More detailed argument for the nature of space buillt from artificial chemistry

There are some things which were left uncleared. For example, I have never suggested to use networks of computers as a substitute for space, with computers as nodes, etc. This is one of the ideas which are too trivial. In the GLC actors article is proposed a different thing.

First to associate to an initial partition of the graph (molecule) another graph, with nodes being the partition pieces (thus each node, called actor, holds a piece of the graph) and edges being those edges of the original whole molecule which link nodes of graphs from different partitions. This is the actors diagram.
Then to interpret the existence of an edge between two actor nodes as a replacement for a spatial statement like (these two actors are close). Then to remark that the partition can be made such that the edges from the actor diagram correspond to active edges of the original graph (an active edge is one which connects two nodes of the molecule which form a left pattern), so that a graph rewrite applied to a left pattern consisting of a pair of nodes, each in a different actor part, produces not only a change of the state of each actor (i.e. a change of the piece of the graph which is hold by each actor), but also a change of the actor diagram itself. Thus, this very simple mechanism produces by graph rewrites two effects:

  • “chemical” where two molecules (i.e. the states of two actors) enter in reaction “when they are close” and produce two other molecules (the result of the graph rewrite as seen on the two pieces hold by the actors), and
  • “spatial” where the two molecules, after chemical interaction, change their spatial relation with the neighboring molecules because the actors diagram itself has changed.

This was the proposal from the GLC actors article.

Now, the first remark is that this explanation has a global side, namely that we look at a global big molecule which is partitioned, but obviously there is no global state of the system, if we think that each actor resides in a computer and each edge of an actor diagram describes the fact that each actor knows the mail address of the other which is used as a port name. But for explanatory purposes is OK, with the condition to know well what to expect from this kind of computation: nothing more than the state of a finite number of actors, say up to 10, known in advance, a priori bound, as is usual in the philosophy of local-global which is used here.

The second remark is that this mechanism is of course only a very
simplistic version of what should be the right mechanism. And here
enter the emergent algebras, i.e. the abstract nonsense formalism with trees and nodes and graph rewrites which I have found trying to
understand sub-riemannian geometry (and noticing that it does not
apply only to sub-riemannian, but seems to be something more general, of a computational nature, but which computation, etc). The closeness,  i.e. the neighbourhood relations themselves are a global, a posteriori view, a static view of the space.

In the Quick and dirty argument for space from chemlambda I propose the following. Because chemlambda is universal, it means that for any program there is a molecule such that the reductions of this molecule simulate the execution of the program. Or, think about the chemlambda gui, and suppose even that I have as much as needed computational power. The gui has two sides, one which processes mol files and outputs mol files of reduced molecules, and the other (based on d3.js) which visualizes each step. “Visualizes” means that there is a physics simulation of the molecule graphs as particles with bonds which move in space or plane of the screen. Imagine that with enough computing power and time we can visualize things in as much detail as we need, of course according to some physics principles which are implemented in the program of visualization. Take now a molecule (i.e. a mol file) and run the program with the two sides reduction/visualization. Then, because of chemlambda universality we know that there exist another molecule which admit chemlambda reductions which simulate the reductions of the first molecule AND the running of the visualization program.

So there is no need to have a spatial side different from the chemical side!

But of course, this is an argument which shows something which can be done in principle but maybe is not feasible in practice.

That is why I propose to concentrate a bit on the pure spatial part. Let’s do a simple thought experiment: take a system with a finite no of degrees of freedom and see it’s state as a point in a space (typically a symplectic manifold) and it’s evolution described by a 1st order equation. Then discretize this correctly(w.r.t the symplectic structure)  and you get a recipe which describes the evolution of the system which has roughly the following form:

  • starting from an initial position (i.e. state), interpret each step as a computation of the new position based on a given algorithm (the equation of evolution), which is always an algebraic expression which gives the new position as a  function of the older one,
  • throw out the initial position and keep only the algorithm for passing from a position to the next,
  • use the same treatment as in chemlambda or GLC, where all the variables are eliminated, therefore renounce in this way at all reference to coordinates, points from the manifold, etc
  • remark that the algebraic expressions which are used  always consists  of affine (or projective) combinations of  points (and notice that the combinations themselves can be expressed as trees or others graphs which are made by dilation nodes, as in the emergent algebras formalism)
  • indeed, that  is because of the evolution equation differential  operators, which are always limits of conjugations of dilations,  and because of the algebraic structure of the space, which is also described as a limit of  dilations combinations (notice that I speak about the vector addition operation and it’s properties, like associativity, etc, not about the points in the space), and finally because of an a priori assumption that functions like the hamiltonian are computable themselves.

This recipe itself is alike a chemlambda molecule, but consisting not only of A, L, FI, FO, FOE but also of some (two perhaps)  dilation nodes, with moves, i.e. graph rewrites which allow to pass from a step to another. The symplectic structure itself is only a shadow of a Heisenberg group structure, i.e. of a contact structure of a circle bundle over the symplectic manifold, as geometric  prequantization proposes (but is a mathematical fact which is, in itself, independent of any interpretation or speculation). I know what is to be added (i.e. which graph rewrites which particularize this structure among all possible ones). Because it connects to sub-riemannian geometry precisely. You may want to browse the old series on Gromov-Hausdorff distances and the Heisenberg group part 0, part I, part II, part III, or to start from the other end The graphical moves of projective conical spaces (II).

Hence my proposal which consist into thinking about space properties as embodied into graph rewriting systems, inspred from the abstract nonsense of emergent algebras, combining  the pure computational side of A, L, etc with the space  computational side of dilation nodes into one whole.

In this sense space as an absolute or relative vessel does not exist more than the  Marius creature (what does exist is a twirl of atoms, some go in, some out, but is too complex to understand by my human brain) instead the fact that all beings and inanimate objects seem to agree collectively when it comes to move spatially is in reality a manifestation of the universality of this graph rewrite system.

Finally, I’ll go to the main point which is that I don’t believe that
is that simple. It may be, but it may be as well something which only
contains these ideas as a small part, the tip of the nose of a
monumental statue. What I believe is that it is possible to make the
argument  by example that it is possible that nature works like this.
I mean that chemlambda shows that there exist a formalism which can do this, albeit perhaps in a very primitive way.

The second belief I have is that regardless if nature functions like this or not, at least chemlambda is a proof of principle that it is possible that brains process spatial information in this chemical way.

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