Mathematics as the Physiology of Mind
This essay regards mathematics as an expression of the physiology of
mind.[i]
Coincidentally it proposes that knower and known are aspects of but one and the
same consciousness in a formulation that provides filling in two areas that
otherwise stand vacant; the physiology of mind which gains a content, and the
nature of mathematics, what it is ‘in itself’. Awareness of this vacancy has
impinged upon scientific consciousness for more than fifty years. The attached
quotation from "Qualia" by Michael Tye in The Stanford Encyclopedia
of Philosophy indicates that it remains in place as current as ever.[ii]
As long as mathematics and the physiology of mind remain unconnected,
each is condemned to remain wanting, mathematics as an immense jigsaw puzzle of
articulated relations whose identity is not yet in sight, and mind as a
presence that, for want of resolution, confronts the earnest student as
impenetrable as a bank of fog.
The task is to show that mathematics is the physiology of mind
and the agent of coherence in the logical function of conscious life. In this
we will link calculation in a base, to thought as the function of the brain
that proceeds transbase.
The subject in view is pure mathematics, known to the Greeks as mathematica,
in distinction from the same as applied to the things and thoughts of the
world, or school mathematics as widely known. Our focus will be upon the
structural bridge between the sides.
We begin then with an unknown, the relation between mathematics and the knowing brain, and our determination to forge a path between them. A question mark falls on the word knowing. What is it to ‘know’? We understand the physiology of every tissue in the body down to the cellular and atomic level, but the labyrinthine pathways in the brain, including the principle that operates within thought has itself remained a closed book.
Thesis
The world we know is subject to a mathematical interpretation, not because it has been trained or coerced, but because, formed by and channeled through the mind it is stamped with the latter’s impress. This can alert us to the fact that mind is a mathematical product. What better then, than to proceed directly to show this?
In doing so, our starting point is our knowledge built upon the math of childhood. To reach our goal we will have to move beyond this to take in the precepts of circlemath, the form that indwells the mind naturally, in whose light the objective form exists.
To go straight to the point, it all begins in the perfect circle, which as a primordial form is intrinsic to every circular expression.[iii] Its undetermined circumference is composed of contrary moments, clockwise and anticlockwise, each of which, in appearance, calls upon the other as its enabling background.
Figure 1
Figure 1 shows 0-circle as the superimposition of clockwise and anticlockwise directions. It has no numerical element, but it already contains the implication of sign, which will continue to feature throughout mathematics. In terms of meaning it thus breaks into two, but it was already a duality, namely an intuitive form that exists in the mind, and the same as determined to expression.
The counting line of counting circles
Number is organized in a twofold way. The first is superficial within the counting line of a base. The second is deep, across the counting lines of all bases. Action within a base is calculation; that across the bases is thinking. Calculation proceeds consciously from number to number within a base. Thinking proceeds unconsciously transbase.
We calculate in a single base, but in the cave of its neural bed the mind thinks transbase. It is no more predetermined to a given base than to a specific language. The inherent and the learned are two stages of one thing, which reciprocally balanced within itself is mind. The question then arises, how do we step through the Carrollean looking glass, from the world of objectivity into the subjectivity of mind? It is easier to do than to explain, so we will just proceed: turn each number into a circle.
Our license for this step resides in the fact just stated, that mind is at home transbase. Numbers do not, in their ordinary expression, suggest a base, but each circular form immediately discloses the set of ideograms that belong to the base it founds. Circularity not only tells us how the mind ‘sees’ numbers; in terms of order it is the medium par excellence wherein mind subsists and exists.
A diagram of the counting line of counting circles follows.
Figure 2
Expanding each number in the ordinary counting line to show the circular form of its internal arrangement takes us backstage to see what number looks like in the mind. Figure 2 shows the formative circles of the bases from 1-circle[iv] to 10-circle.
In terms of thinking each counting number falls into every circle ‘to infinity’. Its negative counterpart, which in conventional math is independent, appears in the circle as a different but related expression. Figure 2 shows that negativity is simply an anticlockwise reading of positive or clockwise numbers. This is ‘in mind’ or thinking as distinct from ‘in world’ or calculation. ‘In-mind’ is primordial and primary; ‘in-world’ is derivative and secondary.
In terms of this difference, we can note results in successive circles such as:
for –1 0 1 2 3 4 5 6 7 8 9… and
for +1 0 1 1 1 1 1 1 1 1 1…
for –2 0 0 1 2 3 4 5 6 7 8… and
for +2 0 0 2 2 2 2 2 2 2 2…
These transbase sequences can be read directly from figure 2 for every positive and negative number in the circles shown, being relations that hold to infinity. Negative numbers give transbase counting lines. Positive numbers iterate, where iteration means ‘falling transbase into the same relative place over and over again’, and this divergent pattern between ‘counting-line non-iteration’ and iteration founds the emergence of knower and known in the function of mind.
Leading digits
Tables 1 and 2 give the results for reading negative and positive counting line numbers into successive transbase circles. Table 1 gives those for the negative numbers –1 to –9. Table 2 gives the comparable results for the positive numbers +1 to +9. The column to the left (‘–1 gives’ etc.) is the feed. Reading the feed numbers into successive circles gets rid of the sign and creates each row of digits to the right, for example, 0 1 0 1 2 3 4 5 6 7… for –3.
We can call the diagonal row of 0s that descends from the top left down and to the right, the foundation. The block to the right is then the product, and to the left is the ground.
1cir 2cir 3cir 4cir 5cir
6cir 7cir 8cir
9cir 10cir
–1 gives 0 1 2 3 4 5 6 7 8 9…
–2 gives 0 0 1 2 3 4 5 6 7 8…
–3 gives 0 1 0 1 2 3 4 5 6 7…
–4 gives 0 0 2 0 1 2 3 4 5 6…
–5 gives 0 1 1 3 0 1 2 3 4 5…
–6 gives 0 0 0 2 4 0 1 2 3 4…
–7 gives 0 1 2 1 3 5 0 1 2 3…
–8 gives 0 0 1 0 2 4 6 0 1 2…
–9 gives 0 1 0 3 1 3 5 7 0 1…
Table 1
1cir 2cir
3cir 4cir 5cir
6cir 7cir 8cir
9cir 10cir
+1 gives 0 1 1 1 1 1 1 1 1 1…
+2 gives 0 0 2 2 2 2 2 2 2 2…
+3 gives 0 1 0 3 3 3 3 3 3 3…
+4 gives 0 0 1 0 4 4 4 4 4 4…
+5 gives 0 1 2 1 0 5 5 5 5 5…
+6 gives 0 0 0 2 1 0 6 6 6 6…
+7 gives 0 1 1 3 2 1 0 7 7 7…
+8 gives 0 0 2 0 3 2 1 0 8 8…
+9 gives 0 1 0 1 4 3 2 1 0 9…
Table 2.
In table 1, the leading digits, from the left, up to the foundational 0, push the lines incrementally to the right. This stagger creates vertical counting line sequences so that the whole block on the right becomes a counting line area, much as cross threads at right angles create a tartan cloth.
Table 2 shows that switching the feed from negative to positive switches the read from counting line to iteration. The descending counting lines connote the independence of different world objects. We are working with very small circles, but our observations, being universal, remain valid in the circle operations into the high numbers that we may conceive as necessary and adequate to the brain’s function, involving billions of cells served by trillions of interneuronal connections.
Our attention will be on the extremes, transbase iteration and non-iteration, as against an unresolved background pattern. The patterns so easily visible to us in the small circles are at the heart of the brain’s thinking and knowing process. If we can logically relate the iteration non-iteration patterns to the difference between knower and known we will have accomplished what we set out to do.
In Broader View
The same digits that, when taken positively and read into transbase circles stack identically, like on like, 1 on 1, 2 on 2 etc, taken negatively fall into the pattern of counting lines, excluding iteration ‘to infinity’. Tables 1 and 2 show this. The result is an ordered in-house differentiation setting the stage for the function of discriminating mind.
The distinction between non-iteration and iteration establishes the mind/ world interface, technically that between subjectivity and objectivity. To us counting lines represent a useful order in number, allowing us to measure and calculate, but in terms of cerebral function it is the exclusion of iteration that marks the insipient establishment of mind.
Iteration is the compaction that underpins our sense of physical reality. Non-iteration is the opposite extreme, which excluding any agglomeration of signal[v] is the retreat into invisibility and immateriality that we call mind, and with this the shell of the mind/ world relation cracks open.
We are seeking to relate observable behavior to the functional process in mind. Our focus can rest therefore on the physiology that underlies behavior’s production, relevant to the being of mind indifferent to its content. The fact that digital computers can reflect the entirety of our intellectual production in science and art greatly aids us in this, allowing us to set that social content aside as a block.
The fact that the entire content can be read into binary, subject to computer management makes it easy to see that organisms, from sea anemones to humans channel the stimuli that impinge them into memory-supervised response. We call discrimination in this response intelligence, whose organization we take back to mind, the latter being invisible in itself except in the outcome.
Genetic memory determines the form of an organism. Neural memory adds a learned component, and together they bridge the gap between mind and material being. We can then see that physiology is mathematically determined to the will of a knowing mind.
The Mechanism In Detail
Each side, non-iteration and iteration, or knower and known arises within the same mathematical formulation. The ‘mathematica’ does not fit the facts the way a glove fits a hand. It is the hand, the essence of mind. It is not attached externally.
Iteration is aggregation. We cannot see raindrops in the distance, but we can see a rainstorm descending from a cloud. We do not notice a speck of dust on the floor, but millions of such specks force themselves upon our attention. The same number dispersed as a film no more than a single particle thick over a wide area would retain the invisibility of the single speck, but spread out as a two-dimensional plane.
We will relate this later to two unlike items, the surface of the cerebral hemispheres greatly expanded by folding, and again to Indra's net of ancient Hindi religion.
The mind is a controlling interface, a monolayer expression that has retreated into invisibility. This gives mind its stamp, to see without being seen, a relation that begins to emerge in Figure 3 below. Positive numbers stack when read into a transbase array, as in table 2. This iteration or condensation and compaction across the bases represents tangibility, the form of shape and visibility of being that we call materiality.
The same numbers, taken negatively (anticlockwise), do not stack. Instead they form counting lines. This contrast, which is maintained transbase to infinity is the inner mathematical dynamic that underpins and founds mind, the relation of knower and known, mind and world. It sustains intelligent being, and in doing so it is the stepping-stone from physical to conscious being.
Iteration or transbase stacking is the essence of materiality. Non-iteration is the essence of mind. Mind ‘knows’ an objective content because it is this content, in reverse direction of reading, and at the same time an ‘other’, external and objective, party to the community of knowing only as a content.
The ‘a’ of ‘a + b = c’ in figure 3, shows the ten circle of the bases shown, Octal Noval and Denal (names modeled slightly for consistency), read positively, i.e. their numerals taken in clockwise order. Choosing three such bases rather than one helps to make the point that the secret and sense of the conclusions we wish to draw relate to universality, which is the stepping-stone from mathematical abstraction to the physiology of knowing. Staying with the universal allows us to jump at any time from the arithmetic of number to the physiology of mind. Only the universal therefore claims our attention.[vi]
Column ‘b’ in figure 3 is the same circles read negatively, i.e., anticlockwise. We then fit the negative readings within the positive as non-iteration within iteration. This gives the column rising above the ‘c’ of the formula ‘a + b = c’, the mathematically coherent transbase pattern in prefiguration of mind. The whole gives us purchase upon the ‘physiology and anatomy’, so the reality of mind, the mandelbrot of conscious being.
Figure 3
Only the ten of each of the three chosen bases is shown. Extend the series imaginatively from three to the infinity of all bases, and apply the ‘mind in world’ operation to their every number, not just the ‘ten’, and we have before us the means we need to understand the mind entire, in terms of its form and content.
Indra is the celebrated God of the Rig-Veda, said to govern weather and dispense rain. There is a wonderful net in her abode that stretches out in every direction. Its knots are jewels, each reflecting its every other. This fits the insect eye almost exactly, whose every facet reflects the same picture from a slightly different angle. Sense is read across the whole and the same principle governs the function of mind in all creatures. It is not difficult therefore to read the pattern of mind and its supporting consciousness into this description. The ancient Hindi religion is an early version of what today we call a theory of mind.
Thought and Calculation
Calculation is monorail. It proceeds consciously in a single base. Thought is unconsciously transbase. Mental function in the myriad pathways of axonal/dendritic interconnection between nerve cells is intrinsically circular, while shape color number quantity etc, are parameters in our discernment of identity in the external world.
The great distinction however, founding all others such as internal/ external, mind/ material, thought/ calculation is between subjectivity and objectivity. This brings us to mind itself wherein such relations occur, and the fact that in mathematical appreciation the sides can be bridged. Math, or more exactly ‘mathematica’ is the bridge. We can see this bridge as the unbroken continuity of contrasting relations in the geometrical circle, which tells us that mathematica and its external reflection, mathematics, is nothing but the physiology of mind.
Mind in turn is ‘nothing but’ the function of the brain, comprehended as logical process. Mathematics expresses the physiology of the brain, but the brain, under the microscope, appears no more distinguishable than the tangled roots of a tree.
Finding a path between mathematics and the brain's physiology then appears as a hopeless task. A lead however emerges in the fact that, in the context of thinking, circular geometry resolves into clockwise anticlockwise order. This corresponds to the great divide between subjectivity and objectivity, taking us to the shore of understanding.
The figurate elements in the circles to infinity, 0, 01, 012, 0123, 01234… then resolve into iteration and non-iteration, wherein the ‘tangled roots’ do resolve. Here as a bridge is the pathway which expands into an uninterrupted highway between mathematics and the physiology of mind.
Mathematics is the abstract image of the pathway between the physically existing brain and its working physiology. Our aim is to define mathematics as this abstraction of the neurology it reflects. Physiology begins in sensation, eye ear touch etc., wherein we appreciate our environment. This is ‘Indra's network wherein thought and action take purchase. The central nervous system dispenses thought much as the cardiovascular system dispenses blood. Each, then, is a circulation. The heart is a pump distributing fuel. The brain is a balance weighing impulse against impulse in the supervision of intention.
It remains to add that the mind, as the thinking side of the central nervous system consists of circular on circular function to infinity. Anticlockwise and clockwise, through non-iteration and iteration then mediate between observer and observed. Bedrock in this is direction, defining knower and known and this is all we need as a kind to head insanely for the stars!
Stephen W. Taylor MbChB © 2004 July 17, updated 05 May 10.