In his "Aufbau" Rudolf Carnap (1891-1970) proposed that logical operations are differentiation from the entire manifold of sense and proposed "recollection of similarity" as the basic relation. This view fundamentally differs from the conventional mechanics of Alan Turing (1912-1954), that treats computable logic as the mechanical integration of logical atoms.
This view, that logic be founded upon manifolds and not discrete logical atoms, is not new and was not uncommon before Alan Turing's Universal Machine model of computation became pervasive, popularized during the critical period founding mathematical logic by the work of Ernst Schröder (1841-1902), influenced greatly by Charles Sanders Peirce (1839-1914).
A common argument against the manifold view is that it makes no difference to computed results. Yet this argument is refuted when confronted by the challenges of general recognition and of locality in contemporary large scale parallel computation.
We present work toward the development of realizable mechanisms for computable logic based upon a re-conception of logic as operations of differentiation upon closed manifolds. This approach requires a unification of conceptions in logic with natural geometric transformations of closed manifolds, combining symbol processing with response potential.
Confirmation of these mechanisms may exist in nature, in dynamic biophysical structure. Our investigation is founded upon the premise that it is the structure of closed manifolds in biophysical architectures thatcharacterize sense and closely bind sense to directed response potential.
The presented exploration is experimental and purely mathematical. The approach argues that the effects we seek to characterize have a natural mathematical basis and that by the elimination of naive assumptions concerning apprehension from geometry a characterization of Carnap's basic relation will suggest itself. We take this approach because it is the action of such apprehension that is the subject of our exploration.
The resulting mechanics suggests the design and physical realization of a new model of computation; one in which structure and the concurrency of action are a first-order consideration. By this model symbolic processing is storage free and closely bound to response potentials, the capacity of symbol representation is combinatorial across these dynamic manifolds, suggesting general engineering principles that offer significantly more symbolic processing capability in biophysical architectures than previously considered.
Carnap, Rudolf. The Logical Structure of the World. Open Court (1928). ISBN:0812695232.
Schröder, Ernst. Vorlesungen über die Algebra der Logik. (1890) [Vol.I-III] (1910)
The above is the abstract of a presentation submitted (and accepted) to CiE Centenary Celebration of Alan Turing to be held in Cambridge, England in June 2012 and fortcoming paper: "Toward the unification of logic and geometry as the basis of a new computational paradigm" by Steven Ericsson-Zenith. It provides the technical details of one of our approaches in current research.