ATOM

Theory

Of Mind

Stephen W. Taylor

M.B.,Ch.B. Otago New Zealand

An Octanary Publication

Preface

Instead of illuminating the structure of the world, mathematics, taken back to its inherent foundation reveals the configuration of mind. In this form, it is metamath, not the ordinary math that applies to the world, but mathematics as a product, itself the focus of attention, not the agent but the object, math taken back to the laws of its origin, to the mind and neural function that produces it. Metamath is not concerned with where math is heading, but where it is coming from.

Two groups come to a river. Those in one group, mathematicians, take to kayaks and follow it downstream to the sea. Those in the other, metamathematicians, trace it to its source in the mountains, from cloud rain and snow. Pure metamath is neurology. The link between math, ordinarily taken, and metamath/ neurology is circle math.

Metamath wells up from unconscious metabolic and genetic depths that constitute the inherited ground of logic. Arising and ascending within the reticular system of the brain, this inborn logikos enters into relation and interaction with another tide of nervous energy whose loop includes the sensory system, inwardly monitoring the body, outwardly the environment. Their confluence is thought.

The two sides then, are the reticular system, and external to this, the sense-to-effector ‘reflex arc’ system, in large the cerebral cortex. Their interdigitation and interaction generates the brain's EEG pulse, the carrier wave of thought. The reticular system is central in the brain stem, from the medulla oblongata below to the thalamus above. The sense-to-effector arc system is central to the evolutionarily newer cerebral cortex. Such, in brief, are the interacting sides.

We will assume that consciousness is a quantitative system in balance, maintained by neural activity, for only as closed is its function accessible to philosophical analysis. Each side, reticular and arc, encloses and feeds the other, the two having evolved from an original sphere of nervous energy, something easier to visualize in primitive organisms such as the hydra. One side is foil for the other in the creation of self-adjusting activity. In our analogy, we can compare this with rain that descends, as against water that evaporates in the process of creating river systems from mountaintop to sea. A single process governs the complexity and diversity that results.

Awakened into action by an input from the deeper brain, —for circularity implies a neural input to sensory fields—a tide of nervous energy arises on the reflex arc side in the special senses, eye, ear, touch, taste etc. Conditioned by environmental impact it makes its way through the nervous system to the body's effectors, but it arrives in passage through the cerebral hemispheres only by leave of a reticular gating influence that facilitates or inhibits synaptic transmission. A state of balance presides at every step on the way. Reflex pathways carry electrical impulses from sense receptors to the body's glandular and muscular effectors, but without a reticular ‘lift’—a gating effect—no signal would cross from one nerve cell to another. This is the interactive theory presented here for consideration.

Comparing metamath in its cerebral arising with ordinary mathematics, metamath in its neural bed is circular in form. Circularity is its principle. In sharp distinction, rectitude, in the sense of straightness is the leading principle in conventional math. This difference sets mind and world apart.

As with rain, river and evaporation in a circular flow, we need to understand their relationships as well as their differences. How do the two principles, circular and straight, stand to each other? The answer is that the straight belongs to the family of curves. It is the zero of curvature, and as such, it belongs to the mind. We have it as an intuitive possession, but as it is the zero in a group relation, it is equally different. It stands apart, and we sense the straight as the foundation of a geometrical world in our surroundings. Chapter 9 discusses the circular and straight geometries together.

Pausing to take breath, for the sketch is compact, the mind is a generation between two poles. First is the zygote-formed newborn, shall we say the flesh; herein find the future unconscious. The other pole is the mother-centered care lavished upon this otherwise helpless creature. A powerful imprinting occurs as a hidden but intensive learning process that generates the subconscious foundation of the mind to come. This creates the new mind as a bipolar mechanism in the combination of these interacting sides, one given in the baby, the other co-opting the mother in the imprinting relation, along with all the psychological constants that will determine the mind's traits in the life to come. The destiny of the individual and the greater society is here determined.

Given these parameters we can fit the circular/ straight math relation into place, and even though it appears as no more than a filigree connection in this initial presentation, it lights the way through, even down to neuron circuitry, allowing us to see exactly how mind and brain relate. In the result, it is its own confirmation.

Externally, the straight is our sense of mind embedded in the world. The circular to straight relation is the inherent form upon which the mind/ world configuration builds, and within which the mother baby imprint takes effect. Every stage unites opposing extremes within the harmony of a presiding unity, and metamath, as the nether root of conventional math will illuminate this internal structure for us.

As the foundation of every measurement, the zero is synonymous with mind. This authority transfers to the straight in geometry. Mathematicians accept the zero's objective existence, an acceptance that requires an intellectual tour de force. To affirm the zero's existence is to affirm the existence of mind, and with it, our sense of the spiritual existence of the world. The chapters ahead will chart this relation in detail.

It is commonplace that the mind reflects the world, but the opposite is rather the truth. The world is the mind expressed. Reality confronts us as a world apart, held within our consciousness. The scholars of old who insisted that the straight is fundamental in mathematics were right. The perfect straight, like the perfect circle is ideal, but for that reason no less real than the world we hold in conscious objectivity. Comparing them will guide us through the maze of relations that connect philosophy, math and science.

The book invokes A Theory Of Mind, whose acronym (ATOM) suits, for it turns upon our conception of the smallest possible (indivisible) thing; that which cannot be cut and remain what it was.

Introduction

To bring the relation between math and neurology immediately to attention, in a number circle each numeral, as an index number, commands a field. Thus, the field of 2 in 4-circle (figure 1b below), is 3 0 and 1. Our postulation will be that neurologically, index cells in the cortex relate to fields of cells in other parts of the brain in exactly this manner, not because mathematics governs the relation between cells, but because the cell-to-cell relation in the brain is the template that creates our sense of reality and number in the first place. Regardless of the vector encased in our viewpoint, this index-to-field relation links neurology and mathematics. The said metamath and worldly math are sides in a single discipline. They dock as accurately as two spaceships built for the purpose, and the more we explore this, the more it sharpens into affirmation.

Once confirmed, this point will establish a bridgehead for the presentation of further work. We will see how the brain's mechanism creates the mind's presence, and this defines our goal and limit. Some dexterity, breaking the mould, is necessary to see mathematics as a product of the mind, rather than its creator. The task is not to apply math to the mind, as if the pot would make the potter, but to explain math by reference to the mind. The latter is the source of mathematics, which is an exteriorized reflection of the mind's activity. The mind illuminates the world mathematically. The math must return the favor. Consider the circular form (figure 1a):

Figure 1

It calls to mind a circle, everywhere equidistant from a conceived central point, but a microscope would reveal that it is far from being the perfect circle upon whose conception mathematical theory rests. We conceive such circles, along with other basic forms upon which mathematical theory builds. The conceptions are ahead of the drawings and diagrams, not behind them.

The forms pictured in mathematics exist in the mind originally. Minds initiate diagrams. Diagrams initiate thinking only in a return volley. Drawing them on paper, we can study, communicate and realize their significance. This ability to define the abstract, defines the human. The human mind sees itself in its own creation, and we call this reflection our understanding. In response, we build our understanding into the world, creating artifacts to our own advantage.

Two kinds of mathematical theory exist, straight and circular. The straight, which is conventional, takes its name from the straight, used as a noun in 19^th century school arithmetic for finite sections of the Euclidean straight to infinity. The circular, essential to our subject, refers to the generation of the straight in the mind, a generation that is as distinct from its product as the mind is from its world, and at the same time, as closely associated. Our task will be to elucidate this generation.

Repossessing the Euclidean Foundation

Euclid's first axiom states that every two points lie on exactly one line. Even though it does not mention it, taken initially and by itself, this axiom defines a straight line, for if it allowed curves, an infinite number of lines would pass through any two points.

Figure 2

In this way we see that the straight is firstly a conception, and secondly a choice, namely that of singularity as distinct from the many. In combination, the conception is rational, for it is subjective and infinite. It is mind-constructed, both internally in the course of nature (the human mind exists), and externally as an exposition or drawn line, something executed.

This, Euclid’s definition, is all we need for our conception of straight: the zero of curvature. It is not only all we need, but what we need, for it establishes the correct order. It tells us that the deal (practicality) is consequent upon the ideal, exactly in line with our experience, that we conceive the straight but nowhere find it in nature.

To find it in nature we would have to alight upon its actuality, and we are its actuality. The Euclidean description expresses this placement. To assert the straight is to assert the zero, and the zero is the authority of the mind.

Circlemath recognizes that the zero stands for the mind, at once the foundation of our freedom—for mind is freedom—as well as its authority and our humanity. To see this is to see that the straight, in the framework of geometry, is its stanchion of externality, and that it imparts integrity and independence to the world and reality to our sensory experience. On this, the human mind stands, but the mind, as well as depending upon this world, is also, in its imprinted, so baby/ infant level depth, its creator.

We may note in passing that the mind possesses a three-level depth: (1) a DNA metabolic level, (2) the birth baby infant level and (3) the learning experience from childhood on. The second level establishes the floor under the third, wherein we regard the world as a matter-based entity, independent of our conscious function. This material independence corresponds to the independence of our consciousness from the unconscious being that supports it. As matter is ‘inert’ energy (and its inertia is an illusion), so matter's independence from our conscious function is illusory. Science depends upon knowledge, and knowledge upon mind.

‘Straight’ in the world is a mental preconception. The statement (in analogy to a taut cord), that the straight is the ‘shortest distance between two points’ is not a definition but a corollary. Given the constancy of the speed of light we can sidestep it, substituting the least time for the shortest distance, but the error sidesteps with us, for we cannot abolish distance from the measurement of time. Astronomical observation tells us that light is not nature's straightedge. Empirical science cannot take possession of the mathematical straight without first coming to terms with the reality of mind.

Euclid's second axiom states that any line segment with given endpoints may be continued in either direction. This implies that there is no upper limit to distance, and mathematical teaching spelled this out as its emphatic doctrine until the 20^th century. This sense of infinite straight distance supports the idea of an infinite counting line. “No matter how big the number,” the saying went, “you can always add one.”

There is no error in this, but because the operation depends upon our action, it is conditional. It is not the absolute principle it intends to convey. It takes mathematics back to mind, which although infinite, still awaits our analysis.

Euclid's two axioms, in tandem, define three things; the straight, the counting line to infinity and Euclidean geometry. Against the latter, which is a spatially configured logical system non-Euclidean geometries now exist. These are the Riemannian and the hyperbolic geometries whose lines delineate a sphere in the first instance and a pseudosphere in the second. For further discussion of these, see Chapter 10.

Conventional mathematics pivots on the straight and goes back to axioms. Euclid's straight to infinity belongs to this axiomatic foundation. The new vision, which goes back through circlemath and metamath, does not abandon the straight, either in its foundation, or in the terms of its existence, but takes it back to circularity, and circularity to mind. The straight rules the world, and the circular rules the straight. The task now is to show how this takes effect.

Subjective and Objective Number

The physical world falls under the influence of gravity, whose polar representations are weight and inertia. Division in the psychical or mind world is into subjectivity and objectivity. These govern attitude and orientation in the domain of knowledge and understanding. To examine subjectivity and objectivity in number—an idea strange to conventional math—is to begin to examine mathematics within the mind's formative process, for subjectivity and objectivity constitute the mind's environment.

Subjectivity is centripetal. Everything belongs to the viewpoint of the subject, who is the sole criterion and authority in the world it anchors. Solipsism reigns. Objectivity, oppositely, is centrifugal. The subject abdicates responsibility. Accent falls upon the independence of the world; its things and events as these stand in their own right, in a viewpoint abandoned to another and others. Importantly, this subjective/ objective orientation also applies to number.

To see this, we have ten fingers and ten forms: 0 1 2 3 4 5 6 7 8 9. We call them numbers, but this is not strictly correct, for number means more than 1. The first two elements, 0 and 1, do not qualify. They are conceptual moments at work in the mind's construction of its number-based understanding.

We can demonstratively count our fingers: ‘1 2 3 4 5 6 7 8 9 10’. The focus is upon objectivity. We can also count the integral symbols, 0 to 9. The objective row, in figure 3, ‘counts’ the subjective row, 0 to 9, as, “1 to 10.”

Figure 3

It is clear that the numbers in the subjective row, compared with the objective, are pushed one place to the right, by placing the ‘0’, which stands for the mind, before the 1. The 1 is the 0's agent or universal determiner in the world. The 0's presence, in front of the 1 (figure 3, Subjective), skews the two lines.

The objective counting line now counts the symbols. The maverick item in the subjective line is the 0. In the objective line, it is the 10. The true numbers are 2 to 9.

The 0 in the subjective line represents the one who sees and counts the fingers, the spiritual unity of conscious mind, written down and counted as a symbol, smuggled into math as a tangible object. It becomes objective, and like all objects it acquires a meaning. Now the mind bequeaths meanings, which then belong to objects and events that we find in the world, but in this case the object (‘0’) belongs to the meaning, for it represents the mind, something not found in the world. ‘0’ means, ‘nothing there’!

This is because the ‘0’ stands for subjectivity itself. Its referent is the human mind. In its case centripetality prevails. As mental awareness matures into objective being, 0 becomes 1. The 1, as universal, applies to every world object indifferently. It mediates between the mind's subjectivity and the world's objectivity, and passes this on to the true numbers 2, 3, 4… Rectify the skew in figure 3. We get:

Figure 4

Remove the inessential numbers:

Figure 5

This shows the subjective/ objective lines in binary, the first base. The same in denary utilizes eleven elements: the 0 which stands for the perceiving mind, the 1 on the mind/ world interface, the ideograms 2 to 9 inclusive, and the compound 10.

Occam's Razor

The next step is to extend these observations to all numbers and bases along with their relations, including the whole of arithmetic along with its calculations. When we do this we find that, in terms of Occam's razor, mathematics assumes its simplest and most compact or economic form, when expressed in a pan-circular arrangement.

At the point of straight to circular transformation, mathematics changes step and falls into alignment with neurology, the science of its ultimate source in the human mind. Simply following the balance of the subjective objective relation will take us through the steps to the final transformation.

The subjective-objective counting-line skew outlined above is the first sign of the discrepancy between the way number is conceived in the mind and the way it presents in the world, a difference wherein (as the subconscious), all intelligence resides and all meaning and value is traded, visible in the otherwise totally abstract form of number.

In this transformation, we do not abandon the straight. Its place is secure within the new theory. The Euclidean straight is ideal, but not, for that reason, ruled out in an approach that affirms the centrality of subjectivity. Objective science requires its observers, who equally require their ideality. Zero is our symbol for the knowing mind. The straight, as the zero of curvature, carries this meaning into the world we comprehend, wherein it reflects as our sense of reality. This substantiates the world's materiality, the basis of our human consciousness and life’s certainty.

When we see that something is missing, and we know it to be so, we assign it to ‘0’. This expresses our knowledge of the missing object, and in counterpoise, our self-awareness in knowing. We know what is not there, and this knowing is sufficient to tell us that mind is the first reality in the realm of our being.

Beyond 0 and 1

Number sequences originate in the mind as circular sets. These disallow extension, in contradiction to the rule that “you can always add one.” The sets go back, not to axioms but to the mind's perfect forms. Mind, from Latin memoria, memor mindful, takes us back to memory and attention (as in the phrase “mind out,” be careful, or pay attention).

The brain, as a formed organ, is a product of genetic process, and this implies life and consciousness at work at every level from atoms to stars. The term ‘consciousness’ implies mutual influence, wherein everything relates to everything else in the sense of a grand circularity that works in and through all particularity.

Taking the circular and straight together, within mathematical parameters, the former belongs on the mind side of the mind-world relation. Circlemath is then an interface between the brain's function and the conventional mathematics that applies so well to the world of our practical experience.

In circlemath, the in-mind predecessor for our symbol ‘0’ is a geometrically perfect circle. This geometrical form is taken as representing our idea of 0 as this exists in the mind. Its acceptance then rests upon the agreeable result. Arithmetic, geometry and algebra coalesce into one subject, and with this, the barriers between math and philosophy disappear.

We learn math, its symbols and their relations in childhood, but we are born with the brain and mind to do so. The circular form is the indwelling ‘first’ that primes the understanding. Reasoned axioms come later. In other words, the comprehension that transforms our sensory experience into an intelligible world is initial and indwelling.

To touch upon some basic points, the true meaning of number is more than one. The first true number is therefore 2, but it is not typical, for it has 1, a non-number, as its upstream neighbor. The first typical number is 3, but it is odd. After 0 and 1, which set the pattern, numbers mount in even/ odd pairs. ‘4’ is thus the first fully developed number. It is even, like the 2, and it has a true number on either side of it, like the 3.

We cannot say that 0 is even. It is our symbol for the mind, and this is even in the sense that it is balanced. It is just as much infinity as zero, and the movement of thought is always that of a shifting state of balance between opposed tendencies and options. 0 is the herald of evenness, or divisibility by 2, the first and greatest tide of pattern that sweeps through all the bases. Next comes triplicity, and after that, each prime initiates a new pattern.

The eye, with its rods and cones, along with its retinal networks can identify vertical, horizontal and parallel lines. Taking this inherent ability as an indication, and extrapolating from it, we can hypothesize that the brain, dedicated to pattern recognition, can determine circles and all manner of forms in virtue of its own in-built zero patterning. Zero patterning here means, ‘an inherited matrix that supports every aspect of reality that we are capable of sensing and understanding’. To take mathematics back to its zero patterning is to take it back to that into which it resolves of its own accord, its original nature, undisturbed by any use or imposition. It will then prove to be the zero patterning of mind that informs our logical comprehension of an intelligible world.

Unavailable to our unchallenged sight, the zero patterns appear to us only when elicited by experiential data. Thus we see a full moon as round, but only because ‘roundness’ is built into our perceptive mechanism. We judge that an external thing is, or is not circular, or straight etc., but in origin, the forms are indwelling, products of neural activity. The logical axiom that relates a circular form to a radius and a central point is secondary. It is the result of logical thought. The primordial circles, and the patterns they initiate, are deep to this thought.

The full moon, its roundness and everything we see, think and suppose about it, including its externality and materiality must necessarily find its explanation in terms of our own being and consciousness. Mind and world are separate only in our judgment that the first is internal and the second is external. Making this decision is the work of the thinking brain, which science cannot leave out of account.

Conventional mathematics is quite correct in tracing its own existence back to axioms, which as postulations of thought are superficial, for it applies to the world, but this is consequential. Circlemath, whose native element is subjectivity, seeks its origins internally as an unfolding of mind from within brain-engendered depths. Given this internal source, circlemath identifies the circle as the original perfect form, from which the straight and all others are developed. It is not consequence but cause, whose ground we must seek in neurology.

Subjectivity and objectivity, ‘in mind’ and ‘in world’, box everything between them. Diagrams convey a sense of something in the world, as distinct from the indwelling forms, but without the latter, we would have no understanding. Circularity, as opposed to rectitude or straightness belongs to this polarity, which orientates us in respect of everything we perceive and conceive. Every meaning we hold in the mind is subject to the governance of this polarity. The first thing to assess in the analysis of an idea is therefore its subjective/ objective orientation, much as a navigator checks a ship's direction against a compass.

On a Wider Canvas

To extend the above associations from mathematics to mind, we can compare the simplicity of 0‑circle, its perfection coupled with its lack of content, with the state of mind in the fetus that exists at approximately six months in utero. This gives us a benchmark for the untried mind in another field, namely our life cycle. We are asking, what form does the mind take when, like a car at the end of the plant production line, it is complete but unused. Its cylinders have not fired. It has never functioned as a car. This is the mind's primal state, a piano never played, an integrity and perfection given, not tabula rasa but tabula new, to each oncoming generation.

The mind returns to this original state, recurring to it briefly in the sleep wake cycle. As a benchmark, a functional zero state of mind, its initial occurrence is in the life cycle as mentioned, but it then recurs in every twenty-four hour sleep-wake cycle. This fits the pattern of the sleep-wake cycle into that of our life cycle, and further, it allows us to see that the pattern of our thought follows in the same way. In the words