Two Types of Mathematics

A collection of five titles

There are two types of mathematics. The first, in its natural form is present as the physiology of mind, which gives us the world we know. Our familiar mathematics as learned and taught then comes to the fore as the scaffolding that holds our known world together.

This duality within mathematics is the parent form of the duality that pervades all knowledge. The Bohr model as identified in particle physics could be the strongest contender for the most advanced form of this duality known. It invites a further step, namely to see that thought itself is objective. This is not new for it is the standpoint of religion. The whole however would shine with a brighter light if it were endorsed in science, not as a concession, which science does not allow, but as a fact and finding in the wake of examination and reason.

Dualism within the Unity of a Digit

The stage of ordinary mathematics is a counting line to infinity. The stage of circlemath is a counting line of counting circles to infinity.

To create the ‘counting line of counting circles’ we transform each number in the ordinary counting line into its base. So ‘2’ becomes base 2. ‘3’ becomes base 3, and so on ‘to infinity’. We then take the ‘below 10’ numerals in each base, and form them into a circle. So, 012 ‘makes’ 3-circle. 0123456789 ‘makes’ 10-circle.

We now notice that we can read each circle (and this is across all the bases), clockwise and anticlockwise. Clockwise the numbers are positive. Anticlockwise the same numbers are negative. When circle is superimposed upon circle to infinity like falls on like clockwise, and this condensation and compaction is the stuff of materiality and world. Anticlockwise the numbers do not stack, but ‘hide’ in counting line differences, ‘to infinity’. Combine the two readings and immediately we have the configuration of ‘mind in world’ and ‘world in mind’, which fits into the picture known in particle physics as Bohr complementarity.

Other significant consequences follow. For instance, every number, positive and/or negative is Janine in itself, an internal duality that fits perfectly into the function of mind. Then, because clockwise anticlockwise direction of reading is an observer function, this (observer function) is impressed upon all math-based physics, a point that has been a subject of debate for some years.

Stephen W. Taylor MbChB © 050513

Janine Numbers and the Rift in Science

Janus was the Roman god of doorways arches and bridges. Pictured with two faces looking in opposite directions he symbolized the ability to pass through an arch from one side to the other.

More profoundly this symbolizes intention in thought and action. For example, clockwise and anticlockwise, as opposite directions in a circle define materiality and mind, which as known and knower within a unity translate intention into activity. This in-mind process comes into play as thought when we assess the opposed sides of things and actions. Given this formulation we can begin to shadow-in the topology of mind itself.

The circles, which replace numbers as the ‘unit’ or entity in thought and mental activity, are through and through Janine in their clockwise anticlockwise direction, as are their strings, for those that signal the known, in reverse direction signal the side or aspect of knowing. Lastly, every numerical symbol is Janine in itself, because each serves in both capacities according to the direction of reading that falls upon it. Importantly, the direction of reading in the context of thinking and logic is a function of the observer.

To review this, for it dwells in the heart of mental function, numbers exist as members of short or long sequences, which in turn are circles or parts of an arrangement of circular systems. The numbers, whether taken positively or negatively are Janine in themselves, for in circular continuity each is a node point or hinge from which clockwise and anticlockwise direction springs. Their identity within the context of difference is their Janine nature.

That which appears in any instance depends upon whether the read is taken in a clockwise or anticlockwise direction. This co-opts the observer into the existence of physical reality, making what we see conditional upon what we are looking for. As shown in the Copenhagen interpretation, this reduces certainty in particle physics to probability, which does not prevent quantum mechanics from being the most successful theory ever, in describing Nature at its most fundamental level, from the structure of atoms to the behavior of stars. Beginning its life as the Heisenberg uncertainty principle, probability is simply the joint hinge or transition between matter and mind in the construction of consciousness.

The unity of mind within the pervasive duality of its Janine setting is preserved in the fact that the opposed directions are built multitudinously into the same circles. Every numeral in every circle is a hinge in this clockwise anticlockwise expression, each side a function held in place by another, sides that, as knower and known perpetually turn into each other in the ceaseless activity of the brain's function.

Finally, the function of the brain builds in the likeness of the right-angled form that serves for radiation. We can see this in the fact that in the analysis of a calculation a single sheet of paper is sufficient. It sits before us on the desk and we can draw our circles on its plane surface. But when we switch to the configuration of thought we see that it needs the superimposition of the same in all bases to infinity, for it operates 3D, and our paper morphs into the layered form of a book.

Meanwhile, the odd-even structure in counting lines comes forward as the mathematical explanation of wave particle duality which I have described elsewhere, but touched upon in ‘Light’, and more systematically in ATOM, a Theory of Mind. As a thumbnail, 1 2 3… falls on 1 2 3… in every circle in every base. -1 -2 -3… falls on -1 -2 -3… in every base. But -1 -2 -3… in Octal is 7 6 5… In Noval it is 8 7 6… and in Denal it is 9 8 7... In this we have the pattern of mind within the function of the brain.

The Copenhagen Interpretation

Building on Heisenberg uncertainty, Niels Bohr and colleagues fashioned the Copenhagen interpretation. In background, the question was whether materiality could be ascribed to subatomic particles (quanta). It found in the negative.

Analysis at this level is exacting. Its logic sets out from the Newtonian physics, which beyond our subjective assessment of reality goes back to universal relationships such as place position mass motion and momentum. In this context our ability to measure a body's position and momentum is not incidental, but rather that which qualifies an entity as an object within an observable community of objects. In the case under consideration the quantum is our conjecture, the unknown. Can we apply these criteria to it?

In early life, given the love and care of a mother, every object, beyond and impinging upon this, begins as a configuration in sense, an entity otherwise unknown. Later, ideas arrive, setting a boundary to the known and unknown. The quantum is such an idea, proposed in physics as the ultimate instance of the known, its ‘atom’ we might say in the subatomic world, and pivot upon which the universe turns.

Given a new identity cross hairs are needed to determine its location. Wishing to go past the atom, for physics has found a new world of activity, a subatomic domain akin to the subconscious behind the conscious, it endeavored, following custom, to impose old ideas upon the new horizons coming into view. It therefore proposed the quantum as a subatomic particle, or smallest possible lodgment of material being. Had it spelled particle quarticle, after the Carrollean quark that gave the quantum its name, subatomic physics would have unfolded much more easily.

Calling a halt to the Jonathan Swift progression (Great fleas have lesser fleas upon their backs to bite 'em…), the Copenhagen interpretation declared that quanta cannot be detected because the radiation used in the proposed examination must knock them out of the position or momentum it needs to determine. Like trying to catch a sparrow with a shotgun the investigation destroys its object.

Given that that which cannot be detected cannot take its place as a member of the family of observable objects, the quantum and quanta in general, like sparrows in the wild, remain as conjectural propositions. Their reality extends no higher than supposition, a pall that spreads its mantel over the entire subject.

Our means, which belong to the macroscopic world, overwhelm the minimalist features we seek to determine. Observation of such objects as cars trains and tennis balls, and calculation of their position and momentum comes to us automatically. A division is thus drawn between the material world and its existence in our conscious awareness, a division that then reflects in world and mind alike. In the world it is between the actuality we confront and the materiality we attribute to it. In our mind it is our conscious appreciation of an external world over and against us, and the explanatory energy infused into this by our mind.

Dismissing the existence of quanta as independent material entities, the Bohr model dismisses materialism in general, unless, overturning the whole, we attribute objective being to mind itself. Particle physics then takes its place within the framework of physics as a whole, which in turn fits within the pattern of the unity of all knowledge.

The Rift

Motion and change are generally accepted as intrinsic to, if not the ground of all being. The other side of this assumption is a failure to address the objectivity of mind. In illustration of this, a rift opened between relativity and quantum mechanics in the nineteen twenties. In 1949 Niels Bohr wrote:[i]

(In 1927) … I advocated a point of view conveniently termed "complementarity," suited to embrace the characteristic features of individuality of quantum phenomena, and at the same time to clarify the peculiar aspects of the observational problem in this field of experience.

Complementarity holds that wave and corpuscular theories are complementary, and that both are required in microphysics to fully describe subatomic events. In his paper in reply Einstein wrote:

… to me it must seem a mistake to permit theoretical description to be directly dependent upon acts of empirical assertions, as it seems to me to be intended [for example] in Bohr's principle of complementarity, the sharp formulation of which, moreover, I have been unable to achieve despite much effort which I have expended on it.

In the background we have classical or Newtonian physics, now qualified by a new physics whose sides are relativity and quantum mechanics. These agree in rejecting the old, but are themselves incompatible where they should come together harmoniously.

They are complementary, but each is frustrated in being unable to complete the arch. The missing element is mind, and along with this, circular mathematics, whose very backbone is complementarity. Neither Einstein, Bohr or anyone else dreamed that the fault could rest in the clipped artificiality of school-taught math.

In the pre-19^th century, just before Planck and Einstein launched their new ventures, science rested upon the centrality of matter, and this in turn upon the ponderability of property and possession. Relativity transferred this centrality to the constancy of the speed of light. In the wings quantum physics began to concern itself with the nature and reality of quanta and the subatomic domain. We need to see however, the centrality of mind, the facility that makes the human human. From this every new development seems to retreat, but is bound to return in the fullness of its completion.

Organization in quantum physics exists in mind per naturam. Complementarity describes the mind's intrinsic form. It is the stuff of logic and the basis of mind. It comes to the fore in circlemath wherein it is fully explicit. Meanwhile the lack of insight that plagues school mathematics, unaware of its circular in-mind origin has acted as a blindfold.

In his concluding remarks Bohr states:

… the astounding simplicity of the generalization of classical physical theories, which are obtained by the use of multidimensional geometry and non-commutative algebra, respectively, rests in both cases essentially on the introduction of the conventional symbol sqrt(-1).

This reference to the square root of minus one brings the culprit into view, the egg on the face of classical math, the sleight of hand wherein it hides its shame, the smoking gun no one dares challenge. In his reference to it Bohr gives no hint of approval or disapproval. Indicating the differences, he adds:

Although in more recent years I have had several occasions of meeting Einstein, the continued discussions, from which I always have received new impulses, have so far not led to a common view about the epistemological problems in atomic physics...

Relativity does away with the idea of space and time as fixed determinations. It has bodies shrink in the direction of their motion and clocks slow down while keeping correct local time, but it remains trapped in the denial of the objectivity of mind.

It admits the observer into physics but only as the reading of a dial. Particle physics on the other hand deals with the mechanism of observation, showing that its means, the shining of an energetic wave upon a target disrupts what it sought to determine, the position and momentum of quanta. The subatomic, like the subconscious, is relevant to but excluded from perception. It rules the quantum out of account, but stopping in this negativity it fails to say what takes its place. The answer we seek, the answer we must give, is the reality, the objectivity of mind, linking observation to reality not as something incidental, but its ground.

That which goes around comes round. The past is contained in the present. The future is hidden in its womb, as the scholastics said, in the infinite eternal and perfect.

Broglie Bohr Matter Waves

Reality presents in three categories, macroscopic, microscopic and submicroscopic.

1. The macroscopic contains objects such as chairs tables trees and oceans.

2. The microscopic, visible through optical and electron microscopes reaches down as far as the atomic level.

3. The submicroscopic is subatomic. It equates in this respect with the subconscious.

The microscopic (2) is clearly a transitional zone between the macroscopic (1), which obeys Newtonian mechanics, and the submicroscopic (3), which falls to particle physics. Together, as contributing to knowledge, they fall within the bounds of the subjective objective divide, whose polarity, as ‘mind and matter’ corresponds to ‘subjective and objective’.

1 and 2 together comprise the observable known. This leaves the submicroscopic (3) as conjectural, a stratum known to us through inference alone. It is beyond the unaided eye and undetectable by instrumental means. This is because the subatomic is the functional basis of mind.

Its mechanism is known in the Copenhagen interpretation as the Bohr model. Paradoxically only its interpretation, as the ‘model’ within material and mental being, has remained wanting. Physicists have but to realize this to see the conundrum that has long perplexed their field begin to unravel and vanish. A Nobel Prize for this? No. More likely to be crucified. :-)

The quantum is a creation of mind. If this seems to reduce it in terms of reality, the remedy is to advance our sense of the nature of mind itself, which is intrinsically objective, the substantial foundation of all existence. There is no need to expand on this objectivity, for as an awareness it has been embedded in human knowledge since our kind began.

Mathematics is natural in the mind, which it generates. It is its foundation, as and identical with the brain's physiology. Now, physiology is the science of function in living organisms, as distinct from their anatomy. There, in this physiology it has dwelt unknown, until over thousands of years aspects have come to light, now this way, now that.

Mathematics also occurs in the mind as a taught and learned expression because we put it there, “burning the midnight oil”. In this it appears as an end product, developed, as said, as a formulation within this physiology. The two forms are mutual opposites and complementary.

This brings us to de Broglie matter waves and Bohr's complementarity, whose symmetrical structure is the operative unit in every living creature and facet of consciousness, the functional ‘cell’ of mind upon which consciousness and conscious function depends. ‘Mathematics’ is thus the heart and soul of thought and life, its intuitive logical machinery, its bolts and chains as the physiology of mind, along with its material expression, the being and solidity of table and chair, forest and hill.

The Cheshire Cat in the Copenhagen Interpretation

Circlemath throws light upon complementarity

In the matter/ motion/ mind arena light presents as a waveform of energetic packets called photons, but the terms alone introduce the study of advanced theory fraught with unknowns and uncertainties. Matter and motion are involved in objectivity, along with the radiation or light that brings these factors to our attention. In a triple mix the interaction of matter energy and observation must engage our attention, while stepping back once again, our understanding becomes its own object in terms of the physiology of mind.

The focus historically has been on the dual nature of mind and matter. Mind corresponds to our nature as conscious beings, matter to the known as understood. The whole is a process, each side of which is a process. The task is to bring the sides, each of which, excluding its other, is two sided, together within the unity of consciousness.

The foundation for doing so rests in the work of Louis de Broglie, who in his 1924 Ph.D. thesis proposed that the matter/ wave nature of light is reciprocal, and that matter waves also exist, an analysis confirmed by later research. Matter is not ‘still’ but vibrates in itself. This work became the foundation of Niels Bohr’s model of relational being and universal complementarity.

Identity in difference

Our subject here is his model of identity through reciprocal difference. Because it is basic, our approach must also be basic, grounded that is, in our interpretation of mathematical findings as these arise and exist within the sphere of the mind. Take any complex thing in objective existence along with our subjective comprehension of it, and ask how the sides, its being and our awareness of that being are related. The object may consist of myriad points, each of which our mind may reflect. Turn this ensemble of known and knowing inside out and we have a two-sided entity, composed of point-by-point identity in difference.

The simple form of identity, like that of peas in a pod, is direct. The other form is indirect through mutual exclusion, which Bohr called complementarity. We now have our object in view. It could be called the cement of mind, the logical complex that holds opposites in the understanding. Expressed in pure form this cement is the circular mathematics that we can now fit into place as a numerical bridge within Bohr complementarity.

According to Bohr, as expressed in the Copenhagen interpretation the logic and laws that hold in the atomic world do not apply in the subatomic realm. We could jump the gun here and say, “That is because ‘atomic realm’ refers to the world in objective reality, and ‘subatomic realm’ to the same within the subjectivity of knowing mind.” Bohr's theory or model points to this. Applied quantum mechanical theory itself shows that quanta are mental, not material entities.

In short, the energetic means we use to obtain results in the macroscopic world fail in the subatomic. The quantum cannot be weighed in ponderable scales. Like Lewis Carol's quark it evades capture, or like the Cheshire cat, only its smile can be seen. The circular relations below indicate the nature of the mathematical bridge between objective being and our subjective understanding.

Complementarity in Circles

Figure 1

Horizontal opposites in a circle, as simple complements, add to their circle number. Their sum counts back into the circle as 0, or jumps out of it to register as the 10 of 1 full circle turn. Using 10‑circle as our example, 8+2 = 10, = 0, = the ‘1’ of the 10-circle. This bifurcation of paths, jumping out and staying in, corresponds to the ‘in mind’ ‘in world’ relation of thought and being.

The relation through difference of the vertical 6,9 then fits into place as Bohr complementarity. 9+6 =15, which is 10/2 in the circle. Note that there are no vertical opposites in odd circles. Taking odd and even into account, all circular relations map to mental configurations on the one hand, and worldly objects on the other.

The 6 and 9 are vertical relations in 10-circle, but we could have chosen 8+7 0+5 1+4 or 2+3. The polar numbers in all these sets, and their like in every even circle to infinity, equate negatively through exchange of their fields (now to be explained).

The field of a circle number is its reciprocal relation with all the other numbers in its circle, or ‘what it is not’. The field of 6 is thus, 7+8+9+0+1+2+3+4+5 =39 =9 in 10‑circle. The field of 9 is, 0+1+2+3+4+5+6+7+8 =36 =6 in 10‑circle.

This vertical pairing introduces a new version of negativity. The simple form is given by 4‑2 =2 and 2-4 =‑2. To keep the two forms of negativity separate, the new form is called ‘nots’ (the negative taken as a noun). It is circlemath complementarity, and it maps to Bohr complementarity. For example, we have taken 4‑2 =2 and 2-4 =‑2 as our simple example. Looking at the 6, 9 relation, 6‑9 =7 in 10‑circle, and 9‑6 = 3 in 10‑circle. They are equal to the same thing in the 7 and 3, which are horizontal opposites. However, taking the vertical relation directly we say that 9 is the not of 6, and 6 is the not of 9.

Some other points to note are that changing every number in an odd circle to its ‘not’ expression reverses the circle clockwise/ anticlockwise. Adding the odd circle to its ‘not’ then annihilates both, just as 4‑4 =0.

Changing every number in an even circle to its ‘not’ reverses it clockwise/ anticlockwise, but also turns it upside down. Adding the even circle to its ‘not’ turns every number into its 10/2 value. This, in physiology, ‘plants’ our understanding, and the mechanism is precisely through difference (Bohr complementarity).

In the counting line, the odd number circles drop out, and the now halved counting line of circle bases to infinity is replaced by the even sequence. This reinstates the counting line to infinity, shedding light upon our mental function in the process.

The above is a synopsis, discussed more fully in ATOM a theory of mind. The ‘number and not’ formulation distinguishes one entity from another and the ‘inside and outside’ of things and thoughts. Figure 2, dealing with our sense of differentiation and identity gives a numerical representation of compounding something and its not.

Figure 2

We have used 10-circle, but for a ‘pixel dot’ simulation of objective reality we would, like the television, have to call upon circles with numbers ‘to infinity’. The ‘uniform result’ on the right (fig 2) ‘plants our understanding’. It tells us that the mind builds its every identity (here the ‘5’) from diametrically opposed sides through iteration, here shown in a second form in the circle. The anatomy of identity is contradiction.

In postulating the quantum, particle physicists held to the expectation that, given fine enough means, it could be detected. Based on quantum mathematical theory itself, the Bohr model rules this out. Detection is logically excluded. Falling short of the criteria for material existence the entire corpus of particle physics theory is not about the mechanism within matter, but the mechanism within mind itself insofar as this expresses itself in the creation of our sense of material existence.

‘Subatomics’ is the mind's modus operandi, or means of proceeding. To see this advances our understanding by a step of magnitude. It does away, not with the quantum, but with its existential independence. Figure 2, in a mathematically based mechanism, shows how an external visible world can build within the mind as its own expression, the mind's creation of itself to itself, the mind in its primal form as its own creation.

Taken within the framework of its larger setting this interpretation shows how, within reason, we can dream hallucinate and figuratively imagine scenes, colors and all manner of sensory expression, whose sum is materiality independent in external existence. The mind, in reciprocity, creates the world. The world does not create the mind.

The Bohr (Copenhagen) model fits into this picture as a reasoned outcome in particle physics. It rests entirely within particle physics, and in showing in a mathematical setting that the quantum cannot be detected as a world object, it assigns reality as a whole to the mind, not reducing reality by an iota in stature in doing so.

Chirality as the Machinery in Quantum Mechanics

Latin machina, from Greek makhana device or pulley, gives us machinery mechanism and mechanics. Thought is involved but the focus is on matter, materiality. The key division in consciousness is between subjectivity and objectivity. Subjectivity points to within; the knowing individual. Objectivity, from Latin ob against + jacere to throw means over and against. Together, subjectivity and objectivity convey the idea of inside/ outside within the framework of thinking.

Turning then to content, as distinct from frame, we have atomic and subatomic domains. The ‘sub’-atomic is the logical ground upon which the atomic stands. Take the terms (atomic and subatomic) indoors so to speak, and they transform into conscious and subconscious as their parent or in-mind form.

In consciousness we sense a world of things, defined as independent entities, each with an inner being and outer appearance. The subconscious mediates between the conscious and the unconscious, both of which reflect into it, so that it becomes the cauldron of thought, the inner reserve upon which the conscious stands. The ‘unconscious’ is our viewpoint on the materiality of the body, being that which suffers the experience, and equally, from its genetic reserve (the living body), comes forward to meet it.

Quantum mechanics builds where the physics of machinery leaves off. Its unit is the quantum. To bring this out, naked and trembling, it is only a figment of the imagination. The latter however, as the infinite arena of thought, is like thought, objective in itself.

We have then ponderable bodies in the macroscopic world that obey Newtonian physics. Below is the world of subatomic particles. The difference is more profound than we might think, for its sides are de Broglie Bohr opposites.

Defining these opposites brings the essential unit of mind into view. Known as the Bohr model within the Copenhagen interpretation it was developed during the course of long debate with Einstein. It is not that one was right and the other wrong. Along with colleagues they cooperated in examining the same thing from opposite sides. The fact that they could not come to grips with each others point of view indicates that there was, or is something else there still to be discovered. The relation between the sides is difficult to describe, let alone define exactly. We can but try.

The Snark we are looking for is identity in difference, and in reverse, difference in identity.

Our first example is met in the phrase, “as like as peas in a pod.” One pea is like another, even as they stand independently apart. This is ‘identity in difference’ and ‘difference in identity’. Clearly, the shuffle is between being and appearance. “One pea is like another in appearance, even as it stands in its own right as different in being.

Our second example is that of a mirror image which reverses two of three orientations. It reverses us front to back. If we face north looking into a mirror, our image faces south. It also reverses us right to left. If you raise your right hand your mirror image raises its left hand. To confirm this, instead of looking in the mirror, face another person. Say, “I am going to raise my right hand. You do the same.” They do so and the result is not what you see in a mirror. The hand they raise is right to them. Observation is the key here. The person observes. The mirror image does not. However, if you say, “touch your head,” they do not touch their feet! This distinguishes the asymmetry in the mirror image from chirality.

Our third example is chirality, abundantly described in the insect eye. The insect's eye is its brain, or putting it the other way around, the function its eyes perform belongs to the cerebral cortex in the brain of larger creatures. The bridge we now have to cross is to see that the ‘unit’ in this function is the de Broglie Bohr model as developed in the Copenhagen interpretation. If we expand this model to cover the gross function of the brain, otherwise out of sight in its complexity, it maps to that of conscious appreciation in animal life generally. It is the unit, in other words, of consciousness.

Bivalves exhibit a simple pairing of shells, but the identity of sides in the Bohr model is not physical but functional, and it is so in an intricate and complete point-for-point configuration. The latter is properly described as chirality, which steps above the instance of the mirror image in the same way that life steps above mere reflection.

The intricacy belongs to chirality in that every point on one side has its matching point on the other, but this matching is also mediated by, or in the system of the whole. These words are correct, but will also be cryptic to those unacquainted with the in-mind form of mathematics only recently introduced to the world. A pointer to the relation given in “The Cheshire Cat in the Copenhagen interpretation, subsection “Complementarity in Circles” might help just here. Complete refers to possessing the whole without omission.

Finally, we need to know where this intricate patterning fits into the world of being. Briefly, it is the identity in consciousness that governs sensibility in all living creatures. It is the key to thought and embraces the functional whole of an organism responsive to its environment. The path through is open, but we should not be surprised if it appears complicated, for it is the jewel in all knowledge, the function of mind within consciousness.

To Homepage Contact Us