My primary interest is “computing with space”. Some remarks by Allen Francom make me believe that it might be a way to give a “space” to the internet of things by using graphic lambda calculus. By that I mean the following: the internet of things is a huge collection of objects which are somewhere on Earth, each carrying a very small processor. So the objects are in the physical space, but they change their places all the time, and their processors are tiny, so one better use tiny programs and simple executions for syncronizing their relative places one with respect to the others.

That is a kind of “computing with space” and it fits well with the following problem with e-books raised by by Mark Changizi, who says that e-books lack space and gives as an example his (physical library). He says he knows that book A is the one close to that calculus book he uses when he need some formulae, and that’s the way he is going to find book A. While, in a virtual library there is no space, the relative positions of a e-book wrt to others is irrelevant, because when you click a link is like you teleport (and ignore space) from one place to another. My interpretation of this is: the way Changizi finds a book in his library is different from the way a robot Changizi would find the book, because the robot Changizi would need the coordinates of the book A with respect to a corner of the bookshelf, or some information like this, then it would go and fetch the book. On the contrary, the human Changizi (‘s brain) acquired a geometric competence by accumulating these simple bits of knowledge about his library, namely tht book A is near the calculus book, and the calculus book is used when he needs formulae, etc.

As you know, the set of things is the same as the trivial groupoid of things , with arrows being pairs of things and with composition of arrows being . Now, let us define first what is the “space” around one object from T, let’s call it ““. (The same works for each object).

For defining this you take a commutative group (could be or could be the free commutative group over an alphabet (containing for example “NEAR” an “FAR”, such that , etc). Call this group the “scale group”. Then to any pair of objects and any scale , associate a “virtual pair at scale epsilon” , where is like a term in lambda calculus constructed from variable names “a” and “b”, only that it does not satisfy the lambda calculus definition, but the emergent algebra definition. Of course you can see such terms as graphs in graphic lambda calculus made by the epsilon gates, along with the emergent algebra moves from graphic lambda calculus, as in

- Dictionary from emergent algebra to graphic lambda calculus (I)
- Dictionary from emergent algebra to graphic lambda calculus (II)
- Dictionary from emergent algebra to graphic lambda calculus (III)

Moreover, you can mix them with lambda calculus terms, as it is possible in graphic lambda calculus.

In conclusion, I am very interested in practical applications of this, and I can at least formulate a list of needs, some of them with an IT flavour, others applied math. As for the pure math needs, they are pure fun and play. All this together would lead to really interesting things. One of the things to keep in mind is that would also endow the internet of things with a synthetic chemical connectome, as an extension of The chemical connectome of the internet .

___________________________

Could we denote any ” each object from the internet of things has a relative representation of some of his neighbours, under the form of a small graph” as a Monad? I ask this as it seems to match what Leibniz had in mind in his Monadology. 🙂

This idea has to be treated carefully as it implies a non-well founded logic of relations. http://plato.stanford.edu/entries/nonwellfounded-set-theory/