Dear Chris,
Of course I am modeling. I do not regard this as an epistemological error. In fact I regard the unrestricted identification of physical reality with mathematics as a serious epistemological error.
This does not mean that there are no places where the mathematics and the physical are indistinguishable. When they are indistinguishable, it means that there is an awareness present gluing them together and not distinct from either. I say this because, for me, mathematics is fundamentally conceptual, concept requires awareness/thought, and yet that thought almost universally needs to be grounded in a physical experience. There is another speculation that is very fascinating: that elementary physical entities are purely mathematical. One may find oneself thinking that perhaps the form and the content of an electron are identical. Take care and look at what you really mean. Maybe this is so and maybe you are just living in the model!
You say "I'm not sure you are doing anything more fundamental than "modeling reality"." If you allow that the model could be something like Escher's Print Gallery
http://www.planetperplex.com/en/item/print-gallery/
then I would agree with you.I suggest that a right way to look at the world is to understand that what observes the world is the world. Any attempt to articulate this idea is likely to be incomplete.
Just because one says that the world is observing herself, does not give one license to relax and stop working out whether one's meanings and sayings are well-formed. In fact it makes the articulation much more difficult since one wants to know whether one is finding out what is dependent on the point of view (the way the world is divided into viewer and viewed) or perhaps independent of that. But in in physical science we use all sorts of different viewpoints and we want them to cohere. Bohr and Heisenberg showed us that what were thought to be objective properties of 'physical things' were often the results of the type of splitting (the type of observation) chosen. I wish my understanding were universal. But what universality it has seems to come from hard work!
Ah! Now! Calculus of Indications. I do not say (or did not intend to say) that it is an extension of Boolean Arithmetic. It is a RADICAL COLLAPSE of Boolean Arithmetic.In Boolean Arithmetic there is a firm separation of the operator of negation, ~, and the possible values, T and F. In the Calculus of Indications the marked value { } and the operator (of negation if you will) are IDENTICAL. There is no distinction between the operator and the operand. Processes are Things and Things are Processes (there is only one Thing/Process at this early stage). The remarkable point about the formalism is that even though it is so collapsed, it is still possible to maintain sufficient distinction to capture the patterns of boolean mathematics and unfold them. It is possible to go from the collapsed state to un uncollapsed state by regarding the mark as an operator as in { { } } = (unmarked) and as a value. In fact the most profound equation of the Calculus of Indications is: { } = { }. On the left hand side the mark is seen to operate on the unmarked state. On the right hand side we just have the marked state. They are identical and they are distinct! Just so in observing 'physical reality', we bring forth 'its' properties.
You suggest that I move toward a system without axioms. Maybe. I like the assumptions to be as simple as possible. But you have to ask, if you think that we are getting toward physical modeling, whether starting with very simple structures of distinction will naturally lead to the complexities of physics. Can you tell a good story for this? Stories people tell usually use some accepted complexities. For example, we may take on qubits, and then we have taken on the basics of quantum theory from the beginning. There is a long story from a bit or a mark to a qubit. I think that looking at the mark with its structure of interaction (like a Majorana particle) is a good step to take before letting it become a qubit Qubit means superposition of possibilities and the apparatus of the complex numbers. I want to slow this down and look carefully at the Metaphor of the Imaginary. When you do this you find that the square root of minus one is a clock! But this is too much for one reply.
Lets end in the question of what is possible in 'modeling reality'. We are deeply involved in the models we make of 'reality'. The reality we come to observe and experience is a function of our models and these models (constructions of language and mathematics) are functions of the reality.
It is circular. In physics we demand repeatability and independence of particular points of view.
The insures that this recursive game of producing and knowing will appear as objective as it can. It also insures that there will be surprises. We shall not know it all at once or even in the course of time.
