## Saturday, August 02, 2008

### Physics theory vs. telecom theory

When I was just starting my book, my colleague Andrew Wheen suggested that I might try to derive some fundamental mathematical principles underlying telecoms network architecture and design. I was sceptical as to whether there were any.

The obvious comparison is with physics. The paradigm in physics is that one establishes foundational equations, such as Newton’s laws, Maxwell’s equations, Einstein’s field equations. Then by adding boundary conditions and doing some algebraic manipulation, a vast diversity of non-obvious and useful results can be derived.

To understand how this can happen, we need to under stand two things: the nature of inference, and the structure of physical theories.

At university, in logic classes, we learn inference syntactically. We are taught to start with a set of formulae (axioms) and then derive consequences by application of the inference rules. But this barely passes the ‘so what? test. When you program a computer to do this (an automatic theorem prover), it simple generates billions of useless formulae.

The semantic story of inference is this: we operate in the model space, not the formula space.

There is some intricate structure which is too complex for the unaided human imagination to encompass in its entirety. The magic of mathematics ensures that if the axioms which describe definitional properties of this object have a completeness property, then everything interesting about the structure (everything “true”) can be derived via calculation, which is really inference. Gödel would say ‘almost everything’.

What we do in calculation/inference is move systematically from partial description to partial description (every equation is a partial description of the underlying structure) until we get to a description in the area of interest to us. The space of ‘interesting descriptions’ is a miniscule subset of the space of all possible partial descriptions, which is why automatic theorem provers haven’t replaced scientists.

The entities of physics (Newton's point-objects in 4D Euclidean space + forces; special relativity's objects and fields in 4D Minkowski space; configuration space in quantum mechanics, etc, etc) are impossible to visualise in their totality by human beings. So we have to describe them partially, a step at a time, using inference in the theories which capture their fundamental structure.

In telecoms architecture and design, we also deal in structures - objects and relationships. The structures are frequently graph-theoretic (nodes and links) and the attributes are things like protocols, QoS, traffic and so on.

However the abstract spaces in which these entities reside are simple by physics standards, and quite visualisable directly by diagrams. So we don't have the conceptualisation problem of physics and we don’t have the distance between foundational principles and phenomena-of-interest which have to be bridged by non-trivial mathematics.

My book did not, in the end, develop a deep theoretical structure. It divided the area into subareas, and dealt with each fairly directly.