# THE MEANINGS OF COMPLEMENTARITY IN PHYSICS AND PHILOSOPHY AND THEIR INTERSECTION WITH COMPATIBILITY THEORY - Ephraim Stephen Essien

**Abstract**

This paper comparatively examines the meanings of Niels Bohr’s and Innocent Asouzu’s complementarity theories and how they differ from or relate with my compatibility theory, **C={(≡)|(≈)|(~)|(∩)|(****c****)}**.

**Introducing the order of analysis**

This paper deals with the meaning of complementarity principle by Niels Bohr and of complementary reflection by Innocent Asouzu respectively, and how they contributed to my theorizing on compatibility. While Niels Bohr’s principle of complementarity falls under the domain of physics, Innocent Asouzu’s theory falls under philosophy. Compatibility comes from the provinces of logic, mathematics (set theory, algebra and number theory) and philosophy of physics; yet its practical applications go beyond the frontiers of these disciplines.

I shall, in this paper, locate Bohr’s theory within the context of the new physics of quantum theory, especially within the wave-corpuscular problematic of the nature of electromagnetic phenomena, such as light. I shall present a firsthand meaning of Innocent Asouzu’s complementary theory (having been privileged to be one of his pioneer postgraduate students and his supervisee while the theory blossomed in his *opus magnum* in 2004 (*The Method and Principle of Complementary Reflection in and beyond African Philosophy*). Thereafter, I shall connect Bohr’s and Asouzu’s theories with the compatibility theory.

Meanwhile, Bohr suggested that comprehensive knowledge of micro phenomena, such as light, requires a description of both the wave and particle aspects separately, not simultaneously; due to the indeterminacy presented by uncertainty principle. There is complementarity when both aspects are examined, but separately, not simultaneously (if simultaneity exists in the strict sense of the word).

Innocent Asouzu’s complementary theory avers that *anything that exists serves a missing link of reality*. In other words, no entity is self-sufficient and exclusive in itself in relation.

Compatibility theory stipulates that the order in a unit of a system in relation to the order in (an)other unit(s) within the same system is the order of the system. Compatibility theory states that compatibility exists if units are equivalent, or approximates, or similar, or intersected, or complements (Essien: 2011c).

Hence, the formula:

C**={(≡)|(≈)|(~)|(∩)|(****c****)}**,

where

C is compatibility,

≡ is equivalence,

≈ is approximation,

~ is similarity,

∩ is intersection,

c is complement.

**Bohr’s Complementarity within the matrix of quantum mechanics**

Quantum mechanics is the branch of physics that studies phenomena of the microcosm. Quantum mechanics has made it more apparent that a researcher cannot have adequate knowledge of a system of interacting objects without active interference in it (Frolov, 1, 348). This theory holds that energy exists in units that cannot be divided. Max Planck ushered in quantum theory into the province of physics when he offered a solution to the problem of black body radiation in about 1902.

While working on the thermodynamic theory of thermal radiation, Max Planck introduced ‘quantum of action’ (Frolov 327). According to Planck, light was radiated and absorbed discretely, by definite portions-quanta (h=6.62 x10^{-27}erg/sec).

This discovery marked a paradigm shift in the history of thought. A transition was made from macrocosm to microcosm. This was the birth of quantum theory, which established the fact of discreteness in the energetic processes and thus extended atomism to all phenomena of nature (Frolov 327). Planck demonstrated that the experimental observations in radiation could be explained on the basis that the energy from such bodies is emitted in discrete packets known as energy quanta of amount ‘hv’, where ‘v’ is the frequency of radiation and ‘h’ is a constant known as the ‘Planck constant’ (Okeke 253).

Planck’s breakthrough was an apparent solution to the problem of black body radiation. While awhite body is an idealized physical body with a "rough surface [that] reflects all incident rays completely and uniformly in all directions” (Planck 1914), a black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A black body is a body that absorbs and emits wave-length and temperature. A black body does not necessarily connote a solid body that is black. It is non-reflecting, perfectly absorbing, non-glossy (Alozie, 98). According to Alozie,

a metal box that is completely sealed, but with a tiny hole pierced would appear black if you look into the interior of the metal box through the hole. If the metal box is subjected to intense heating by probably a blacksmith to a glowing red colour, the hole will be showing a red colour. This was the type of phenomenon that Max Planck studied (Alozie 98-9).

Black bodies are physical abstraction and have no concrete, tangible, tactile existence.

According to Alozie, Planck conceptualized a model of an ideal black body as a large number of atomic oscillators that emit and absorb electromagnetic waves. Planck assumed that the energy of E, of an atomic oscillator could have discrete value of E=0, hf, 2hf, 3hf, 4hf, etc.

This implies that E=0, hf, 2hf, 3hf, 4hf, etc, are the only permitted values of the energy “E”.

Energy quantum is “hf” of “hv” and “n” is the quantum number of the oscillator.

For Planck E =nhf,

where

n= 0,1,2,3,4,etc.

h = constant (6.6260755 x 10^{-34} J.S.)

f=frequency of vibration(measured in Hertz)

There are no energies with discrete values. Thus they are quantised. Hence energy quanta.

Following the quantum theory of radiation, Albert Einstein postulated that light of frequency v contains quanta of energy hv which he called photons.

Einstein’s version of quantisation is useful in explaining photoelectric effect.

Electrons are emitted from an insulated metal surface when light of sufficiently high frequency falls on the surface. This phenomenon is called the photoelectric effect. Under the wave theory of light the kinetic energy of the emitted electrons should increase with the intensity of the incident light. The photoelectric effect, also, should occur for any frequency of the incident light provided the light is intense enough. Under the wave theory, for a very feeble source, the part of the wavefront that is intercepted by an electron in the irradiated material will be so small that it will take the electron a long time lag between the incidence of the light and the electron emission. (Evwaraye 229).

Reacting against the failure of the wave theory of light to explain the observed effects in photoelectricity, Einstein enunciated a mechanism for the phenomenon based on Planck’s quantum theory of black body radiation.

Based upon Max Planck's theory of black-body radiation, Einstein theorized that the energy in each quantum of light was equal to the frequency multiplied by a constant, later called Planck's constant. Einstein postulated that light of frequency v contains quanta of energy hv which he called photons. A photon of light of frequency v carries an amount of energy E= hf, where h is Planck’s constant 6.6260755 x10^{-34} J.S).

In Einstein’s explanation of the photoelectric effect,

(i) The entire energy of a photon is transferred to a single electron in the metal, which gets emitted instantaneously. This immediately removes the difficulty regarding time lag.

(ii) When the electron comes out of the metal surface it will have a maximum kinetic energy given by

½ mv2 = hv–f.

This equation shows that no electron can be emitted if the frequency of the incident light is so small that hv < f.

Thus, there is a threshold frequency in agreement with experimental result.

The particle nature of radiation was further confirmed by the *Compton Effect*.When x –rays were scattered by a target with loosely bound electrons, e.g. carbon, the scattered radiation was found to consist of two components. The one was having the wavelength as the incident beam (unmodified line) and the other having a slightly longer wavelength. But there should be no change in wavelength or frequency following the classical electromagnetic theory. The incident radiation was expected to set the atomic electron vibrating with the frequency of the incident radiation, and then produce radiation emitted in all directions with the same frequency (scattered radiation).

Compton, in 1923, provided the explanation to the observed effect by treating the incident radiation as a stream of individual photons each of which could interact with a single electron.This is the *Compton Effect*. The wavelength of the scattered x – rays is greater than the wavelength of the incident radiation (Evwaraye 230).

In photoelectric effect the photon ceases to exist when all the energy of the photon goes into the energy necessary to remove the electron from the surface. However, in* Compton Effect*, the photon continues to exist, but does not lose energy as shown by the change in the wavelength or frequency of the incident x –rays (cf. Okeke 256).

Huygen proposed a wave theory of light. The wave theory sees light as a wave fronts spreading out from a light source as spherical or circular waves. The wave fronts travel out with the speed of light. A sensation is produced as these wave fronts reach our eyes. A long way from the source, the circular waves appear as plane parallel waves.

James Clerk Maxwell showed that light was an electromagnetic wave. Light consisted of electrical and magnetic vibrations (Okeke 256). Though the electromagnetic theory’s calculation of the speed of light was approximately equal to 3 x 10^{8}m/s, the theory failed to account for certain properties of light such as emission and absorption of light and radiation of energy by heated bodies.

Louis de Broglie later postulated in 1923,, that since light waves could exhibit particle – like behaviour, that particles of matter equally exhibit wave-like behaviour, (Alozie, 104). Since nature, in his reasoning, is symmetrical in many ways and our sensible universe is made of energy and matter; again, since light has a wave-particle nature, Broglie concluded that matter does also. In predicting the wavelength of a particle, Broglie stated that the wavelength of a particle is given the same relation that applies to a photon. Put in other words, the wavelength of the predicted matter waves was given by the same relationship that held for light.

Broglie’s wave nature of matter was confirmed few years later by Clinton J. Davisson and Lester H. Germer through an electrons-diffraction experiment. The duo demonstrated that electrons exhibit wavelike properties of diffraction and interference by passing a beam of electrons through a crystalline solid. Thus there is also a wave-particle duality in matter. That is to say that matter behaves in some circumstances like a particle and in other circumstances like a wave.

The wave-particle duality of light and matter bequeathed to Erwin Schrodinger made possible the coinage of the concept, wavicle. Schrodinger observed electrons as patterns of standing waves. Those standing waves are quantized as particles in a discrete pattern. Based on his observation, Schrodinger gave a formula which the electron wave shape would obey if the electron was part of the hydrogen atom. By using his equation to deduce the light spectrum of hydrogen, the idea that electrons are waves was confirmed. Yet the motion of particles as well as waves is relative motion.

Schrodinger observed electrons as standing waves. What was waving was not certain, but that something was waving was sure. He designated this “wave function”. There was an indeterminate perception of what was waving, but a certainty that something was waving.

Max Born considered Broglie’s and Schrodinger’s standing wave as the wave of matter and not of particles as unsatisfactory. For Max Born, the latter’s interpretation of wave-function is an indicator of the probability of finding an electron at some point in space (Alozie 107). This wave-function probabilistic interpretation ushered in indeterminism in quantum mechanics. Quantum mechanics reached the zenith of its indeterminism in the uncertainty principle.

The uncertainty principle is one of the principles of quantum mechanics put forward by Werner Heisenberg in 1927. Simply expressed, there is a basic uncertainty in our knowledge of particles. It was observed by Heisenberg that the very act of measuring physical parameters like position and velocity of an electron disturbed the electron because of interaction between the apparatus and the electron. This invariably introduced uncertainties in the precision of measurement. On an atomic scale, Heisenberg established that it is in principle impossible to obtain an exact measurement of both the position (x) and the velocity (v) of a particle. We are unable to specify precisely the position (x) and the velocity (v) of a particle, but the likelihood of its being located at a certain point.

There is always an uncertainty (x) in the position of the particle and an uncertainty (v) in the velocity of the particle. The product of the uncertainty (x) in the position of the particle and the uncertainty (v) in the velocity of the particle (x.v), according to Heisenberg, must be greater than the Planck constant (h) divided by the mass of the particle. The uncertainties concern the nature of matter and not related to errors introduced by the limited precision of the measuring device. Hence, x.v > h/m,

since m.v = p, the uncertainty in momentum p,

then p.x > h

Also, E.t > h, where E is the uncertainty in the energy of the particle andt is the uncertainty in the time measurement (Okeke 257).

The uncertainty or indeterminacy principle implies that to know the world we must observe it, and in the act of observation, uncontrolled and random processes are initiated in the world (Pagels 83). Uncertainty principle introduced an unavoidable element of unpredictability or randomness into science (Hawking 60). We cannot at the same time know precisely both where a particle is and how it is moving. The more accurately we can observe its location, the less accurately we can observe its momentum, and vice versa. If, at a particular time, we cannot know with precision both the position and momentum of a particle, we are deprived of data needed for predicting where it will be later on. The future is thus indeterminate (Hoffmann 185).

Due to the contradictory, corpuscular–wave nature of micro-objects, uncertainty principle posits the impossibility of simultaneously determining their exact coordinates and impulse. Commenting on the principle, Alozie remarks that the uncertainty principle of quantum mechanics is difficult to swallow in the light of the reality of the macroworld (109).Yes, of course, in our everyday world particles can have a well-defined position and velocity of momentum.

**Bohr’s Complementarity Principle**

Niels Bohr’s scientific interests lay at the junction of physics and philosophy, in the sphere of analysis of conceptual apparatus of physical theories (Frolov 48).

Bohr put forward the principle of complementarity, a method of description that was applied to various fields of knowledge in the analysis of alternative, contradictory situations (Frolov 48).

In point of fact, Bohr’s principle of complementarity as a method of description was suggested to interpret quantum mechanics. Here is the thesis of this principle: To reproduce the wholeness of a phenomenon at a certain ‘intermediate’ period of its cognition, use must be made of mutually exclusive ‘complementary’ and mutually limiting classes of concepts which can be used separately, depending on specific conditions, but only taken together cover all definable information (Frolov 77).

Bohr advocated the admission of the contradictory positions in quantum theory via complementarity. Thus the principle of complementarity helped to bring out the dual, wave-corpuscular nature of light. This principle submitted the equivalence of two classes of concepts describing contradictory situations. Bohr’s principle seems to contain elements of dialectical thinking.

In physics, complementarity is a basic principle of quantum theory closely identified with the Copenhagen interpretation, and refers to effects such as the wave-particle duality, in which different measurements made on system reveal it to have either particle-like or wave-like properties. Niels Bohr developed this concept at Copenhagen with Heisenberg, as a philosophical adjunct to the recently developed mathematics of quantum mechanics and in particular the Heisenberg uncertainty principle. In the narrow orthodox form, it is stated that a single quantum mechanical entity can either behave as a particle or as wave, but never simultaneously as both; that a stronger manifestation of the particle nature leads to weaker manifestation of the wave nature and vice versa (en.wikipedia.org).

A profound aspect of complementarity is that it not only applies to measurability or knowability of some property of a physical entity, but more importantly it applies to the limitations of that physical entity’s very manifestation of the property in the physical world. All properties of physical entities exist only in pairs, which Bohr described as complementary or “conjugate pairs” (en.wikipedia.org/wiki/complementarity(physics)). Physical reality is determined and defined by manifestations of properties which are limited by trade-offs between these complementary pairs.

The emergence of complementarity in a system occurs when one considers the circumstances under which one attempts to measure its properties. As Bohr noted, the principle of complementarity “implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear” (en/wikipedia.org/wiki/complementarity(physics). By ‘complementary’, Bohr submits that “any given application of classical concepts precludes the simultaneous use of other classical concepts which in a different connection are equally necessary for the elucidation of the phenomena (Bohr 10).

According to Heisenberg, “by the term ‘complementary’ Bohr intended to characterize the fact that the same phenomenon can sometimes be described by very different, possibly even contradictory picture, which are complementary in the sense that both pictures are necessary if the ‘quantum’ character of the phenomenon shall be made visible. The contradictions disappear when the limitation in the concepts are taken properly into account” (Heisenberg, Remarks on the origin of the relations of uncertainty, 6). The method of complementarity, according to Heisenberg, is represented as a tendency in the methods of modern biological research which, on the one hand, makes full use of all the methods and results of physics and chemistry and, on the other hand, based on concepts referring to those features of organic nature that are not contained in physics or chemistry, like the concept of life itself (Heisenberg, Quantum Theory, 94).

Bohr summarized the principle as follows:

*[H]owever far the [quantum physical] phenomenena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms.* The argument is simply that by the word "experiment" we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.This crucial point...implies the *impossibility of any sharp separation between the behaviour of atomic objects* [i.e., objects governed by quantum mechanics] *and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear....* Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as *complementary* in the sense that only the totality of the phenomena exhausts the possible information about the objects (Bohr 1961).

**Asouzu’s Complementarity and its Critique of Bohr’s**

Innocent Asouzu says that there are a trace of exclusive dualism in Bohr’s reasoning (Asouzu 99). He, however, submits that the substance of Bohr’s theory is that “nature has “complementary aspects” (Asouzu 100). Asouzu observes that philosophers, in many cases, have found it difficult to come to terms with the dualism existing between opposites. Many thus tend to interpret this in the manner of independent interacting opposites. According to this observation, most philosophical traditions that are modeled according to the demands of Aristotelian Greek dualistic conception of the world seem to understand contraries as antagonistic opposites that are destined for parallel existence. According to Asouzu, this is the point of difference between this approach tagged “complementary refection” (Asouzu 2004).

Complementary reflection, while being anchored on true African experience, seeks to reach out to the ultimate foundation of reality as a universal philosophical task. Bearing this fact in mind, Asouzu establishes that African philosophy in a complementary perspective is the systematic ambient methodological reflection about reality with the aim of explaining and understanding reality authentically in a way that portray the totality of the factors and actors that influence the thinking of the thinker. This means that complementary reflection transcends the immediacy of traditional African ambience to include all the factors and actors that are constitutive towards the emergence of the true and authentic nature of reality. In this way, complementarity has a universal appeal and relies on a method of universal applicability for its reflection. This systematic methodological approach is like providing a key with which to enter a building. Whoever has this key has access to the contents of the room (Asouzu, 2004:271). Inspired by the ideas of traditional African philosophers of the complementary orientation and relying on all useful inspirations emanating from the complementary nature of reality *in globo*, Asouzu presents the principle and imperative of complementarity.

**The principle and imperative of complementarity**

Complementarity contains two principles: the principle of integration and the principle of progressive transformation (cf. Asouzu, 2003:58). Principle of integration serves a complementary ontology, while the principle of progressive transformation is a theory of rational praxis. Whereas the principle of integration specifies the general metaphysical implications of the theory, the principle of progressive transformation specifies the relevance of the theory to human action.

The principle of integration as the metaphysical variant of the principle of complementarity states, that anything that exists serves a missing link of reality (Progress in metaphysics: The phenomenon of “missing Link” and interdisciplinary communication (Asouzu, 1990:82 – 91). The practical variant states that all human actions are geared towards the joy of being. The principle of integration and the principle of progressive transformation engender the imperative of complementarity which states: “Allow the limitation of being to be the cause of joy”. The imperative of complementarity is the condition for the realization of the principle of complementarity in the relative and fragmented moments of existence. Otherwise stated, allow all world immanent realities, in their fragmentation, to be the cause of your joy.

When Asouzu says, “Anything that exists serves a missing link of reality” and “allow the limitation of being to be the cause of your joy” what does he mean?

I begin with the latter. Meanwhile, one must note that Asouzu employs the category ‘being’ here in its traditional philosophical understanding as the unifying foundation of all existent realities (271). By the expression ‘limitations of being’, Asouzu affirms the fact that fragmentation is a constitutive characteristic of being in history (273).

The experience of fragmentation as an integral part of historical existence forms the methodological point of demarcation between all kinds of extensive reasoning and complementary reasoning. Complementary consciousness harmonises fragmented world immanent historical experiences in a manner that conveys authenticity to them (277). “Missing links”, in Asouzu, are the diverse units that make up an entity within the framework of the whole and as they are complementarily related. They are the imaginable units, fragments, components and combinations that enter into our understanding of any aspect of reality. They are also all the units and combinations necessary in the conceptualization of an entity or of the whole. They are thoughts and thoughts of thoughts, categories and categories of categories, units and units of units, entities and entities of entities, things and the things of things, ideas and the ideas of ideas (278).

Operating from his ambient, Asouzu is of the opinion that the idea that anything that exists serves a missing link of reality reflects the central paradigm of the anonymous traditional African philosophers of the complementary orientation. All existent realities, Asouzu maintains, relate to each other in the manner of mutual service (278).

How does a thing serve a missing link of reality?

When a unit is abstracted from the whole and viewed as completely isolated, discrete and discontinuous, it is seen as an independent non-relational entity. At this moment it could be said that they are missing in relation to one another. They are missing in the sense that, as discrete units, each can be viewed in isolation to each and in total disregard to one another. When this happens, a unit can be unaware of the other and in this moment, the one that it is unaware of is missing (Azouzu, 2003:59). This idea of complete isolation is, in Asouzu’s view, counter-intuitive. In this counter-intuitive mind-set, they stay in relation to one another, and for this reason, they serve a missing link (Asouzu, 2004:278).

Asouzu frankly maintains that to conceptualise discrete units in an authentic systematic way involves conceptualizing them as completely complementary to one another. This means that one must understand or conceptualize units implicitly and necessarily in complementary anticipation of the whole that gives them their legitimacy (278-9). Such consciousness that is complementarily oriented is a transcendent consciousness that is capable of ‘sublating’ (aufgehoben) the contingent immediacy of the moment and views units as existing now yet in a proleptic, future referentiality. Authentic complementary reflection and actions are motivated by an experience of transcendent complementary unity of consciousness.

**Ibuanyidanda**

Central to this Complementary Philosophy is the theme: *Ibuanyidanda.*The concept,*Ibuanyidanda,* is drawn from the Igbo language and has, as its nearest English equivalent, the idea that “complementarity”, in the sense of togetherness, is greatest (*njiko ka*), *igwe bu ike *(strength in togetherness) (*Asouzu 2007:*11). The word,*Ibuanyidanda,* is a combined word made up of three parts: *Ibu,* which means “load or task”; *Anyi,*meaning not insurmountable for, and *Danda*, which is a species of ants.

This concept *Ibuanyidanda* draws its inspiration from the teachings of traditional Igbo philosophers of the complementary system of thought. For the traditional Igbo, “danda” (ants) can surmount the most difficult challenges if and only if they work in a harmonious complementary unified manner (Asouzu, *2004:*108). This implies the idea of mutual dependence and interdependence in Complementarity. Complementarity (*Ibuanyidanda*) is the moment of reflection between the choices of isolation and teamwork. As I see it, clear instance of *ibuanyidanda *in action is found among players in the football field, who carry their load together to attain victory. They do so through team work and cohesion and mutuality. By*Ibuanyidanda,* we make recourse to such an ontological state of mutual service in complementarity as the horizon of our reflection (*Asouzu 2007:*12).

Asouzu’s complementarity rejects all forms of exclusive dualism and polarization, by its concept of truth and authenticity criterion. According to the truth and authenticity criterion, no entity is an absolute. Asouzu suggests complementation as a *modus vivendi* (mode of life) and mode of being.

In complementarity, Asouzu argues that incompatible opposites and contraries can co-exist (Asouzu 304). In authentic complementary reflection, the mind, Asouzu says, concedes to the co-existence of opposites and tries to see how they can be related to each other in non-contradictory ways. In the resolution of the incompatibility posed by opposites, we are offered the possibility towards extracting the harmony that unifies them towards eradicating all forms of disharmony. Through the determination of the reasons for incompatibility between opposites, we can create room for their compatibility. An important dimension of complementary reflection would be the determination of the conditions of possibility for compatibility of opposites.

Asouzu’s query about the determination of the conditions of possibility for compatibility of opposites led me to develop the conditions for compatibility in the compatibility theory, wherein I employed mathematical logic, algebra and set theory to derive the compatibility formula, C**={(≡)|(≈)|(~)|(∩)|(****c****)}**,

where

C is compatibility,

≡ is equivalence,

≈ is approximation,

~ is similarity,

∩ is intersection,

c is complement.

**The Compatibility Principle**

The compatibility principle affirms that entities are often compatible by their common property and yet by implicate order when they appear dissimilar. The order in each unit of a system is defined by the order of the system. In other words, the wholeness of a system is the common property and the implicate order which exist in the system. Compatibility principle thrives on the premise that entities which share a common property are often compatible and that there is compatibility by implicate order in entities that appear to be dissimilar. Entities, units or systems share common property if they are equivalent, or approximates, or similar, or intersected, or complements.

There are thus three interdependent postulates of the compatibility principle

Postulate 1: The order in each unit of a system in relation to the order in another or other unit(s) within the same system is the order of the system.

Postulate 2: There is compatibility by implicate order.

Postulate 3: The all are in the one and the one is with the all, such that a set is defined only by the elements that it contains.

We can demonstrate the interconnection of the three postulates of compatibility principle by relating it to set theory. Thus the property in each element of a set in relation to the property in another or other element(s) within the same set is the property of a set. Yet set theory does not exhaust conjunctive thinking.

**Discursive Analytic of the Compatibility Principle**

In the section that follows there shall be attempts to demonstrate the compatibility principle. It is claimed that this principle applies to all branches of learning. But the demonstration is beyond the scope of the present work. In this book, the compatibility principle shall be related to logic, arithmetic, set theory and human physiology. Here, we are concerned with conjunctive thoughts.

**Human Physiology and Compatibility Principle**

Let us consider the human body as a whole system. This system is composed of parts, such as the eyes, hands, legs, nose, mouth, tongue; the organs such as the heart, lungs, kidney; and the tissues of the human body. All these parts are connected with the brain, which is the “central processing unit” in the human body. All the parts of the body function, each in itself and with one another, towards the soundness of the whole body system. All tissues, organs and parts of the human body function toward maintaining the whole body in a state of equilibrium, homeostasis. The body is actually a social order of about 100 trillion cells organized into different functional structures, some of which are called organs. Each functional structure provides its share in the maintenance of homeostatic conditions in the extra-cellular fluid, which is called the internal environment. As long as normal conditions are maintained in this internal environment, the cells of the body continue to live and function properly. Thus, each cell benefits from homeostasis, and in turn, each cell contributes its share toward the maintenance of homeostasis (Guyton and Hall 7). This reciprocal interplay provides continuous automaticity of the body until one or more functional systems lose their ability to contribute their share of function.

When this happens, all the cells of the body suffer. Extreme dysfunction leads to death, whereas moderate dysfunction leads to sickness (Guyton and Hall 8). The logic of the human physiology is that the soundness of each part of the body in relation with the soundness of other parts of the body constitutes the soundness of the body as a system. The compatibility rule relevantly applies to the soundness of the human physiology, and emphasizes that the order in the unit of a system in relation to the order in other units within the same system is the order of a system. In the human physiological system, if any part is in disharmony or disorder, say as a result of the effect of a pathogen, this disharmony and disorder affect the holistic well-being of the human physiological system. This is why it is said that a sound mind is in a sound body (mens sana in corpore sano).

In a state of disharmony in the human physiological system, one cannot claim to have contentment or peace of mind.

**Logic and the Compatibility Principle**

We relate the compatibility principle to conjunctive propositional form in logic.

We use the variable, p,q,r,… to represent propositions. In conjoining two propositions there are four possible values to be considered, since a proposition is a statement that may be assigned a true or false value. Proposition p may be either true (T) or false (F). Similarly, proposition q may be either true (T) or false (F). A truth table works by setting up reference columns for all the possible combinations of truth values of p and q and then filling in the truth value of the compound position by referring to the reference columns, such that:

Case p q p ^q

1 T T T

2 T F F

3 F T F

4 F F F

a conjunctive proposition is true if and only if both of its conjuncts are true, and false if and only if both or any of its conjuncts are or is false.

When the principle of compatibility and order is applied to a conjunctive propositional form, like the truth table above, it is observable that a co-existence of variables of the same property produces an orderly system. For example, cases 1 and 4 prove that the order in the each unit of a system in relation to the order in another or other unit(s) within the same system is the order of a system. In terms of case 4 above there is order by virtue of the rule of double negation, such that,

~ (~ p) p

The compatibility principle confirms the De Morgan’s Theorem. The negation of both p and q is equivalent to the negation of p or the negation of q; the negation of either p or q is equivalent to the negation of p and the negation of q.

Hence

~ (p^q) (~pv~q),

~ (pvq) (~p^~q),

**Arithmetic and the Compatibility Principle**

We here apply the principle of compatibility and order to arithmetic. Here we shall see that this principle in one way or the other applies to only the conjunctive elements of arithmetic, such as addition and multiplication.

The sum of two positive integers will give a positive integer, such that:

3 + 3 = 6

The sum of two negative integers will give a negative integer, such that:

- 3 + (-) 3 = - 6

However, the sum of a negative number and a positive number will be equal to naught (e.g., -3 + 3 = 0).

We have some exceptions in multiplication when two negative numbers are conjoined to produce an effect.

Meanwhile:

3 x 3 = 9

and -3 x 3 = -9

But -3 x (-) 3 = 9

One may have to say that the rule of double negation makes it possible that the product of two negatives is a positive answer.

**Set Theory and the Compatibility Principle **

The compatibility principle prevails in set theory.

Axiomatic systems have been developed by logicians and philosophers of mathematics for set theory in the hope of avoiding contradictions and paradoxes. Some of these axiomatic systems are: the Zermelo-Fraenkel-von Neumann system, the Godel-Hilbert-Bernays system, and the Russell-Whitehead system (see Anderson 51). But set theory was basically initiated by Georg Cantor and Richard Dedekind in the 1870s. Set theory was, in fact, founded by a single paper in 1874 by Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers". Cantor focused his study on infinity and number theory, and thus gave a modern understanding on infinity, a new shift from Zeno of Elea’s understanding. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. However, around 1900, it was discovered that Cantorian set theory produced several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself, and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his *Principia Mathematica*.The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the standard axioms for set theory.

What is compatibility in set theory? How does compatibility theory work within the provenance of set theory?

In a affirming that entities are often compatible by their common property and yet by implicate order when they appear dissimilar, it is implied that compatibility entails sets and the universal set. In other words, entities that are compatible by their common property entail elements of a set and entities that are compatible by implicate order entail the universal set. With reference to the first implication of compatibility, we reaffirm that,

If “a” is an element of set “A”

i.e., a ∈A, it means that “a” possesses the same property possessed by other elements of set “A”.

Again,

A set “A” is a subset of a set “B” i.e. A⊆B, if every element of “A” is an element of “B”, i.e. if x.A, then x.B.

Hence {p,q,r)⊆{p,q,r,s}

Moreover, if A={x:x is an animal} and

B={x:x is a living organism},

Then A⊆B.

Again,

If A and B are sets,

Then A=B, whenever,

For any x, x.A, if and only if AB

In other words, sets are equal if they contain exactly the same elements.

If A is the set {4,6,8} and

B is the set {x:x is an even positive integer less than 9),

then A and B are equal sets.

With reference to the second implication of compatibility, that is apart from compatibility of entities by common property, there is yet compatibility of entities by implicate order, we have the following to say still within set theory; That,

The universal set U is a set which has the property that all sets under consideration are subsets of. In number theory, the universal set is usually the set of all integers. It may thus be said that the universal set connects all integers by an implicate order.

There is thus compatibility in sets and in the universal set. The common property of a system retains the wholeness of a system, while implicate order harmonizes apparently existing antagonisms and makes a system whole. This is a similar idea in Asouzu’s complementarity, which seeks to harmonize antagonisms.

**Onto-Logical Determinations of the Compatibility Principle**

A unit is what it is and cannot be another, “*simpliciter”*. Yet for any unit to be it must have an internal unity in substance, indivisible in itself, but divided from another (Essien: 2008a; 2008b).This condition is designated “Unit-Identity-Condition” (U-I-C). A unit must not have an internal disorder, lest it contradicts itself and loses identity. Since some units share a common property with other units they can be cognized as constituting a class. The identity of a class is defined by the common property that holds all members of that class in unity. A system is thus constituted by a common irreducible factor. However, a unit cannot qualify to enter into a class if it does not have first, an internal unity and then a common property with other units that make up the class. This condition is designated “Unit-Class-Identity-Condition (U-C-I-C). The onto-logical determinations of compatibility principle are the Unit-Identity-Condition (U-I-C) and the Unit-Class-Identity-Condition (U-C-I-C).

The Unit- Identity-Condition (U-I-C) and the Unit-Class-Identity-Condition (U-CI-C) are founded upon an ontological and logical ambience of identity. The ontological identity of a unit determines the logical identity of a unit. Hence the ontological status of a unit as well as the logical principles of identity and contradiction is foundational in the compatibility principle. It is clearer to say that the compatibility principle depends on onto-logical identity. For anything to enter into any relation, such a thing must first of all *be* (by its ontological substance and per identity and contradiction principles). While ontological substance satisfies what I refer to as Unit-Identity-Condition (U-I-C), logical identity relation satisfies what I refer to as Unit-Class-Identity-Condition (U-C-I-C). Hence, the onto-logical conditions or determinations in compatibility principle.

**Compatibility Principle observable in the History of Philosophy**

The idea that entities are compatible by their common property is attested to in the history of philosophy.

Gottfried Wilhelm Leibniz’s “identity of indiscernibles” is an ontological principle which states that two or more objects or entities are identical (are one and the same thing), if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa. In other words,

for any x and y, if x and y have all the same

properties, the x is identical to y x y[ P(Px Py) x=y.

Again, for any x and y, if x is non-identical to y, then x and y differ with respect to some property x y[x.y P(Px Py)].

The principle of identity of indiscernibles is related with the principle of indiscernibility of identicals. Indiscernibility of identicals affirms that,

for any x and y, if x is identical to y, then x and y have all the same properties x y [x=y P(Px Py)].

In otherwords, for any x and y, if x and y differ with respect to some property, then x is non-identical to y x y[ P(Px Py) x.y.

The principle of indiscernibility of identicals states that if two objects are in fact one and the same, that they have all the same properties.

From the above, it is observable that the antecedent of identity of indiscernibles is the consequence of indiscernibility of identicals, while the antecedent of indiscernibility of identicals is the consequence of identity of indiscernibles. Copleston observed that a fundamental idea in Leibniz’s philosophy was probably that of the universal harmony of the potentially infinite system of nature (Copleston A History of Philosophy. Vol. 4. The Rationalists: Descartes to Leibniz, 293).

Martin Heidegger would have come against the principle of identity in his lecture, “The principle of identity”, which begins by opposing a common assumption that identity simply means the unqualified and uninterrupted self-sameness of A as A. According to Heidegger, the logical formula A = A conveys only the belonging togetherness of A with A. To say A = A is first to presuppose that there is a distinction between the first A and the second A, and then to assert that they are alike. Heidegger thus understands the principle of identity as A with A, which enables him to say that there is difference anterior to either unity or sameness. At the heart of identity, in other words, there is difference. To identify one thing with another or even a single thing with itself, it is first necessary that the one who identifies them experience them in time and space. What matters is that meaning occurs through an act of identifying, and identifying presupposes a difference between instances of the phenomena identified (Heidegger “The principle if Identity”, *Identity and Difference *27f/90f). For Heidegger, therefore, there is difference in identity. For Leibniz there is identity in difference.

Besides the idea of compatibility of entities by their common property, there is also compatibility by implicate order whenever entities appear or tend to be dissimilar. Compatibility principle also allows for identity in difference. David Bohm systematically lends credence to this belief.

According to David Bohm, “in the enfolded (or implicate) order”, space and time are no longer the dominant factors determining the relationships of dependence or independence of different elements. Rather, an entirely different sort of basic connection of elements is possible, from which our ordinary notions of space and time, along with those of separately existent material particles, are abstracted as forms derived from the deeper order. These ordinary notions in fact appear in what is called the “explicate” or “unfolded” order, which is a special and distinguished form contained within the general totality of all the implicate order (Bohm Wholeness, xv).

Bohm give primacy, in his conception of order, to the undivided whole, and the implicate order inherent within the whole, rather than to parts of the whole. For Bohm, the whole encompasses all things, structures, abstractions and processes, including processes that result in (relatively) stable structures as well as those that involve metamorphosis of structures or things. In this view, parts may be entities normally regarded as physical, such as atoms or subatomic particles but they may also be abstract entities, such as quantum states. Whatever their nature and character, according to Bohm, these parts are considered in terms of the whole, and in such terms, they constitute relatively autonomous and independent “sub-totalities”. The implication of this view is that nothing is entirely separate or autonomous. With application to the compatibility principle, entities can be compatible by the implicate order inherent within the whole, if not by the common property they share. Bohm said: “the new form of insight can perhaps best be called undivided wholeness in flowing movement. This view implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form and dissolve in this flow” (11). According to Bohm, a vivid image of this sense of analysis of the whole is afforded by vortex structures in a flowing stream. Such vortices can be relatively stable patterns within a continuous flow, but such an analysis does not imply that the flow patterns have any sharp division, or that they are literally separate and independently existent entities; rather, they are most fundamentally undivided. Thus, according to Bohm’s view, the whole is in continuous flow and hence is referred to as the holomovement (movement of the whole).

A key motivation for Bohm in proposing a new nation of order was what he saw as the incompatibility of quantum theory with relativity theory (Wikipedia.org). Bohm summarized the state of affairs he perceived to exist:

…In relativity, movement is continuous, causally determinate and well defined, while in quantum mechanics it is discontinuous, not causally determinate and not well defined. Each theory is committed to its own notions of essentially static and fragmentary modes of existence (relativity to that of separate events connectible by signals, and quantum mechanics to a well-defined quantum state). One thus sees that a new kind of theory is needed which drops these basic commitments and at most recovers some essential features of the older theories as abstract forms derived from a deeper reality in which what prevails is unbroken wholeness (xv).

Bohm maintained that relativity and quantum theory are in basic contradiction in these essential respects, and that a new concept of order should begin with that towards which both theories point: undivided wholeness (67).

He argued that each theory was relevant in a certain context, that is, a set of interrelated conditions within the explicate order, rather than having unlimited scope. And that apparent contradictions stem from attempts to over generalize by superposing the theories on one another, implying greater generality or broader relevance than is ultimately warranted. Thus, Bohm argued:

…in sufficiently broad contexts such analytic descriptions cease to be adequate…‘the law of the whole’ will generally include the possibility of describing the ‘loosening’ of aspects from each other, so that they will be relatively autonomous in limited contexts…however, any form of relative autonomy(and heteronomy) is ultimately limited in holonomy, so that in a broad enough context such forms are seen to be merely aspects, relavated in the holomovement, rather than disjoint and separately existent things in interaction. (156-167).

Bohm’s theory is compatible with Bell’s theorem. Bell’s theorem implies that the apparently “separate parts” of the universe could be intimately connected at a deep and fundamental level. Bohm asserts that the most fundamental level is an “unbroken wholeness” which is, in his words, “that–which-is”. All things, including space, time and matter are forms of that-which-is (Zukav 323-324).

**Conclusion: The relationship between Compatibility Principle, Principle of Complementarity and Complementary Reflection**

What relationships are there between Bohr’s principle of complementarity, Asouzu’s complementary philosophy and the compatibility principle? It must be noted that the three ideas: Bohr’s complementarity, Asouzu’s complementarity and my idea of compatibility share something in common, namely, the fact that they all have to do with relations. Bohr’s complementarity admits of a co-existence of mutually exclusive concepts, the complementarity of wave and particle. Asouzu’s complementarity relate all existent realities to each other in the manner of mutual service. The compatibility principle thrives on the onus that entities are compatible if they share similar properties and that if they tend to be dissimilar, that they remain compatible by implicate order.

In complementarity, Asouzu argues that incompatible opposites and contraries can co-exist (Asouzu 304). In authentic complementary reflection, the mind, Asouzu says, concedes to the co-existence of opposites and tries to see how they can be related to each other in non-contradictory ways. In the resolution of the incompatibility posed by opposites, we are offered the possibility towards extracting the harmony that unifies them towards eradicating all forms of disharmony. Through the determination of the reasons for incompatibility between opposites, we can create room for their compatibility. An important dimension of complementary reflection would be the determination of the conditions of possibility for compatibility of opposites.

The compatibility principle gives a solution to this problem by affirming that entities with similar properties are compatible, and as well compatible by implicate order. Asouzu’s complementarity which stresses co-existence of opposites as well as harmonious complementation suffices for implicate order. The implicate order is non-manifest at the physical realm, such as when we could observe similar physical features between entities. Co-existence is emphasized by the three theories of relations.

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