Dharma as the Number or Code of Existence and Life: Hence Dharma is both a thing and its law, its number in the pythagorean sense, or logos (a cognate of the samskrt Loka) as alluded to by Heraclitus in his 50th Fragment:  “It is wise to heed not me but the Logos and to confess that all things are one”.

The word Muni which applies to the Buddha as well is related to the greek menein (stable in greek) and monas: unity which in English has given rise to the word monk. Monas is the First Principle, as Pythagoras and Leibnitz defined it. The seed (bindu, dot in Samskrit) of a thing: ekatvam.

This introduction to the notion of Dharma and its virtual synonym Rta reveals that science insofar as it is a quest for truth and for the laws of Nature, has Dharma as both object and subject. Science draws laws from the observation of nature by exploring it in order to learn about it. It draws maps of nature while exploring it and inducts laws from observation; as we see in mathematics as in physics, astronomy and even in biology, representation is what enables us to make sense of what we study.

We don’t see the universe as it is but rather through pictures, charts and diagrams formed with the help of various instruments such as telescopes, radio-telescopes, microscopes, computer programmes, particle accelerators, bio-magnetic imagers, radio-scanners, X-ray machines and so on. As Heisenberg explained: “the smallest elements are not physical objects in the common sense, they are shapes, ideas which only the mathematical language allows to describe unambiguously”.

Indeed reflection leads us to conclude that not just the “smallest” (to us) elements but all things share this feature of essential inscrutability. Galileo stated four centuries ago: “Philosophy is written in this huge book which is always open in front of our eyes – I mean the universe – and we cannot understand it without having first to know the language and interpret the characters in which it is written. Indeed it is written in mathematical language”.

Like all equations, Phi, the Golden Ratio, is written in mathematical terms as a relation between numbers but it must be recalled that mathema in greek meant (mental) awakening and related to (omni)science, so that it is consonant, both in phonetically and semantically with the Sanskrit expression: Mahatma. The members of Pythagoras’s “Hemicycle” (his initiatic fraternity) were called mathematikoi, which means the all-knowing or awakened ones. Numbers are not only integers as we have noted in our reference to irrational numbers which, like the often paradoxical notions of relativity, non-Euclidian geometry, quantum theory and fractal quasi-crystal physics, always extend beyond the frontiers of our knowledge that they reshape in counter-intuitive ways.

The discovery of irrational numbers (which they called mute or ineffable) led post-pythagorean philosophers to separate geometry from arithmetic and the cognitive gap was only bridged by Descartes.

A good example of a counter-intuitive mathematical revelation is the fact that any number divided by zero is not, as the spontaneous guess would be, the number itself but the infinite. The Indians who conceptualized the zero in mathematical terms realized it, thus fully comprehending the character of that hitherto elusive notion. Infinite numbers like Pi etc… are other objects of meditation and computation is a synonym of belief so that mathematical thinking is much closer to “mystical” or metaphysical speculation than rational materialists think.

It is another matter that “western” science set as its goal since the Renaissance to acquire “physical” or mechanical mastery and control over the universe. This peculiar course has led modern scientists away from the traditional or Perennial Philosophy whose ultimate goal is to contemplate or celebrate the glory of creation by unravelling (without disenchanting or desacralising) the miraculous secret of its maker who is not separate from his/her creation (4). But when we examine the evolution of the natural and physical sciences and the insights they have gained in recent decades we will see that the nature (the bhutatathata or suchness of the Buddhists) of all things they investigate exhibit the properties enshrined in the polysemic and multifaceted concept of Dharma.

The latter may be equated, as we noted earlier, with the logos of Heraclitus, which expresses the unity of all things whose “conspiracy” (Vitruvius) between the parts and the whole is perceived as beauty both in natural and human creation. Adharma, the negation or Dharma is anrita; the cessation of rta, which is tantamount to death as a result of the breakdown in the “pre-established harmony”. Anrta has its remote echo in the latin english word inertia, which describes the features of non-living objects.

As the Scriptures say: Dharma when it is destroyed destroys and when it is protected (or upheld) it protects (or upholds):

Dharma anahato hante dharmoraksati raksita (Mahabharata, Vanaparva, 313. 128)

A brief study of the aforementioned Golden Ratio and its arithmetical expression, the Fibonacci’s series which was known to many ancient civilizations, including the Indian, Chinese, Mesopotamian, Egyptian and Pre-Columbian American ones, reveals that it expresses in mathematical and geometric terms the very notion of Rta and Dharma, in all the meanings we have described hitherto.

Dharma, Science and the Golden Ratio

Wittgenstein, who seems to have known only western philosophy, warned in his Tractatus that in order to study science, one should forget about philosophy. However that caveat does not apply to Eastern thinking which sees all aspects of knowledge within a truly unified field of investigation and contemplation. A practical illustration of that convergence which reflects the very notion of Rta as “agreement” or “consonance” and further implying a circular or spiralic motion around a centre may be found in the underlying significance of the equation of the Golden Number (phi) – also the initial for the greek physis or Nature, written 

?= (1+   v5)/2

That relation expresses the proportion around which the universe as we know it is organized in its growth and stability laws, at all scales, from the infinitely minute to the infinitely large. The logarithmic or “golden” spiral that reflects this ratio manifests the reconciliation of harmonious though asymmetrical opposites by revolving around the point of origin: the bindu, aksara (primal sound) and stambha (pillar) of Creation. Like Dharma it is ever renewed and permanent in all its variations: eadem mutata resurgo.

The Golden Number illustrates the correspondence between Reason, both as a natural faculty and a rule of justice and ratio as a mathematical proportion. The same property defines Dharma which lays down the equation between all things, including ourselves and the universe that contains and includes them, called by Plato the “Just Mean” intrinsically related to Nagarjuna’s madhyamapratipad. Both Pythagoras and Plato related Law to the harmonic relations regulating the Cosmos and inherent to music.

In her remarkable work entitled “Divine Proportion: Phi in Art, Nature and Science”, Priya Hemenway defines the Golden Section (thus called by Fibonacci and formalized by Martin Ohm) as a “formula that serves as a key to a Great Unifying Principle” and she notes: “when we ponder our place in this equation, we discover that we are at once the whole, the largest part of the whole and also the smallest, in an unvaryingly balanced ratio of the whole to the largest part and of the later to the smallest”.

She further explains: “There is a mathematically demonstrable relation, generating patterns and dynamics diffused all over in nature. The laws of proportion used by artists are derived from it and also in the spiritual field, those principles of harmony are seen as a fundamental truth… They even determine the proportion of our physical bodies” and were defined by Jay Hambidge as a dynamic fundamental symmetry of Nature which lies at the core of traditional art forms worldwide.

The American philosopher, scientist and artist Walter Russell illustrated that cosmic law of harmony in his monumental life work (5). Harmonia in mythology is said to be the daughter of Ares (Arya: aryadharma - as Dharma is presented as father to Hari (Visnu) - or of Hephaistos (Tvstir), the architect and builder of the Cosmos: Visvakarma.

The Mathematics and Geometry of Dharma

It is important to point out that Phi is an irrational number, illustrating the fact that natural law is not reducible to the ratio of an integer and as such has no definite digital representation. It can only be represented by the operation that determines it, because it is a process and not an object. That suggests something very profound by analogy about the character of natural laws. The innumerable dharmas of all beings and things, as well as their common essence or principle the Sanatana or Samanya Dharma may be described as fractal patterns (Mandelbrot sets) in which all parts are similar to the whole (self-similarity principle) though they retain their individuality and symmetry.

Mathematics enunciates its laws through theorems which can be seen as translations of the concept of dharma. A word like ‘triangle’ for instance describes a  geometrical object shaped according to the mathematical laws applying to a flat surface containing three angles; however the specific shape of a triangle  reflects its particular properties (i.e. isosceles, equilateral, rectangle). Likewise there is a more general dharma for a species and a more specific one (visesa) for each sub-species, family and individual. Further for the human being who wishes to attain enlightenment and liberation from the trammels of cyclical mortal existence, there is a transcendental (ekantika) dharma.

Indeed, theorems (a word derived from the greek verb “consider, examine, observe”, related to theoros: spectator and theater) are defined as statements proven on the basis of previously established statements (other theorems) or accepted but not necessarily proven statements (axioms). Theorems, at least in mathematical language are necessary consequences of hypotheses and are hence deductive though they almost always involve elements of empirical evidence and speculation. In physics too, theorems or laws inevitably involve assumptions and intuitions.

Theorems can also generally be described as laws or principles and cannot always be formally proven to be true. They are used as building bricks for formal theories or systems which in turn can be defined or characterized by “wider” theorems known as “metatheorems”. Therefore, Dharma, especially in its timeless sanatana form (Pali akaliko dhammo for Theravada Buddhists) is a metatheorem about the universe defined as a formal system such as, for instance, the “Net of Indra” or “Golden Egg of Brahma” (hiranyagarbha).

The cosmos is defined in contemporary physics as a “network of interactive events”, rather like the mayasamsara of Hindu and Buddhist metaphysics and we can try to show how the patterns and processes mapped out by contemporary research are quite consistent with the concepts laid out in the cosmological sastras of Indic literatures which have influenced the spiritual and scientific heritage of most of East Asia.

The Divine Proportion, The Fractal Universe and the Web of Dharma

Until recently it was thought that chaos and order were two separate realities, the former denoting the amorphous and inanimate world while the latter was characteristic of living organizations. However in recent decades, the discovery of the properties of fractals by Mandelbrot, Penrose and others has led to a gradual unification of chaos and order which was already central to many ancient philosophical systems of Asia such Hinduism, Buddhism and Taoism.

For instance, the basic notions derived from the study of crystallography were anticipated by the Indic descriptions of Indra’s jeweled net (indrajala or brahmajala for Buddhists), the Tantric Vajrayantra (tantra means network, web) in which “diamonds” are packed and ordered periodically and symmetrically like knots or links in a fabric or like the atoms and molecules of a crystal. Alan Watts describes it as “a multidimensional spider’s web in the early morning, covered with dew drops. And every dew drop contains the reflection of all the other dew drops… And so on ad infinitum…” (6).

Analogies with contemporary research in this area of geometry and physics are provided in the book Indra’s Pearl, The Vision of Felix Klein (2002) by Mumford, Series and Wright and the image of Indra’s net was used by Douglad Hofstader in his well-known work Godel, Escher and Bach to explain the structure of complex interconnected networks.

As Priya Hemenway explains in her book, the discovery of quasi-crystals, which appear to be the fundamental structures for life (such as biotic cells) and are packed in an orderly, aperiodical but non-amorphous sequence, reveals a hitherto unknown state of matter and, even more significantly, enables us to peer into a wondrous cosmic process: the interaction between neighbouring atomic clusters can lead to the formation of stable and dynamic solid born out of sharing between them.

The creation of such new structures that do not diminish or disrupt their co-creators takes place through a series of “discrete atomic jumps” or phasons, illustrating the property described in Isopanisad as purnasya purnamadaya purnat purnam udacyate and provides a tangible parallel to the Vedic concept of yajna or sacrifice of the One to give birth to the many and of the many to reintegrate Oneness, in a cyclical process of expansion  or translation of the One into the plural (Brahman, from brih) and of return of  the many dharmas into the Dharma at the source that pervades (Visnu, from Vis) all.

Mathematics, physics and biology can therefore be understood as tools to observe and describe the ceaseless rotation of the wheel of Dharma at all levels of manifestation.

Paper presented at inaugural seminar of Bhopal Dharma-Dhamma University, July-August 2012

Notes:

1)      Article dharma in wikipedia, last checked on 12 August 2012, 3 p.m.

2)    “Virtue, Success, Pleasure and Liberation –The Four Aims of Life in the Tradition of Ancient India” (1993) by Alain Danielou, quoted in Sadhu Vivekjivandas “Hinduism – An Introduction” Part II,  (Swaminarayan Aksarpith, Ahmedabad 2010).

3)     “Nagarjuna, The Philosophy of the Middle Way”, David Kalupahana, SUNY Press (1986).

4)     Jon Lilly: “The miracle is that the universe has created a part of itself meant to study the other part and that this part, by studying itself, ends up finding the rest of the universe in its natural and internal reality” (quoted in  “Divine Proportion” by Priya Hemenway, Stirling Publishing House, 2008).

5)     “The Secret of Light” (1947) and “The Message of the Divine Iliad” (1948-49) by Walter Russell.

6)    “Following the Middle Way”, Alan Watts, podcast.com 2008.08.31, quoted on Wikipedia article Net of Indra, checked on 12 August 2012, 3,30 pm.

(Concluded)