The Mathematical Foundation of Mind
This article supports ‘Mathematics as the Physiology of Mind’, which
proposes a working mechanism in the void between subjectivity and objectivity.
The word ‘void’ here refers to the fact that no such mechanism for the brain
has so far been proposed and endorsed.
By subjectivity we mean the state of conscious knowing in a thinking subject, as distinct from objectivity or that which is known. To put it simply, the mechanism we seek is mathematica, or that which joins the sides, and which, projected into the world as known, is mathematics.
The context which allows this representation is the theory that the spheres of knowing and known, which constitute the mind, are reciprocally united and divided in a way that matches opposed aspects across the subjective objective divide. This is a splitting apart of something at root undivided. When describing the cycle of their relationship, we will, for the sake of system, adopt the convention of taking the side of knowing first, and move from there to the known. This is the order of objective idealism.
Further aspects are that mathematica on the side of knowing has as its foundation the ideal or perfect circle. The idea of perfection here echoes the scholasticism of medieval times, which proposed omniscience infinity and perfection as the three attributes of God. Here this ideal perfection is within the foundation of mathematics, reflecting its ideal or in-mind side, as distinct from its in-world projection.
Mathematics, being the same as mathematica but in extroversion, has as its foundation the ideal perfect straight (the Euclidean straight to infinity). In determining order we can see that as a circle increases in size its curvature becomes less.
This leads us to think that, taken to infinity any section of its circumference becomes or is straight. This is exactly what happens when the ideal introversion of (knowing) mind, in the subjective objective axis, converts into the projection of a known world. The perfect straight of the world is thus the opposite partner in counterpoise to the mind’s perfect circle.
The converse does not apply. Imaginatively taking the perfect straight to infinity does not bring us to think of it as circular. In logic, therefore, the pathway is from the circular to the straight and the circle is first in the hierarchy of our understanding.
The two sides (mathematica and mathematics), taken together as in a sense identical, and in another different, constitute the keel of math. With this preamble we can now reconsider our intent to build a functional bridge between subjectivity and objectivity.
We tentatively assume that the brain’s operation is mathematical, which in turn is two-sided, mathematica and mathematics; the test then being whether, in completion, our assumption is affirmed or denied. The plan of the brain’s function should come into view in the process, as represented in ‘Mathematics as the Physiology of Mind.’
The current lack of a global explanation for the operative mechanism at work between subjectivity and objectivity heads our determination to establish this bridge within the function of mind. To speak of mind is to speak of this mechanism which underpins the generation of thought. Focusing upon it brings the determination of consciousness into view, which defines the being and nature of philosophy. If this examination is pursued scientifically the result is philosophical science. To define mind we must look at its sides, which are, (a) memory, and (b) the thinking function of the brain.
(a) dovetails with the mind’s etymology, Old English (ge)mynd mind; related to Old High German (gi)munt memory.
(b) defines the role of memory in the life process of the brain, and the two sides together sustain our sense of the independence of mind.
Having found the division in mind, we now need to correlate it with an external partner. This brings us to Hegel’s speculative jump. It is a jump because it is something not at first obvious, but once named we cannot mention or think about it without its affirmation being confirmed. Mind’s external partner is mathematics, and as part of the same jump we must (or should) make a two-point landing: mathematica and mathematics.
Mathematica is the physiological process in the brain. It is not mathematics as such, but something deeper, the generative process that underpins thinking altogether. It is or ‘happens’ to be that whose reflection, falling upon the sensory process, has emerged over centuries of analysis of experience as that which know as mathematics. We see our self in other words, in the world we sense, and we see it there as that world's meaning.
This internal, not-sensuously-known form is based upon the perfect circle, not one circle but countless circles and their relations, in the ‘10 billion’ cells and their ‘10 trillion’ intercellular connections in the brain, and it stands in balanced relation to the mathematics that we have gleaned across thousands of years of experience.
The mathematics we know is the same, but as taught to and learned by each new generation. It is external in form and founded on the straight. It is mathematica’s shadow, an impress of the brain’s thinking process cast upon objectivity, upon what the brain projects, and we see as an external world apart from the psychical spirit of our conscious thinking being.
For centuries the heart was thought to be the center of the emotions. The great vessels, arteries and veins, along with the blood, were said to constitute a cooling system, conveying the heat that arose from the emotions, to the lungs, there expelled in the breath. The demonstration of blood circulation unseated this idea. The problem of the psyche still remained, and it was simply retired, without further explanation, onto the brain and nervous system.
The question then, is, “how does the brain balance our central knowing (and so adjust our behavior) to the state of the world that we know in context of our sense appreciation?” Or to put it more simply, how does the brain think?
The answer will dot the ‘i’ of a controversy
that has whirled around our sense of self and reality for centuries. In meaning
the center of the storm has always concerned the nature of the self and its
world. In location it always refers to the polar opposites, subjectivity and
objectivity. Plato described it in his Timmaeus as, “The Battle of the Gods and the
Giants.” The latter pulled up trees by the roots to use as clubs and would have
nothing to do with anything they could not touch. The Gods, in response, fired
intellectual shafts at the Giants from the clouds.
The Socratic philosophy, as another front of the same battle, opens the
debate between idealism and materialism. Marx charged that Hegel, in putting
the mind first, “stood things on their head.”[i] Hegel, referring to his Science of Mind, had
already pointed out that, “for naďve consciousness, to give itself up
completely and straight away to science is to make an attempt, induced by some
unknown influence, all at once to walk upon its head.”[ii]
‘Mathematics as the Physiology of Mind’ sees the division between subjectivity and objectivity, not as a tearing apart, but the essence of unity. To ask, “which is first?” is like asking, “If you cut an apple in half which side comes into existence first? There cannot be a knower without a known, or a known without a knower. However, the problem is mathematical, and we can begin with the fact that mathematics proceeds in a given base, while the brain ‘thinks’ transbase.
This is not something we have to prove for it is a universal observation. The brain is no more preconditioned to work in a single base than it is preordained to expression in a given language. Children speak Japanese in Japan because that is what they hear, and the same for children born in other countries speaking their native tongue. Our language is not biologically predetermined, and it is the same with mathematics. The brain is pan-determined from the beginning, both as to language and mathematical base, and this observation will serve as a steppingstone as we begin to unravel the functional balance within it.
The transition
Every number in a counting line can be turned directly into a circle. Numbers invite external operations such as counting, adding to or taking away. Circles, as runs of numbers, invite perception of how such operations proceed in the mind.
Figure
2
The counting line 0 1 2 3 4 5… in Figure 2 is that of our ordinary numbers. The circles represent their pre-configuration in the mind. Each counting line number occurs in each circle, but it is obscured by the circle 0 (on a clock face the 0 is obscured by the 12).
Numbers are conclusions. Circles are the process of arriving, together with the result. Instead of 0 1 2 3 4 5… we have 0 01 012 0123 01234 012345… Instead of the ‘+’ and ‘–’ numbers or integers we have direction (clockwise and anticlockwise) in circles.
We now come to the big jump. To see how numerical operations do proceed in the mind, and taking account of the fact that the mind ‘thinks’ uniformly transbase, we can take each ‘worldly’ number, say the 2, 3 and 5 in 2+3=5, and express it, not in our chosen base that maps to our fingers, but in every base to indefinity. This seems awfully complicated, but we will then have it as the mind has it. To anticipate, the mind takes a snapshot of the whole as if through a thousand lenses, an explanation that finds its explication in the multifaceted insect eye, but of this later.
Now, we might think that expressing each term in an indefinitely large number of bases would dilute the result. Instead of an image of a plum or cherry we have an indefinitely large number, a veritable atomic reproduction of the same. Resolution comes in the fact that each of the many individualities we start with (and numbers are serving as our examples here), transforms uniformly into a precise clockwise and anticlockwise direction in a circle. For the onlooker this is a complication, but for the mind engaged in the process it is the basis of resolution.
Examining the way successive numbers feed into a multibase array, as described in Mathematics as the Physiology of Mind, reveals that exactly the same content appears on one side as qualia, and on the other as materiality (read, “on one side as ‘non-iteration or subjectivity’, on the other as ‘iteration or objectivity’ ”). We know when a jigsaw puzzle piece fits into place correctly, even though the design and fit does not yet reveal the end picture.
The specific mechanism in the calculation-to-thought transition is not repeated here, for it is central to ‘Mathematics as the Physiology of Mind’.
Translated into multibase array, clockwise expresses as iteration, anticlockwise as non-iteration. Order in a sequence, which interprets as clockwise/ anticlockwise in a circle, takes over from the identity of individual expressions, be the latter numbers that lend themselves to tracking, or sense identities which posit the identification of kind. This is the bridge we need to cross the line from mathematics to memory-based discursive understanding.
This transbase expression finds a parallel in the compound eye of an insect, which is composed of tens or thousands of ommatidia, each comparable to an independent eye. The same qual or picture falls on every lens, but as these are modified parts of the insect’s exoskeleton the focus of each is fixed in relation to the others. The compound eye is thus set to scan the whole sky or entire scene. The same image falls upon each lens (iteration), while the difference (non-iteration or knowing) is geometrically set in terms of angular orientation. In this way the insect’s eye performs a task, which in larger creatures is assigned to the cortex of the brain. The advantage of this in terms of economy in structure and function to a tiny creature is self evident.
The transition between sense and understanding is obviously a sea change. Our approach to logic proceeds on mathematical rails, but the train itself, our understanding, requires us to interpret the results in terms of physiological meaning. We seek, not numerical sums, but guidance as to how thinking proceeds. All we have to do (and this is not easy) is to rid our mind of the counting line as something useful in terms of buying and selling, estimating the passing of time etc, and think of it instead as non-iteration to infinity, which sums in the brain’s mill as knowing.
In essence consciousness hinges upon the difference between knowing and being. Along with knowing, gathered up in the non-iteration of anticlockwise order, there is iteration, the known in consequence of clockwise order. Mathematics is the perfection and systemization of this order, but we need to take it back beyond its abstract expression to the function upon which it reflects, the physiology of our mind and being as developed across (a) evolutionary and (b) personal time.
Iteration applied to the qualia of color tone touch taste odor, and balance builds our sense of a surrounding world in consciousness. The transition from order in a base to order across all bases is unconscious. We come to a table which is not only set but loaded, and this is the key to the different interpretations given by idealism and materialism on the subjective objective axis.
To sense something and think in its relation we only have to be awake. Materialism takes this as a free meal. According to it our sense of sight, balance etc, is given to us along with our birth. Idealism, as in philosophy and science says that there is no free meal. Everything has to be studied and understood from the beginning, starting from zero.
First there is unconscious brain function, then the creation of a sensed and seen world, a process and production that begins when we are conceived and born. We then take what we find in consciousness and make it our starting point for knowledge, of who and what we are, where and when. This is first. The Giants win against the Gods, Marx against Hegel. Given the establishment of a world, we then begin to learn how we, as thinking beings exist, and what we are as body and mind. But chromosomes and qualia, nervous system and brain were there first, and in this respect the Gods win over the Giants, and Hegel over Marx.[iii]
We are on the high seas now. The slightest storm could collapse our frail ship. We need to touch base, to see how mathematics in the operation of its two sides links the ‘clockwise iteration known’ with the ‘anticlockwise non-iteration’ act of knowing.
Here the doors open and the void gives way to a chasm, for there is a tremendous amount of detail to be filled in, but we can see that it has two sides and is technically negotiable. The needed bridge will be a vast structure, but it will span the gap knitting mathematics and the physiology of mind together, and the function of the brain in all its detail.
Mathematics will disappear in the physiology of mind, only to reappear as the latter’s presence. Simultaneously, human knowledge will come of age in the amalgamation, and this alone will give each side the foundation it requires for the whole to be navigable.