In this presentation we propose realizable mechanisms for computable logic founded upon a structural theory of logic, sensory characterization, and response potential in closed manifolds.

Briefly, the mechanics of differentiation, in which the mechanisms of sensory characterization play a role, upon the surface of a closed manifold characterize logical elements (signs) and naturally covary with the mechanical response potential of the structure. The manifold provides a natural, continuous, and unifying dynamics binding these elements. Inference is a transformation of the manifold.

We suggest that these mechanisms are observable in nature. In biophysics it is structure and the concurrency of action that are first-order considerations. It is the shape of single cells and multicellular membranes ( “closed manifolds” in mathematical terms) that characterize sense and modify action potentials that produce behavior.

A generalization of the existing evidence suggests that symbols form directly upon the surface of these manifolds in cell and membrane architectures, the processing of which constrains biophysical action potentials associated with the structure. This close binding of symbol processing and action potential is naturally formed by the evolutionary process.

Symbolic processing in the biophysical system is profoundly efficient. Storage is free and the capacity for symbol representation is combinatorial across dynamic sensory manifolds. This simple efficiency suggests general engineering principles that offer significantly greater symbolic processing capability in biophysical architectures than previously considered.

In contrast, parallel computation as we understand it today is decomposable, a second order consideration of the Turing model. Parallelism can be semantically removed from computer programs with no discernible effect upon the results. Therefore it contributes nothing algorithmically, providing only performance semantics.

The parallelism that we consider here makes a difference. As in biophysical systems, structural parallelism is not decomposable without impact upon the results. It plays a role algorithmically, providing the mechanisms of recognition and memory in the surface conformations of the processing architecture. Large scale differentiation appears in the dynamics of these closed manifolds and result in measurable characteristic behavior suggesting new architectures for recognition and prediction.

Two opposing views concerning the nature of Logic will concern us. The first, represented in the variety of models of computation considered by Alan Turing[1][2], is the view that logical operation is the integration of symbolic elements. The second is the view, suggested by Rudolf Carnap[3], that the basic relation is “recollection of similarity” (recognition) and computable Logic is “differentiation from the entirety of sense,” in which symbolic elements are continuously bound by the originating whole.

Our goal here will be to show that these two views, and the realizable mechanisms that they represent, are distinct and that their operation produces different results. In particular, the models of Alan Turing represent a metaphysical view in logic that has no capacity for the basic relation of Carnap and results in prohibitive storage and value distribution requirements.

To effectively construct such machines we require the development of a new computational logic, one that deals with differentiation, structural conformation, and related action potentials. We will outline our first steps toward such a logic.

This approach suggests a new pragmaticist foundation for logic (and potentially a new mathematics to be built upon it) since it eliminates the integration of traditional truth values in favor of symbolic differentiation upon closed manifolds and the transformation of the associated structure.

References

1. Turing, Alan. On Computable Numbers, With An Application To The Entscheidungsproblem. (1936).

2. Turing, Alan. Intelligent Machinery, A Heretical Theory. (1951).

3. Carnap, Rudolf. The Logical Structure Of The World. Open Court (1928). ISBN:0812695232.

2. Turing, Alan. Intelligent Machinery, A Heretical Theory. (1951).

3. Carnap, Rudolf. The Logical Structure Of The World. Open Court (1928). ISBN:0812695232.

*This presentation is an invited talk for the conference The Incomputable, an Isaac Newton Institute for Mathematical Sciences event held at the Royal Society's Chicheley Hall in England in June 2012. The Incomputable*

*is a major workshop of the 6-month Isaac Newton Institute programme - "Semantics and Syntax: A Legacy of Alan Turing."*