebra setting;
  • the notion of an agent 'knowing x' could be correlated to having access to a name nx for an agent that has the capability x.
My idea was to pursue this in the reflective framework i was setting up via the \rho-calculus. It is not essential to the core idea, but provides a number of computational advantages. In particular, since names are conflated with the codes of agents in the reflective setting 'knowing x' can now be interpreted directly as knowing a name code(x) which is the code of an agent exhibiting the capability x.

i mentioned some of these ideas to Bob Coecke and Alexandru Baltag in an email exchange. Later, Radu Madare -- whom i knew through my connection with Corrado Priami -- worked with Alexandru. He and Alexandru did some work on epistemic logics. Then Radu considered with Corrado the connection between spatial versions of HMLs and epistemic logics with HMLs -- a consenual validation, imho -- of my intuitions that HMLs make a reasonable -- if not better -- framework for ELs.

Connecting the dots

How do these two lines of research fit together? The key point is that by realizing (quantum) gravitational characterizations of spacetime as processes in the same framework as one can write down properties of agents that believe and know and make statements about their beliefs enables one to check -- computationally -- whether -- or better which of -- the processes of spacetime satisfy properties that characterize conditions of belief and knowledge. This makes good on a part of the proposal. Here is a framework in which one can reason about configurations of spacetime in a way to verify what sort of agency it has.

As to life... well, one of the most interesting aspects of the recent flurry of activity in using process algebras to model biological systems is more abstract characterizations of life. In What is Life? Schroedinger investigates this as a physicist would -- looking for an abstract theoretical framework in which to characterize what we mean when we say life. It is recognized by many that Schroedinger's little essay can be seen as predicting the existence of DNA before it was discovered. Likewise, i think that the process algebraic accounts of biological systems are a step towards finding a more abstract characterization of life that will ultimately be predictive.

In particular, i think we can expect to be able to debate in a very precise way about formulations of the notion of life expressed as properties written in some HML. Such properties can be applied to the processes representing spacetime. We can ask specifically which configurations of space have life. In fact, because there is likely to be considerable debate about which of many properties we want to include in the notion of life we can expect gradations of notions of life. We can ask -- as Alexander does -- which of these configurations exhibits more life -- meaning which configuration satisfies more of the properties we wish to associate with life.

While it is certainly satisfying and exciting to think that the profound intuitions of such a great architect as Christopher Alexander really can be put on firm mathematical footing, i find myself even more intrigued by what happens to my conception of the universe in which i live in the wake of these investigations. It is not just that the very fabric of spacetime comes alive, but that life is foliated onto many levels. Such an account of life as certain kinds of information processing systems -- certain kinds of processes or agents -- enables a view into a world in which everything -- from the internet to pop songs -- comes alive to the degree it is able. Suddenly, i find myself living in a larger, more lively world. This is exciting.


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