Over 20 years ago I generated a web site with some ideas about time and consciousness. An understanding scientist named Julian Troianov wrote a kind review of the work.
Let me begin with citing the following words, regarding rigor and intuition from the new hypertext book "Now and Trajectory" by Dr. Ellis Cooper: "Rigor cleans the window through which intuition shines". In my opinion these words are not yet put into practice by the researchers in the field of consciousness studies and a lot of work in this direction must be done. I have my own intuitions about consciousness and since my first encounter with the relatively new branch in mathematics called Category Theory in which so many mathematical concepts can be described in an elegant way I was trying to make them rigorous. I must say that I am not a professional in the field of consciousness, but coming from the control systems community and having a mathematical inclination I am convinced that the word of mathematicians, working in pure mathematics must be heard in the discussion of the Mind-Body problem. The first and only attempt I found of applying the concepts of category theory in the consciousness studies is made by Dr. Ellis Cooper. The book is designed as a hypertext document utilizing the Windows Help format (.HLP). One can use the familiar buttons that appear in "Help" files such as "Search" button. Let me warn the readers that this part is really a hypertext document one can spend a lot of time on browsing it. The book is divided into two main parts, called "Person" and "Theory". In the "Person" part of this hypertext document one may find dated diary notes, which altogether tell the story of the author and his family.
The "Theory" part is of main interest for us. In the "Mathematics" section of this part one can find the "Lecture for High School Students" subsection, which consists of fourteen parts (talks). Talks from 1 to 7 are a very good introduction to the main concepts of category theory such as category, functor, natural transformation, limit, adjoint functor. I really enjoy reading these talks because they are easy reading yet with illustrative examples. They are designed to guide the reader to the final definition of the concept of topos (a cartesian closed category with equalizers and sub-object classifier). Unfortunately there are some typos in the rigorous definition of topos in the end of Talk 3, but the meaning may be easily restored.
Talks from 8 to 14 explain the essence of author's theory of consciousness. The central idea one needs to understand is the concept of "Now" which is inspired from the concept of the specious present as used by William James. The concept of "Now" as it is defined in the book is very general and on its basis a new rich and coherent theory of subjectivity is built. In Dr. Cooper's theory the "Now" is an element of a category called "Nature", in which all of nature is modeled. This category is a topos with natural numbers element and infinite colimits. Adding the axiom of infinite depth, which says that every directed relationship in Nature is also an element of Nature is crucial for the formalization of the intuitions behind the concept of "Now". This axiom allows also to define for every directed relationship its function and structure in truly categorical fashion. The capture in Nature of the idea that the directed relationships of a Now have subjective feelings is done in Cooper's theory in an elegant way, defining the functor Feel: Nature -> FEELINGS, where category FEELINGS is a free monoid. The definitions of object and subject in Talk 10, which are very general, are based on this idea. Now when one is equipped with Dr. Cooper's interpretation of subjective feelings he is ready for the "Law of Motion for Consciousness", which is explained in Talk 9 and says that consciousness flows to difficulty. The relation between the feeling of difficulty and the consciousness is well known to psychologists and many facts, that confirm it may be found in the literature. So the rigorous definition of "Law of Motion for Consciousness" that may be found with the appropriate diagram in Talk 10 is a valuable achievement. In the same Talk the following general problems are discussed: the problem of formation of an unified subjective time for the individual out of various nows, the problem of the physical basis for this integration and the problem of unifying the subjective times together into one objective time. These are really hard problems, but when the proper formal definitions are introduced the formulation of each problem becomes clearer and this gives the author the opportunity to share some thoughts about its possible solution. He introduces the element in Nature called Spacetime, which may have directed relationships to the nows of the subjects. Assuming a subject may postulate any such directed relationships, which are intuitively plausible, the author asserts that nature has enough room for disagreements. Postulating that for every subject (or object) there is a motion of the subject (or object) in the Spacetime the author infers that for the subjects that are not moving very fast relative to each other there is one objective time. In the general case the objective times of the subjects are related by the Lorentz transformations and the interested reader may check the Physics Theory section, where an original derivation of the Lorentz transformations can be found along with good introduction of the basic math used in quantum mechanics. After the Spacetime element is introduced one may understand Dr. Cooper's solution of the Mind-Body Problem, which can be found in Part 11 of the Lecture for High School Students. The author grasps the very essence of the Mind in his statement that the Mind is neither material nor immaterial, but it is a virtual machine and the patterns in his behavior are elements if Nature with directed relations to physical spacetime. The idea of emergence of patterns and sensitivity to patterns needs a the concept of a virtual machine as a pattern in the behavior of a machine, which may itself be a virtual machine. That chain ends in neural machinery, for which there are biophysical mathematical models, which are efficacious because physics is distinguished among mental models of nature in that it has objective procedures with numerical variables whose values are consistent with values computed from the mathematical models.
The author claims that there are numerous intermediate levels of virtual machinery between the physics of fundamental particles and their interactions at the base of the tower, and Mind at its pinnacle and the patterns of behavior of all of these intermediate levels, which include proteins, cells, neurons, and neural networks are described by systems of impulsive differential equations. The notion of a system of impulsive differential is used because it is a convenient means of representing mathematical mental models of units which time out, send signals, and trigger each other. A detailed explanation is given in the section named "The Timeout, Signal and Trigger Model of Parallel Computation in Neural Machinery". A formal definition of the concept of Neurality may be found in Talk 13 for High School Students and it surprisingly turns out to imply that in a way the "self" may be considered colimit on his flows. A reader with a mathematical inclination will see this definition in its proper categorical perspective.
Throughout the hypertext document the concept of input-output serial machine is used and the construction of an input-output machine in the Nature is explained in a subsection of the section named "A Mathematical Theory of Nature", where also the author asserts that every input-output machine can be simulated by a neural machine. Moreover using the concepts of compiling and imaging Dr. Cooper gives explains how input-output serial machinery may be considered to support an intuitively resonant and rigorously definable notion of emergent downward causation. In terms of the impulsive ordinary differential equations the author's explanation of the downward causation corresponds to the existence and uniqueness of solutions to a system of impulsive ordinary differential equations.
If the reader wants to see the descriptive power of author's theory put in action he must read Talks 13 and 14, In the end of Talk 13 Dr. Cooper's interpretation of the classic psychophysical experiments of Robert Effron may be found. The preceding two talks contain some introductory remarks on this interpretation. Talk 14 consists entirely of interpretations of various notions of temporality from the works of Julius Fraser.
If one carefully reads the book he will notice that Dr. Cooper's theory of subjectivity has one unique virtue of metaphysical value. Suppose a man (the author) has a mathematical mental model of nature, called Nature. Then if this model is what it is supposed to be in the author's theory, the Nature must have a model of him. Dr. Cooper writes that this kind of circularity is crucial to his "feeling of rightness", which I share too. In my opinion this circularity is due to the generality, embodied in the language used in category theory. One may suspect this language to be conducive to incorrect speculations about consciousness, but if rigorous arguments are given along with the translation of the intuitive notions in this language, the danger will be avoided. The book "Now and Trajectory" shows how this goal may be achieved, but yet an enormous work is to be done. I hope this book may give the consciousness studies community the foundations to build upon. The most controversial idea in the book in my opinion is the model of parallel computation in neural machinery. It is obvious (at least to Dr. Cooper and me) that the common notion of a neural network as we know it, is not appropriate as a serious foundation of the science of subjectivity. But I think that the arguments in the book in favor of the new model of parallel computation are too intuitive and it is easy for one to overlook the power of this new model and consider it to be a mere twisted variant of the classical neural network.
This book is unusual in both aspects of contents and form. The ideas presented are absolutely original. Offering connections between different fields of knowledge these ideas have the power of inspiration. The material is dispersed in many relatively autonomous pieces of information and this organization makes reading difficult, but in strange way it is an illustration for the theory the book teaches. I highly recommend everyone interested in the study of consciousness to buy and read this book and I hope a new exciting book from Dr. Cooper will be available soon.
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