Steven Ericsson-Zenith will be speaking on his approach to computation and logic in England during June 2012 at The Incomputable conference (12th to 15th) and at Computability in Europe (18th to 23rd). He will also attend the ACE Alan Turing Birthday Party also at Kings College, Cambridge.

Steven spoke at the Stanford University Mathematical Logic Symposium on May 8th, at the Stanford Mathematics Department. A video recording of that talk is available to IASE access subscribers.

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The elusive mechanics of natural logic is by definition the mechanics of biophysics, describing how sense is characterized and how the biophysical structure is moved from apprehension to action. I will argue that this mechanics is fundamental to the inquiry of logic, determining the natural laws of logic, and that it is time for logicians to return to these foundational issues as theoretical biophysics, a field in which a wealth of new data promises to inform us.

I present the current state of my inquiry: a new logic and model of computation based upon the function of flexible closed manifolds describing how sense is characterized, symbolic processing, and covariant response potentials, the analogs of biophysical cells and multicellular membranes and their associated mechanics. The mathematization of this approach formally requires a unification of logic and geometry. I will present steps toward the specification of such a logic and its geometric implementation in dynamic structure designed to enable the explanation and reproduction of biophysical function. And I will speak to the predictions of the theory concerning the mechanisms that remain to be discovered.

"Logic And Computation As Biophysics" by Steven Ericsson-Zenith is the abstract for a presentation at the Stanford University Mathematical Logic seminar on May 8th, 2012.

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Is the logical assembly of the parts sufficient to determine the whole? Or is it, rather, the inverse, that the parts are determined as logical differentiations of the whole? And if so does anything remain of the whole that is not characterized by the logical relations between the parts? Are these two views equivalent, i.e., does the fact of the matter make any difference to logical results?

These questions relate to the mechanics of the structure that is the natural basis of logical operations. A way to view the question is to ask if this base mechanics is either differentiation in dynamic manifolds of sense, as suggested by our biology and experience, and in which the mechanisms of sensory characterization may play a role; or the construction of discrete logical parts and the relations between them. Before the current era, resplendent with computing machinery constructed upon the latter view, versions of the manifold argument were often preferred, notably in the algebraic logic of Peirce-Schröder.

The necessity to provide a computable explanation of sensory characterization and bound response potential in biophysics justify a review of the matter. A review is further justified by the search for a computable solution to general recognition, a ready operation in biology that is imperfect and costly, and may be intractable in its general form, in current computing architectures.

Central to the question is the nature of the inference mechanism determined by the basis structure of logical systems. Can we say, for example, that presented with A and B that the logical combination of the two loses nothing of their natural implication? Can we extend such a claim when presented with A and B and C etc..?

We may also question whether it is always reasonable to assume that the logical product of A and B combined is immutable in the presence of all other terms? In other words, are logical compositions side-effect free as the atomic view insists, or are there hidden factors that are not accessible to it?

In the atomic, analytic, view all derived logical values have equal standing to their premises and a logical expression describes the composition of atoms. The interpretation of combining these atoms in a logical expression is either independent of any a priori binding or it is considered to capture or impose such a binding.

The manifold view can be given clear geometric correspondents, “shapes” ( “symbols” or “signs” ) upon the manifold characterize the parts. The manifold provides a natural, continuous, and unifying dynamics binding them, inference in this view is a transformation of the manifold. We must consider therefore whether the atomic view is sufficient to characterize the behavior of such manifolds and to do so we must identify the difference that this makes to results.

Certainly, Charles Sanders Peirce (1839-1914) did not consider the atomic view sufficient, unlike Frege he was not concerned with mere statements of truth but rather the differences the apprehension of such statements identify in the world, and he was critical of directions being taken by his contemporaries (notably Bertrand Russell) that ignored these considerations. Of Russell's algebra, as of 1904, Peirce wrote to Victoria Welby:

❝ The criticism which I make on that algebra of dyadic relations ... is that the very triadic relations which it does not recognize it does itself employ. For every combination of relatives to make a new relative is a triadic relation irreducible to dyadic relations. 
Charles Sanders Peirce. Letter to Victoria Welby. (October, 1904)

Peirce is essentially arguing that the interpretant manifold of the logician (called “thirdness” in Peirce's vocabulary, hence “triadic” ) is a necessary addendum for the correct interpretation of such logical text, i.e., the effective results, to be viewed as behavioral outcomes according to Peirce's Pragmaticism, between a purely mechanical interpretation of the dyadic relations according to the principles of modern computable inference differ from the natural manifold interpretation because there are factors inaccessible to the interpretation of purely atomic (dyadic) relations.

Christine Ladd-Franklin (1847-1930), Peirce's doctoral student at John Hopkins University, continued this criticism against the “Russellisation” of logic long after Peirce passed. We will explore Peirce's view in some detail and briefly discuss the contributions of his students.

In this presentation we will consider the question and the diverse historical views related to it during the period from Charles Sanders Peirce (1839-1914) to Alan Turing (1912-1954), especially including a review of the contributions of Ernst Schröder (1841-1902) and Rudolf Carnap (1891-1970) compared to Gottlob Frege (1848-1925), Bertrand Russell (1872-1970), and Alfred North Whitehead (1861-1947), and the influence of Clarence Irving Lewis (1883-1964) and Cooper Harold Langford (1895-1964).

The above is the abstract of an invited presentation for the Special Session "The Universal Turing Machine, and History of the Computer" of CiE Centenary Celebration of Alan Turing to be held in Cambridge, England in June 2012 and fortcoming paper: "Conceptions Of Locality In Logic And Computation, A History" by Steven Ericsson-Zenith. It provides the historical context for logic as manifolds v. logic as atoms.

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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.

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.

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.

Th presentation is an invited talk for the conference The Incomputable, an Isaac Newton Institute event to be held at 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."

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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 that characterize 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.

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Three theories

This work is concerned with the development and operation of sentient biophysical structure. The vehicle of our inquiry is an investigation into the foundations of logic and apprehension with respect to the mathematical characterization of such structure, its behavior, and its computable reproduction.

Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. By extension our inquiry brings together the traditional concerns of cosmology, computation, and epistemology.

Underlying this investigation is the broad range of contemporary biophysical observation and experimentation. Much of this observation and experimentation was impossible before the current era. The results are voluminous and often narrowly specialized. Theorists have, as yet, had little time to consider the broader implications.

These theories are based upon a new explanation of experience in nature, the construction of senses, and the operation of spontaneous biophysical behavior. This new approach is developed from first principles to enable a rigorous and systematic explanation of the variety of associated behaviors. 

The nature of our inquiry

Alongside this development is a further inquiry that focuses upon the nature of our work. It discusses the existential aspects of scientific inquiry, its epistemology and logic. It seeks to clarify the nature of the mathematical characterization and computation of natural behaviors, dealing with questions in the foundations of logic. It explores methodological issues related to reduction and the refinement of ideas from intuition to formal logical structure.

This second inquiry is the necessary complement to the first because it is an explanation that deals with its own foundation.

A calculus for biophysics

In support of this broad inquiry we work toward the development of a calculus for biophysical construction and its dynamics. The focus of this calculus is the structural dynamics for the range of single cells, multicellular architectures, and membranes. In our model it is the shape of these biophysical elements that characterize sense and modify action potentials producing motility, if successful this mechanics mathematically characterizes sensory and motile behavior.

Upon this foundation we propose a model of apprehension and explore how its products are processed by the organism. Finally, we propose  a probabilistic theory that enables us to reason about inaccessible factors in group behavior.

Three mathematical approaches

We follow three mathematical directions in anticipation that they inform each other. The first of these is the simple assertion that the basis of sense and spontaneous biophysical action is universally present and by this simple presence structure assembles against it. This approach can only serve us if the mechanism characterizes a structural dynamic that is a consequence of this presence. The second direction is more conventional and follows a similar line of reasoning to the first except that it suggests the mechanism is the result of a covariant field effect upon the geometry of closed structures.

The third approach is radical and a purely mathematical exploration. It argues that the effects we seek to characterized have a natural mathematical basis and that if we eliminate naive assumptions concerning apprehension from a logical geometry then a characterization of the effect will suggest itself.

Rejection of emergence theory

You will note immediately that our approach is differential upon closed smooth manifolds and not founded upon a discrete particle theory. This is due to a recognition that neither construction from atoms nor the magic of emergence are viable existential explanations of our continuous and unfied experience, of sense.

A new computational mechanics

The mechanics we propose 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. Symbolic processing in the biophysical system is storage free and the capacity of symbol representation is combinatorial across dynamic sensory manifolds, suggesting general engineering principles that offer significantly more symbolic processing capability in biophysical architectures than previously considered.

Proof in practice

We identify opportunities for experimental verification of the theory and we suggest a proof of our results in practice by the identification of this mechanism, allowing the construction of machines that experience.

Explaining Experience In Nature: The Foundations Of Logic And Apprehension is a series of technical volumes authored by Steven Ericsson-Zenith and published by IASE. Available by subscription.

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Dr. Steven Ericsson-Zenith - Principal Investigator, IASE

Bob Krysiak - Corporate Vice President, STMicroelectronics

Prof. Suresh Jagganathan - Professor of Computer Science, Purdue University

The board of directors provide oversight for our efforts. We welcome expressions of interest from academics in all disciplines and industry leaders. If you wish to put yourself forward as a potential board member or member of our scientific or development advisory committee please contact one of the board members and ask about opportunities.

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