## 2. Complementarity vs. Supplementarity

The term “complementarity” first appears in William James’s book

Principles of Psychology [

10], in the context of the idea that human consciousness consists of two parts:

“...in certain persons, at least, the total possible consciousness may be split into parts which coexist but mutually ignore each other, and share the objects of knowledge between them. More remarkable still, they are complementary...”

There is a great similarity between the concept of complementarity that James introduced into psychology in 1890 and that Bohr introduced into physics about four decades later. Whether Bohr’s complementarity was influenced directly or indirectly by James’s notion of complementarity is a challenging question for future philosophers of science to answer.

The concept of complementarity emerged in 1926–1927 from intense discussions that transpired between Bohr and his then-assistant Heisenberg in the wake of the latter’s discovery of the

matrix mechanics and

uncertainty relations [

11]. Bohr discussed his philosophy of

complementarity in public for the first time at a meeting held in Como, Italy, in 1927 and published the first paper on complementarity in 1928 [

12,

13]. In 1958, Bohr summarized the concepts of

supplementarity and

complementarity as follows:

“...Within the scope of classical physics, all characteristic properties of a given object can in principle be ascertained by a single experimental arrangement, although in practice various arrangements are often convenient for the study of different aspects of the phenomenon. In fact, data obtained in such a way simply supplement each other and can be combined into a consistent picture of the behavior of the object under investigation. In quantum mechanics, however, evidence about atomic objects obtained by different experimental arrangements exhibits a novel kind of complementary relationship. Indeed, it must be recognized that such evidence which appears contradictory when combination into a single picture is attempted, exhausts all conceivable knowledge about the object. Far from restricting our efforts to put questions to nature in the form of experiments, the notion of complementarity simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instruments and the objects forms an integral part of the phenomenon...” (emphasis added)

The

supplementary and

complementary relations defined above can be conveniently represented as triadic relations among three entities labeled A, B, and C.

Supplementarity refers to the relation in which the sum of a pair equals the third:

As an example of supplementarity, Einstein’s equation in special relativity, E = mc

^{2}, may be cited. Energy (A) and matter (B) may be viewed as extreme manifestations of their source (C), which can be quantitatively combined or added to completely characterize C. As already indicated, there is no common word to represent the C term corresponding to the combination of

matter and

energy. Therefore, we will adopt in this article the often-used term “mattergy” (meaning

matter and ener

gy) to represent C. Through Einstein’s equation, matter and energy can be interconverted quantitatively. The enormity of the numerical value of c

^{2}, namely, 10

^{21}, justifies the statement that

In contrast to supplementarity,

complementarity is non-additive: i.e., A and B cannot be combined to obtain C. Rather, C can be said to become A or B depending on measuring instruments employed: i.e., C = A or C = B, depending on measurement. We can represent this complementary relation symbolically as shown in Equation (3):

where the symbol ^ is introduced here to denote a “complementary relation”. Equation (5) can be read in two equivalent ways:

Statements (4) and (5) should be viewed as short-hand notations of the deep philosophical arguments underlying complementarity, as, for example, discussed recently by Plotnitsky [

5] and Camillieri [

13]. The principles of

complementarity and

supplementarity defined above may operate not only in physics but also in biology, as first suggested by Bohr [

14,

15]. In other words, it may be said that

Statement (6) has been referred to as the

Symmetry Principle of Biology and Physics (SPBP) [

1]. SPBP appears to be supported by the symmetry evident in

Table 1.

In

Table 1, two new terms appear, “mattergy” (see Item 2) and “liformation” (Item 7), whose meanings are explained in footnotes. One of the most significant conclusions resulting from

Table 1 is the assertion that

life and

information are intimately related in biology, just as

matter and

energy are related in physics (see Items 1, 2, 6 and 7), leading to the coining of the new term “liformation” in analogy to “mattergy” (see Items 2 and 7). Another important insight afforded by the symmetry inherent in

Table 1 is the “liformation–mattergy complementarity” (see Item 9), which may be related to the view recently expressed by Lloyd ([

16] p. 38), if

computation can be identified with

liformation or information processing:

“...The computational universe is not an alternative to the physical universe. The universe that evolves by processing information and the universe that evolves by the laws of physics are one and the same. The two descriptions, computational and physical, are complementary ways of capturing the same phenomena.”

## 5. The Planckian Information: A New Measure of Order

As illustrated in

Figure 1f,i,m,p,r, the Planckian distribution equation (PDE) contains a symmetrical equation that overlaps with PDE in the rising phase of the curves. This symmetrical function is referred to as the Gaussian-like equation (GLE), Equation (13), because it is identical with the Gaussian distribution equation, Equation (14), except that the pre-exponential factor is a free parameter, A, independent of σ:

Using the areas under the curves (AUC) of PDE, Equations (8) or (9), and GLE, Equation (13), a new function was defined called the

Planckian information, I

_{P} [

2,

3]:

Our current interpretation of the Planckian information defined in Equation (15) is that it represents the degree of organization (and hence the order) of a physical system resulting from symmetry-breaking selection processes applied to some randomly available (and hence symmetrically distributed) processes, whether the system involved is atoms, enzymes, cells, brains, languages, human societies, or the universe [

3]. Therefore, to better understand the meaning of I

_{P}, it may be helpful to compare it with the Boltzmann—Gibbs and Shannon entropies which are, by contrast, measures of disorder. As is well known, the meanings of the terms “entropy” (and its derivative “negentropy”) and “information” are controversial, perhaps because of their lack of precise, mathematical definitions. However, this is, fortunately, not the case for the phrase “quantum of action” or “quanta of action”. Hence, if “entropy” and “information” can be shown to be related to “quanta” mathematically, such a triadic relation may contribute to clarifying the true meanings of “entropy” and “information”.

The concepts of entropy, quanta, and information all share the common property of being characterizable in three distinct ways—(i) experimentally; (ii) statistically and mechanically; and (iii) mathematically, as shown in

Table 4.

There may be an irreducibly triadic relation among THERMODYNAMICS, QUANTUM MECHANICS, and INFORMATICS, thus forming a mathematical category (the TQI triad or category?). This idea is represented diagrammatically in

Figure 2.

The derivation of PRE (see row 4, column 2) by Planck in 1900 utilized the concept of thermodynamic entropy (see row 3, column 1) [

47,

48], which establishes a

paradigmatic (to borrow the concept from linguistics) [

49] relation between Entropy and Quanta. Therefore, Quanta seem paradigmatically related to both Entropy and Information, which indicates that Quantum mechanics mediates the interaction between Thermodynamics and Informatics (e.g., energy dissipation under-writes all communication of information), thus suggesting the possible irreducible triad or a category as depicted in

Figure 3:

As already indicated, Boltzmann’s entropy equation played a major role in deriving the blackbody radiation equation by Planck in 1900, which in turn led to the derivation of the “Planckian information equation” (PIE) in 2015 [

2,

3]. PIE is shown in row 4 and column 3 in

Table 4 and the related equation, the Planckian distribution equation (PDE), is shown in row 5 and column 2.

Thermodynamics, quantum mechanics, and informatics constitute an irreducible triad of communication (also called semiosis by Peircean scholars [

50,

51]. That is, it is impossible to communicate, at all scales from microscopic to macroscopic to cosmological without energy dissipation (thermodynamics), discrete objects to choose from (quantum mechanics), and selection of one or more objects from the message source (informatics). The idea of “communication as an irreducible triad of thermodynamics, quantum mechanics and informatics” may be algebraically (?) represented as follows:

where C = A^B reads “A and B are the complementary aspects of C”.

## 9. The Fourier Language as the Cosmic Language (Cosmese)

In his book

Quantum Reality ([

8] p. 79), Herbert states that Fourier discovered a new language (i.e., the wave language), referring to the

Fourier theorem. I did not understand the true meaning of this statement until October–November, 2016, when I actually saw the intricate internal structures of water waves as visualized by CymaScope (

Figure 6). Many interesting demonstrations of standing waves generated by vibrating surfaces are available on the Internet. When the solid surfaces employed in generating the so-called Chladni figures (

Figure 5) are replaced by water, amazingly detailed wave forms can be visualized (see

Figure 6 and

Figure 7); this was made possible by the invention of CymaScope [

7]. I agree with Reid that the invention of CymaScope may be akin to the inventions of the telescope (essential for astronomy) and the microscope (essential for biomedical sciences), and the area of investigations opened up by CymaScope, I suggest, may be the “cosmic linguistics”, or the study of cosmic language or

cosmese. This term was coined in [

1] and it was postulated then that the medium of cosmese is “cosmic waves”, in which I now include (i) the strong; (ii) electroweak; (iii) sound; (iv) chemical concentration; and (v) gravitational waves. In other words, I am assuming that, in agreement with Herbert [

8], waves are a new language and as such can be both the

medium and the

message, if

the McLuhan equation (asserting that the Medium = the Message) [

65] (can be applied here. Examples of “wave messages” are provided by the last panel in

Figure 6, where the “meaning” of the first “wave message” is the “relaxed waking state” of the human brain, and that of the second “wave message” is the actively functioning human brain, etc.

CymaGlyphs, some examples of which are shown in

Figure 6 and

Figure 7, may be viewed as words or sentences of the cosmological language (which I would be happy to refer to as the “Fourier language”, in agreement with Herbert [

8], who claims that Fourier (1768–1830) discovered a new language, a

wave language. Hence, I would predict that there would be almost infinite number of CymaGlyphs, all obeying the Fourier theorem but reflecting individual situations—just like in the human language, where we can generate an almost infinite number of sentences obeying a small number of grammatical rules and reflecting individual ideas in the human brain. This is of course the well-known principle in linguistics called “rule-governed creativity” [

49]. In other words, I maintain that:

- (i)
CymaGlyphs are the words and sentences of a cosmological language based on waves discovered by Fourier in 1807.

- (ii)
The grammar of the cosmological language is the Fourier theorem.

- (iii)
The linguistic principle of rule-governed creativity applies to CymaGlyphs.

If the above conjectures turn out to be validated by further inquiries, future historians may regard the invention of CymaScope as the beginning of cosmic linguistics, just as Galileo’s and others’ invention of the telescope is viewed as the beginning of astronomy.

Thus I am inclined to conclude that waves are the language mediating communication throughout the Universe, i.e., cosmese, and hence, based on the McLuhan equation, the following relations can be inferred:

**waves** = the medium of cosmic communication, and

**CymaGlyphs** = the

cosmic messages whose meaning may be identified with “BEAUTY” among others, in agreement with Masaru Emoto [

66].

## 10. Triadic Monism: The Universality of the Irreducible Triadic Relation (ITR)

Burgin’s suggestion that the relationship between

information and

knowledge (or

structure more generally) is akin to that between

energy and

matter [

67] motivated the construction of the scheme shown in

Figure 7, the center of which prominently displays his idea (see Arrows 1 and 4 in this figure and Tables 2–21 in [

1]). Since

energy and

matter are quantitatively related through E = mc

^{2}, which can be viewed as a

supplementary relation and, since the combination of

energy and

matter is conserved according to the First Law of thermodynamics, it would be logical and natural to combine these two terms into one word,

matter–energy or

mattergy, as is widely done. Similarly, it may be convenient to coin a new word to represent the combination of

information and

knowledge, namely, “information–knowledge” or “infoknowledge”, more briefly (see Arrows 4/5 relative to Arrows 1/8 in

Figure 7).

Figure 7 can be viewed as a network consisting of eight nodes/vertices and 16 arrows/edges. If one focuses on the system of arrows, ignoring the names of the nodes,

Figure 7 clearly exhibits a

reflection symmetry. “An object has reflectional symmetry (line or mirror symmetry) if there is a line going through it which divides it into two pieces which are mirror images of each other” [

68]. The reflection symmetry is in turn composed of two complementary pairs (Nature–Form and Body–Mind pairs) and two supplementary pairs Energy–Matter and Information–Knowledge pairs). Thus, the network in

Figure 7 can be said to embody both

symmetry when viewed in terms of the organized system of arrows and “

broken symmetry” or “

antisymmetry” as defined in [

69] when viewed in terms of the names of the nodes.

Figure 7 is

asymmetric as a whole, with symmetric and antisymmetric aspects [

69].

Figure 7 is also a diagrammatic representation of the theory of everything (TOE) proposed in ([

1], pp. 633–642), which seems consistent with the theories of everything proposed by Popper [

70], Rosen [

71], Penrose [

72], and Burgin [

67]. The TOE depicted in

Figure 7 is an attempt to correlate and integrate the following three hybrid concepts, i.e.,

mattergy,

gnergy, and

infoknowledge, using

category theory [

73,

74]. The first two terms appeared in [

1,

17] and the last term,

infoknowledge, was coined more recently motivated by Burgin’s suggestion [

67] that:

where the structure includes knowledge and data.

To facilitate possible future discussions, I suggest that Statement (17) be referred to as the Burgin thesis. Since the combination of energy and matter is often referred to as mattergy, the Burgin thesis suggests an analogous hybrid term combining information and knowledge, which is referred to as infoknowledge.

It is instructive to compare

Figure 7 with the

gnergy tetrahedron (

Figure 8). The

body-centered tetrahedron (BCT) was found to provide a useful topological template to organize various sets of related ideas and principles discussed in many fields of inquiries (Table 10-5 in [

1]), including physics, biology, and philosophy.

Both

Figure 7 and the

Gnergy Tetrahedron,

Figure 8, contain 4 nodes, three of which are common (

Energy, Metter, and

Information) and one is unique to each diagram, i.e.,

Knowledge for

Figure 7 and

Life for the Gnergy Tetrahedron,

Figure 8. Thus there is a lack of congruency between these two diagrams, which is somewhat surprising, since both diagrams are rooted in the same symmetry principles, i.e.,

complementarity and

supplementarity. Again,

supplementarity is an additive relation, i.e., A + B = C, and

complementarity is non-additive, i.e., A^B = C, where the symbol ^ indicates that A and B are

complementary aspects of a third entity C or that A appears as B or C depending on the mode of observations employed.

The lack of congruence between

Figure 7 and the Gnergy Tetrahedron (

Figure 8) may have at least two possible explanations:

- (i)
There may be one or more logical errors embedded in the reasoning behind the formulation of one or both of

Figure 7 and the Gnergy Tetrahedron,

Figure 8, and

- (ii)
Knowledge in

Figure 7 and

Life in the Gnergy Tetrahedron,

Figure 8, may be more or less synonymous or refer to the same object.

If Possibility (i) can be ruled out upon further scrutiny, we will be left with Possibility (ii), which in effect asserts that”

If Statement (18) is accepted, the Gnergy Tetrahedron,

Figure 8 and

Figure 7 become logically consistent with each other, although different diagrammatically, which may be considered to be an example of

antisymmetry discussed by Darvas [

69] (see below).

Symmetry is generally defined as “invariance under any kind of transformation” [

68,

75] or as “the existence of different viewpoints from which the system appears the same” [

76]. Darvas [

69] provides a more detailed definition:

The symmetries embedded in

Figure 7 may reflect the symmetric properties of the reality itself. The concepts of

symmetry and

symmetry breaking are fundamental to physics and philosophy [

69,

76,

77]. According to

complementarism [

78], ultimate reality is

irreducibly triadic [

3,

22], embodying

**A**, which is the complementary union of irreconcilable opposites

**B** and

**C**. That is, complementarism asserts that the ultimate reality is

three in one and hence its philosophical framework is here referred to as

**triadic monism** which can be diagrammatically represented as shown in

Figure 8. According to Darvas [

69], the world is

asymmetric, embodying

symmetry and

antisymmetry:

Asymmetry is the absence of symmetry, i.e., those transformations that fail to preserve any structures or regularities (e.g., the human body is asymmetric under a 2-fold rotation). Symmetry is defined in Statement (19). Antisymmetry is exemplified by the yin–yang symbol of the Daoist philosophy: When the symbol is rotated 180°, the shape of the symbol remains invariant but the black and white colors are exchanged.

According to triadic monism, the Ultimate Reality is

irreducibly triadic. This notion is depicted diagrammatically as shown in

Figure 9. The Ultimate Reality encompasses Complementarity. In complementarity ([

1], pp. 24050), A appears as B or C, depending on how A is observed. The transition from A to B or to C may be considered as “symmetry breaking”, as discussed in physics [

69,

76], which is the universal mechanism of diversification and the

structure formation (or structuration) in our universe. In the same vein, the reverse transition, i.e., from B and C to A, may be referred to as “symmetry making”.

Symmetry breaking underlies the evolution of the physical universe as it gradually cooled after the Big Bang. In contrast,

symmetry-making proceeds in the

mental universe as the human mind abstracts

invariances from observed

diversities in the physical and mental universes.

Symmetry breaking and

symmetry making may be considered as an example of the yin–yang pair that embodies complementarity or supplementarity [

79], depending on context [

59].