Someone asked me, “How is it that babies are not born drowned, because the womb is full of water?” When I said they breathe through the umbilical cord the reply was, “I know that, but they are immersed in water, so at birth the lungs will be waterlogged.”
I had to explain that before birth the lungs are not air-expanded but solid flesh. There is no air or air space between the cells. They do not become sponge-like until the baby takes a first breath. There is no place for ‘water’ to get in until the first breath inflates, and so creates the irreversibly expanded air-filled lungs, as we know them. Once this occurs there is no alchemy in heaven or on earth that can reverse the process, returning them to their original solid state.
Apply this idea of transition to the mind. The brain is ‘solid’ against sense until it is ‘ventilated’ by the first imprint triggered by birth. Keeping the terms of the analogy in view, an ‘unexpanded’ system of knowledge in the form of a ‘solid logic’ exists a-priori in the brain. We are comparing then, the solid state of the unexpanded fetal lung, which on taking breath becomes the baby's cushion-like air-filled lung, to the genetically formed logically ‘solid’ brain of the fetus. As taking breath expands the lung, whose pre-existing form becomes the basis of function, the birth experience awakens the brain to that two-sidedness wherein the receptivity of individual mind develops; in each case a necessary and irreversible transition.
The comparison allows us to see that nature, in the form of a genetic predisposition provides the functional basis of breathing and thinking alike, which nurture then proceeds to develop. This ‘solid’ pre-packed mind, equivalent to the unventilated fetal lung, is exactly the content explored in the Hegelian philosophy, a ‘tree of knowledge’ wherein every logical relation is wrapped up in conjunction with its opposite, and the whole consists of (1) an unconscious foundation, (2) an interface, which as the subconscious is the future mind, and (3) a coalface of sensory experience.
This remained a closed book to Hegel, whose philosophy neither intuits nor describes this transformation in physiological developmental terms; nevertheless his analysis of logic as such establishes the precondition upon which our understanding can build. Note that between the unconscious and conscious, the subconscious belongs with the unconscious; in other words the embryonic mind is deep before it becomes focused on a world, so {(unc subc) cs} until the whole translates into consciousness.
We can also consider the unconscious separately, as distinct from the subconscious and conscious, so {unc (subc cs)}, or the subconscious by itself {unc (subc) cs}, where, jam in the sandwich, it is ‘pure mind’. However, brought into this kind of focus it becomes a special area requiring our whole attention, until we understand all the frames of mind involved in attitude and thinking.
Hegel's absolute is the mind unexpanded by ‘ventilation’ with sense, which remains as an unconscious foundation even after this expansion. Comparable to the first breath that expands the lungs, the mother-baby relation, known as bonding, fills the mind, giving it a malleable two-sided aspect. The process is imprinting, as studied by Konrad Lorenz. It was unknown to Hegel, so it is the missing side in his philosophy, whose other defect was his inability to resolve the mental state that provides our understanding of mathematics. His philosophy remains however, our base and foundation.
Ventilation for the lungs, like diving into a pool, is sudden. In the case of the brain the process wherein two sides, mother and baby, come together in the foundation of the generic mind (more fully covered in my manuscript BIRTH), takes as long to develop and more than the nine months of gestational development in the womb, following the ovum sperm coming together that defines conception.
All things to which we give name are subject to number. In this sense the name comes first and number is tagged on as an afterthought. Nevertheless, the entire world in the atomic conception is depicted as numerate in form, so that mind itself is but the surface appearance of a numerically impelled or mathematical undercurrent. In this greater picture there is mathematics applied by the mind, and in another form, the same mathematics, but as constituting the mind. This calls upon us to adjust our ideas from the bottom up to conform with the new model emerging that will find its ultimate roots in neurology.
Conventional mathematics is not prepared for this revolution. It is unable to explain the most simple mathematical relations to children, such as -1 x -1 = 1, for not having the circular counting line it does not grasp the base-determined significance of the - sign to the left of 0, as in -3 -2 -1 0 1 2 3. It is thus an incomplete system full of contradictions that come to a head in the infamous ‘square root of minus one’. This is briefly discussed below. It struggles to compensate for its superficial theoretical grasp by asserting an overweening dogmatism, which it endeavors to hide, and battens down the hatch against the evidence of its deeper in-mind foundation.
If we take the form of numbering ascribed to the mind, which it initiates in its process of shaping its objects of thought (so a chair is ‘one’, a cow is ‘one’ and so on), then this quantifying expression, which folds quality into quantity, is the foundation of all mathematics. Since the mind as the function of the brain is active in the formulation of every ‘one’ of thought, it must itself be mathematical in nature. This, its quality, which is directly quantity, springs from a chromosomal bed, which creates in the brain's output all mathematically subject expression. An inner order and sequence transforms in this way, by design into a mathematically intelligible world.
We have come to the stage where we have to challenge the superficial view of mathematics, which limits it to a cabalistic cult of signs that shuts out the light of its real foundation in the mind's process. To this end we must turn to the mind’s ability to conceive the circular form as the most simple of patterns. We know that children just out of infancy can recognise and reproduce a circle. It is shown along with the spoken word “circle”, and later, when this is repeated along with the presentation of an array of different forms, including the circle, children correctly identify the circular figure.
At this early age, the child has no comprehension of the circle’s definition, for this is a derivative construct known only to the mathematically trained. The circular form, visualised in an object or drawing, is inherent, and the mind’s ability to grasp its essence is there first. Inherent, inherited, it is there before the first sighting of a circular object.
We further know that the filigree network of nerve cells in the eye, immediately behind the photoreceptors (rods and cones) that comprise the retina, is part of the brain. This is so because the retina originates as an invagination of the neural plate that forms the brain. The fact that this network can respond selectively to the images of parallel straight and circular lines falling on the retina supports the thesis that these conceptual forms are built into the neural foundation of the mind at a most fundamental level. From this we can deduce that the circular form is primordial in all recognition.
Evolution, in foreordination of the natural mind, creates an experience-built ability to conceive; the reference here being to experience in evolutionary time ahead of the birth of the individual, while ‘to conceive’ refers to mental awareness or consciousness. This puts us on side with the concept of Platonic forms, taken as the existence of an inherited mind, no less evolution-dependent than the body itself. A child is born. Then comes a period of constructive experience in human life that in refinement leads to the formation and science we call mathematics.
This just says that we come equipped, not only with eyes ears fingers and thumbs (our ‘digits’), but the mental patterns required to use them. This ability constitutes a pre-formed or Leibnitzian harmony, namely an individual mind with a predetermined ability to share in a language and math once we know their unit meanings.
Circlemath is the key to unifying knowledge because it chooses the circular as the first form over the straight. Our current world gives precedence to the straight, and is therefore held in the grip of conflict. How is this?
There is a natural order in life logic and knowing. If it is infringed, the consequence is stress imbalance and fragmentation. The phenomenon has many faces, but to choose a primordial form we can examine the relation between the circular and the straight. Now, scientifically, math is the very foundation of order in knowledge and society, and in math, the distinction between the straight and the curved or circular is fundamental. It is fitting that we should subject it to analysis, but first a question. It may be critical in math, but does it interest society at all?
The facts of history will answer the question for us. For hundreds of years math has taught that straight is the criterion in all math and measurement, and it has clung to this view with tenacity. For instance, it holds and teaches children that the number counting line marches off, by positive and negative numbers right and left to infinity. This is the embedded idea of Euclid's straight, but we get a much richer field, far more satisfactory for all mathematical purposes if we take the counting line as a series of circles (for more on this see, ‘A Theory of Mind’, Introduction).
The Greeks held that the circular form, visible in the heavens, was divine, and so the foundation in all order, a viewpoint our modern interest in space travel is likely to bring into favor again. Nevertheless, until a few generations ago, the question was linked to the shape of the earth, said to be flat. Copernicus delayed publishing his findings until he was on his deathbed. Bruno, after him was burned at the stake. Galileo was tried and placed under house arrest; such was the vehemence that upheld the inviolability of the assertion of the flat earth and the sanctity of Euclid's ‘infinite straight’.
If circularity is primary, conventional math is incorrectly founded. This relation in the deep foundation of mind has a pervasive logical significance. If taken wrongly, all that rests upon it is affected. The math itself, so twisted, channels off into ‘i’, the symbol for an imaginary number, the square root of -1, impossible in that math, to which all subscribers must agree, in step with Dodgson's Queen in ‘The Adventures of Alice in Wonderland’, who insisted on believing three impossible things before breakfast. And this in the 21st century!
To digress for a moment, just to show how circlemath handles the ‘sq rt -1’ problem; -1 in 10-circle is 9. In 10-circle 3*3=9 and 7*7=9, so the square root of -1 in 10-circle is 3 and 7. Add these and they come to 10, which is 0 in that circle. These patterns, which hold in all circles and bases constitute in determinable form the inner framework of the mind's operation, linking knowledge and world in consciousness. Notice that -1 has a square root in some circles, but not in others. In this way the square root of -1 opens onto a whole new mathematical world, whose relations certainly have to do with the mind, but are not, for that reason, imaginary. This term is rather blown right out of the water, unless we take it as shedding light upon the mind's imaginative power, to conceive compare and anticipate relations.
We need to ask mathematics to show us the proof which it otherwise hides in a show of erudition, for to put the straight ahead of the circular is to reverse the order in nature and put rectitude, in the sense of coercion, before harmony and understanding. It asserts the world ahead of the mind, force over nature and man over woman.
Not mislead by assumed identities and arbitrary definitions, take a circle conceived as expanding from a point in the dimensions of a planar surface. As its radius increases, so does its circumference proportionally, which losing curvature tends more towards a straight line. This points to the straight as the end of that journey. The reverse process does not occur. If we start with a straight, and extend it imaginatively to infinity, it does not lead us to think of a circle. Quite the opposite! This logical imbalance where we might expect reciprocity will point us to the next step ahead.
At issue is the relation between an ideal straight and its corresponding circle. Each is a conception that we can clothe in reality, but which of the two is primary, and which is secondary? They are related, not only because the expanding circumference steadily approximates the straight, but because a circle, viewed edge on is already a defined straight. Both of these considerations and a third lead us to think that the circle is primary, for its destiny is the straight, while in viewpoint it is already straight, whereas the straight, in altered viewpoint, disappears in a point.
The third consideration is that, viewed in terms of curvature we have on one side an infinite sequence of curves (circle segments), and on the other only one straight. The straight therefore belongs to the family of curves as a special or limiting case.
In a logical sequence, one thing follows on another, and we accept this order on the authority of our inherent reasoning power. The result is as good as our understanding, but we have to have a starting point.
In this case we are endeavoring to scout the identity of circular and straight within the context of their difference, the supposition being that if we accomplish this completely we will also see that this identity in difference, and difference in identity also defines the transition between ‘in mind’ and ‘in world’, or subjectivity and objectivity.
A mathematical point defined as a location without dimension meets this requirement. It is a pure or logical conception; nothing more. If we want to give it the semblance of reality, we touch the compass point to paper in drawing a circle, and this satisfies our idea of the point's existence. We can assume the point as a logical beginning but we cannot do the same for the ideal straight or circle, because these, although mathematically related, are the principals being compared. Which of them, straight or circle, is logically prior is precisely the question that has to be answered. We will find the means to do this in the relationship itself.
So far we have seen that starting with the point, as the circle grows in size a given section of its circumference increasingly approximates to the straight. After the point, the circle is first, and in its process of expanding it approaches the straight as a result. Conversely, and contradictorily, starting with a conception of the straight (and all, points circles and straights originate as conceptions), extending it, even to infinity, does not engender the sense of a circle. We are left with the result that putting the straight ahead of the circle, making it the first principle of mathematics (strapped in with birch and cane), is a mistake as flagrant as teaching the number line as 1 3 2 4 5… and it signals only this, that compulsion, not reason rules the world.
The more, after Euclid, we extend our idea of an infinite straight in the vainglorious effort to link up with the idea of the ‘circle at infinity’, the more we are forced to the conclusion that we have no way of defining the straight at all. It is a product. It does not serve as a foundation. For the ‘straight’, we are forced back upon the definition of the ‘zero of curvature’ where ‘zero’ is adjectival and curvature is determinative.
Everything in the circle straight relation indicates a sequence that puts the circular first, and our question is answered. The circular comes first, and in Hegelian terms, the mathematical discipline that founds itself upon the absolute straight is ‘standing on its head’.
Beyond using a straightedge, one to make another, how do we know what is straight? A carpenter will use line of sight, but we cannot carry this over into science. A taught wire may appear straight, but it is subject to the influence of gravity as we see in the case of a sagging power line. This tells us that we would have to think away gravity, which also affects light, and this means thinking away the ponderable universe.
There is no natural straight; any more than there is a natural point or perfect circle. These forms are resident only in the mind, and in the mind the laws of logic prevail. When we appeal to logic it tells us that the natural order of these, in the mind, is point circle straight, as we have described.
The dependence of the straight upon the circular corresponds to the dependence of the world, as known, upon the mind. To reverse the order is to make the mind dependent upon the world, and cart before horse we are thrown into the vortex of the materialism/ idealism controversy.
Because conventional mathematics retreats into its castle of assertion, and denies that it has anything to do with logic beyond that prescribed by its own axioms and definitions, this approach, which puts circularity first and maps out a much greater domain as belonging properly to mathematics, will surprise many. In effect it takes mathematics, and with it all forms of knowledge back to its genesis in the mind, as this arises (we could say, ‘like a Phoenix’), from the body brain relation.
Our progress in the analysis so far has been from detail to general, small to great and finite to infinite. If we reverse this order we still come to the same ‘circle first’ conclusion; granted that, in all this, a logical, historical, philosophical etc., convention is assumed, always available to be stipulated or spelled out. A circumscribed area on the surface of a conjectural sphere at infinity is a plane. As the plane appears our sense of the sphere at infinity disappears. The two are logically exclusive in their conjoint status as reciprocally supporting opposites. Similarly, the world comes into view, or sense exists at the expense of mind, which vanishes as the context of that appearance.
Have the plane touch or cut a finite sphere and a point or circle results. Have one such plane cut another and the result is an ideal straight line, which in conformity with traditional mathematical thinking proceeds to infinity. Again the problem arises; how should the three products, point circle and straight be aligned in terms of logical order?
The plane will touch the finite sphere before it cuts it, so the point is ahead of the circle. The ideal straight line to infinity requires the intersection of two planes to infinity, each of which implies a pre-existing infinite sphere. In both cases, circle sphere and curvature antecede the straight, and our judgement has to be that circularity is primal, and that the straight, as derivative, is secondary. We must therefore pull up the anchor, which hidden in an assertive and shallow mathematics constrains all other knowledge to an incorrect and undesirable point of view.
© Stephen W. Taylor MbChB, 040216, Brisbane stetay @ bigpond.net.au
The article is problematic, but the point it is making is firm enough. The idea of a sphere at infinity stumps the imagination, because the word ‘sphere’ implies a limit, while ‘infinity’ excludes limit altogether. This note attempts to approach the problem.
What happens when the mind, either in its internal working or its expression, viz. in its subjective operation or the world this operation presents to our awareness broaches the idea of a ‘sphere at infinity’? We reach the impossible, and then mischievously ‘go a little further’. Surely, this is not allowed!
The answer is not quite the problem it seems. There is another: the transition from ‘in mind’ to ‘in world’, and vice versa. The two are comparable and linked. Accepting the actuality without the least hesitation we say, “I have in mind Hot Cross Buns.” We see the buns sitting on the bench and we accept that both aspects, ‘in mind’ and ‘in world’ are real. This dynamic reciprocal exclusion, each side of the other, constitutes the fabric of our reality. Drawn to the point of subjectivity it is our consciousness of a world. Taken to the other extreme it is the world seen as the foundation of our sense of conscious being overall.
When we speak of the sphere at infinity (the circumference of the Universe, not that of the Earth is engaging our attention), we face this crisis of contradiction (the mutual incompatibility of ‘sphere’ and ‘infinity’). Its resolution depends upon the fact that in our viewpoint we are standing on that other threshold, the switch between ‘in mind’ and ‘in world’, and the two are sides of the one holistic or greater reality. We must ask, however, can there be a holistic or greater reality? The statement, “In the beginning God created the heavens and the earth,” indicates that there is no need to reinvent the wheel; just fit it in correctly with aspects of our modern understanding.
The sphere at infinity bids to include everything known and unknown, everything in space and time, excluding nothing. The transition between ‘in mind’ and ‘in world’, conversely, is hidden within consciousness. It is the inner alchemy of that outer reach. As the key to a more concrete discussion of this, see ‘A Theory of Mind’, chapter 2, figure 9, as further developed in chapter 7. We cannot cross from ‘sphere’ to ‘infinity’ without first crossing from ‘in world’ to ‘in mind’. The one threshold and switch is the reciprocal of the other.
The impossible contradiction that meets us when we take our conception of pure curvature and pure rectitude or straightness to the astronomically great ‘sphere at infinity’, where at the same time in the same sense we have to determine that an infinite plane and an infinite sphere are identical, is the same contradiction that we take in our stride all the time when we say that mind and matter are each a side in our immediate sense of reality in consciousness.
The philosophical approach germane to this situation is already extant in the Hegelian philosophy. We are there. It stands before us, needing but development in the context of our more advanced age. The old math, lacking insight and taught under punitive control is deeply implicated as the principal offender retarding a broad advance of knowledge on a wide front. Circlemath, or mathematics shifted bodily onto a circular foundation provides the vision needed to bring about the desired change. On this path, which is within our reach, we can complete the sphere of our understanding.
Brisbane
Australia
Comment query response etc. welcome.
stetay @ bigpond.net.au (omit the spaces which hide the address from automatic spammers)