Jainism: The Eternal and Universal Path for Enlightenment: 08 Jainism, Mathematics and Cosmology

God did not have to create each and everything.
All He had to do was to make laws and everything followed by itself.

Philosophy, culture and mathematics are intimately related to each other and each of them builds upon the concepts of the other as is well reflected in ancient Indian thought. Jain scriptures mention that the counting system and script (called Brahmi lipi) were evolved in ancient times by their first Tirthankar Rishabh Dev. The roots of ancient arithmetic and geometry are documented in the history of Babylonia (going back to 5700 BC), Egypt (4000 BC), Sumeru (2500 BC), and Greece (600 BC) where calendar, weights and measures etc. evolved. But formulation of numbers (0 to 9) as we use today and the decimal system was conceived in India. These developments gave a powerful tool to ancient Indian mathematicians in developing several new concepts and in attaining high level of precision in calculations. The discovery of Zero, attributed to the Indian mathematician Pingal (about 200 BC) and decimal system, more than anything else, has laid the foundation of modern physics, astronomy, cosmology and computers. Some scholars believe that the discovery of zero owes much to the concept of Sunyavād, an important tenet of Buddhism and Hinduism. Jains were the first to conceive infinity and to recognize that there are many kinds of infinities. Indian mathematicians, specially Jains applied the mathematical concepts of zero, infinity etc not only to cosmology, astronomy and geography but also to philosophy and culture (e.g. see Ganit Sār Sangrah by Mahvirāchārya, 9^th century A.D.). For example, infinity was much used in the spiritual domain as some faculties of pure soul are infinite knowledge, infinite potency (infinite Jnān, infinite Virya) etc. discussed in Chapter 2).

In the ancient times, the concepts of decimal (and the power of 10) systems made it easy to handle large numbers and enabled the mathematicians to comprehend the vastness of the universe. In many cases they obtained realistic ideas such as in the case of dimensions and age of the universe and other large structures like galaxies. However, in the light of the recent progress in sciences and observational techniques, it appears that several ancient deductions related to geography, solar system and planets were erroneous (Appendix-1). It is not the purpose of this chapter to go in to the history and priorities in arithmetic and geometry but to discuss some concepts which may still be relevant.

The number system enabled ancient Jains to define vast sets of units of time and space, although they did not use decimal system and preferred some kind of binary or other adhoc (multiples of six or eight) systems. Jains divided numbers in to three types, enumerable (countable), innumerable (uncountable, Asankhyāt) and infinite (Anant). Infinity is something which has no boundaries. Asankhyāt (innumerable) is a unique concept developed in Jainism and defines a number with flexible boundaries. It is strange that some number are considered innumerable (Asankhyāt), because the power system, already known, enables one to count any number, howsoever large or small it may be, so that there should be only two types of numbers, countable and infinite, as is currently accepted. Value of Shirsh Prahelika determined to be 10250, calculated accurately to 70 digits is mentioned and is probably the largest number we encounter in Jain scriptures, although we do not know its significance. Asankhyat is also used in connection with the units of time (samay and avalikā) which themselves are very small. It is mentioned in Tatvārtha-rājavartikā that it is not in the power of even the omniscient to know the asankhyāt number precisely. We attach deep philosophical and scientific connotation to the innumerable. We take the view that the number of entities cannot be counted, only if the entity is continuously changing its properties and is indeterminable. We interpret innumerable as not necessarily a very large or very small number but as a number which cannot be determined because the number of entity is changing at every instant. Such examples do exist in physics. Due to particle-wave duality, the number of particles in a box cannot be precisely counted; only their probability can be estimated. This will imply it to be innumerable or asankhyāt. This brings us to the Uncertainty Principle discussed in the previous chapter. Heisenberg found that certain parameters (like energy and time; location and momentum of an elementary particle), both can not be measured with absolute precision but within an error related to Planck's constant (h), not because of the limitations of the instruments but because this uncertainty is the fundamental law of nature. Another example of indeterminate number or asankhyāt may be the number of protons and neutrons in a nucleus. A nucleus is made of protons and neutrons and on the average it is said that a nucleus of, say oxygen, has 8 protons and 8 neutrons. However neutrons and protons are continuously changing from one form to another. This is what the Japanese physicist Yukawa found and proposed an exchange meson called pi-meson. So at any instant it is impossible to say exactly how many neutrons and protons are there in the oxygen nucleus. This we consider is the true meaning of innumerable.

Now this innumerable criteria applies here to smallest units of time (avalikā), only bigger than Samay (innumerable Samay make one Avalikā and Avalikā =1.717x10^-4 seconds, according to N.M. Tatia). It is strange that the Jain time units stop at avalika and then jump to Samay, the smallest unit of time. In comparison, Avalika is quite large compared to Planck time (10⁴³ seconds) used in connection with the Big Bang origin of the universe below which there is some uncertainty in the physical processes occurring there. On the other hand, innumerable is also used in case of units of large space. If we consider large size of the universe, the universe is finite with volume of 343 cubic Rajjus but a Rajju is made of innumerable Yojans x10¹⁴) and thus Rajju is indeterminable in absolute sense, although according to some scholars its value varies between 10¹⁵ to 1022

Cosmology

Origins, i.e. origin of life, origin of universe, origin of earth etc are fundamental questions in philosophy, religion as well as in science. We have learnt from science that every physical process is governed by certain laws, which are well defined and can be mathematically formulated with precision. These laws can never be violated. Origin of everything we see in the universe must have followed certain laws. Thus before there was anything, there were laws. Two questions naturally arise, as to how the laws came into effect and, why are the laws as they are and no different. If everything is a natural consequence of certain laws, then the question arises as to who made the laws? Are there other universes where laws are different from our universe?

To understand the process of origin or essentially who came first, the universe or the laws, three possibilities arise

1. Laws were in effect (existing) before the Universe originated.
2. Laws and Universe (and time) came into existence simultaneously, ie at the same time.
3. Laws are eternal and so is the Universe: they have existed at all times in the past and will continue for ever. They did not originate.

The first assumption explains that the universe originated following some laws and not in an arbitrary, ad hoc or lawless manner and is continuing to evolve according to the same laws. If time and laws are created simultaneously, there was no time before the universe began and the question as to what was before¹ can be avoided. In this assumption, it is implied that the laws are spontaneous, swayambhoo, self-created. If laws themselves are created spontaneously, then the law of spontaneous creation is one of the basic law. Once it is accepted that the laws can appear spontaneously out of nothing then there is no difficulty in generating everything else also spontaneously. If the universe and laws appeared together, i.e. they are coupled, arguably laws will also evolve as does the Universe. They are linked and interdependent. Observationally we find that the Universe has evolved over the ages but laws of physics have remained the Universe same. All attempts to see any time variations in basic laws (for example, the law of gravity) have so far not been successful. The laws are universal, applicable at all times and places. The third alternative is that the laws are eternal. If laws are eternal, then why not matter (ajiva) and jiva too are eternal? This is the basis of the steady state concept of the universe. To resolve the problems stated in the first two assumptions, Jains believe that certain "things" (Jiva, ajiva and other tattvas) are eternal and the universe follows the Steady State model. Jiva and Ajiva always existed as they are now. There is no origin or creation. But we know that the Universe, the earth and the life has originated and evolved (see e.g. chapter 1) and one day they will also be destroyed. In this respect a Buddhist concept is very appealing. They postulate that karmas of sentient beings is the motive force for origination of the material Universe (or at least various habitat planets, like, for example, the earth). Appropriate habitats and their environments are automatically and spontaneously created by nature to meet the requirements of the karmas accumulated by the sentient beings so that they can go through their consequences.

To circumvent the problem of origin and destruction of Universe and the changes we see around, Jainism postulates cycles of various types within a steady state Universe. These cycles result in origin, evolution, sustenance and dissolution of various bodies geometry, placed one above the other (left) to match the shape of the Jain Universe (right). The density of the universe determines its geometry. If the density of the universe exceeds the critical density (W₀), then the geometry of space is closed and is positively curved like the surface of a sphere (left top). If the density of the universe is less than the critical density, then the geometry of space is open, negatively curved like the surface of a saddle (left bottom). If the density of the universe exactly equals the critical density, then the geometry of the universe is flat like a sheet of paper (left, middle). Some of the enigmatic points framed above are eternal questions and have not been satisfactorily answered. Whether they will ever be understood remains to be seen.

Fig. 8.1 a, b Universes with closed (W0 > 1), flat (W0 = 1) and open (W0 < 1)

According to Jainism, the universe consists of Loka which is finite and is immersed in infinite space. Jains have given considerable thought to the structure of the universe, its shape, size and units of time and space. Their concept of numbers (a variety of infinities to innumerables to smallest numbers possible) and the precision, with which the calculations have been made, comparable to the present day precision, is amazing. It is not however surprising in the land where numerals and decimal system were discovered. Unfortunately there is much confusion in the units of space and time because inconsistencies have crept in the undocumented (memorized) records over several millennia. None the less, the concepts are still preserved and comparison of contemporary astronomical dimensions of the relevant Jain structures may be able to resolve the discrepancies in Jain units of time and space. An added confusion arises because at places Jains use devgati (divine velocity) and calculate the dimensions which bear the same name as the common units; For example dev-yojan and yojan have been mentioned and used to describe certain aspects of the universe. In spite of these difficulties we make an attempt here to describe and compare the modern and the Jain concepts.

Modern Cosmology

One of the assumptions on which some of the present theories of cosmology have been developed is that the Universe on a large scale is isotropic and homogeneous. It is infinite in expanse and it does not change with time, an assumption known as the Perfect Cosmological Principle.

Theoretical calculations showed that a static universe is not possible and therefore dynamic Universe models (expanding, oscillating etc.) have been proposed. Before we try to understand the way the Universe originated it is necessary to define what a Universe is. One way of defining it is that it is the totality of space, time, matter and energy. The jiva has no place in the modern cosmology although it is considered to be very important in Jain cosmology.

Fig 8.2 Jain Cosmic periodicities

The most acceptable theory for its origin, well supported by precise observations and theoretical calculations is the Big Bang theory but other models are also possible. The Big Bang model shows that the universe started with a big explosion some 14 billion years ago. There was nothing before this time; even the time was born then.

According to the models, the Universe has expanded, cooled and evolved in ever increasing steps of time, sequentially controlled by quantum gravity, electroweak and then strong nuclear forces. The initial 10^-43 seconds, called the Planck time, was epoch of quantum gravity when temperatures were higher than 10³² Kelvin. Then it entered Grand Unification epoch which lasted till 10^-34 seconds. Electroweak forces dominated up to 10^-10 seconds and were followed by radiation dominated era. Thus in the beginning there was only radiation, which quickly converted in to matter as space expanded, Universe cooled and time evolved. It took the Universe about 100 seconds after the Big Bang, to enter matter dominated era when fundamental particles were formed. The first to form was quark gluon plasma (see Chapter 7). They quickly froze into protons, neutrons and electrons, which in turn fused into hydrogen, helium and lithium. These particles combined in definite proportions to form matter as manifested today. As the Universe expanded in space and time the reduction in temperature allowed the formation of structures like clusters. The matter so formed was dominantly Baryonic which is what one sees around today. As matter dominated, the cooling became faster and the radiation got decoupled from matter and the universe became transparent. Inside clusters, individual stars formed due to gravitational contraction of Molecular clouds of hydrogen, which by thermonuclear fusion produced all the heavier elements and their isotopes. As the universe evolved, generations of stars formed, evolved and died resulting in varied objects including the solar system and life as we have, that populate the universe today. The radiation which decoupled from matter around 300,000 years after the Big Bang cooled and reached the present temperature of around 2.7 K, exhibiting itself in the present era as a nearly isotropic background radiation in the infra-red frequency, as discovered in 1964.

One of the major problems in this scenario is that the laws of physics require that matter and antimatter should form in equal amounts but what we see around is only matter. Where has the antimatter gone? It could form another isolated universe, because matter and antimatter together will annihilate each other.

The universe has been expanding and cooling ever since. In this theory most of the matter was formed in a tiny fraction of a second (10^-35 seconds to 3 minutes) since the beginning and evolution is slowing down gradually. The first galaxies started forming in a billion years from the beginning. The theory is primarily based on the observations of Edwin Hubble that all the galaxies in the Universe are going away from each other. He found that the light coming from a galaxy becomes redder the farther the galaxy is from us. The shift of light towards red colour occurs when the source is rapidly going away from the observer. Hubble found that farther a galaxy, faster it is moving away from us. The Universe has been expanding ever since the Big Bang and as mentioned before, the temperature of the initially very hot universe has come down currently to 2.7 Kelvin because of expansion over 14 billion years.

Whereas everyone agrees with an expanding universe, there have been competing theories to Big Bang. Fred Hoyle, Thomas Gold and Hermann Bondi, proposed a Steady State cosmology in 1948 but Hubble's observation of Expanding Universe contradicted it. The expanding universe requires creation of matter to compensate for expansion. To accommodate the Big Bag event within the Steady State theory, Fred Hoyle, Geoffrey Burbidge and Jayant Narlikar have modified the Steady State theory to a Quasi-Steady State Theory. It is possible that the universe is oscillating and the expanding universe isjust the current phase which is ultimately going to enter a contraction phase. Then the Big Bang theory, in this scenario is only the last phase of this cyclic "Quasi Steady State Universe".

Simply stated, the shape of the universe is determined by competition between the momentum of expansion and the pull of gravity. The rate of expansion is expressed by the Hubble Constant (H) while the strength of gravity depends on the density and pressure of the matter in the Universe. The fate of the Universe is then governed by the density. If the density of the Universe is less than the "critical density" which is proportional to the square of the Hubble constant, then the universe will expand forever. If the density of the universe is greater than the "critical density", then gravity will eventually win and the universe will collapse back on itself, the so called "Big Crunch". However, the results of a recent study suggest that the expansion rate of the universe is actually increasing and not slowing down. One way it can happen is if a form of matter exists which applies a strong negative pressure. This form of matter is sometimes referred to as the "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever.

Recently the temperature of the Universe has been measured using space crafts. The observed density of the universe based on the fluctuations of the microwave temperature is found to be close (within 2% uncertainty) to the critical density, and therefore it appears that the geometry of the universe is flat.

Two things are clear from the observational astronomy. Firstly, everything in the universe is rotating, around its axis and around the centre of the system, be it planets, galaxy or any other object. Secondly everything within the universe is expanding and contracting, in howsoever miniscule manner (akin to breathing), for whatever reason, be it the Sun, Earth, or stars. These two points must be borne in mind when we discuss Jain cosmology because these features are not explicitly mentioned in the scriptures.  

Jain Cosmology

Jains divide the universe in two parts Loka and Aloka. Loka, the visible universe is finite, defined by the existence of the six reals, Jiva, matter, ākāsh, dharmāstikāya, adharmāstikāya and kāl (chapter 7). Beyond the Loka is Aloka, the invisible universe, which has infinite expanse. Mahendra Muni has discussed the Jain cosmology and compared it with modern cosmology in great depth in his treatise "Enigma of the Universe". Briefly, Jain cosmology is a Steady State cosmology. It assumes that the universe has been always like this, without beginning and without end. The Jain concept of a Steady State-Oscillating Universe is as follows.

The shape of the Jain universe (Loka) is very peculiar (Fig 8.1b). This kind of structure is surely not stable unless it rotates around its vertical axis. But rotation can only be inferred with respect to a fixed frame of reference. If the coordinate system rotates with the Universe, we will not notice it. Also, sharp edges and corners are not permitted in large structures by physics. It may be borne in mind that there is no direction (up or down as mentioned in the scriptures) in space. It is difficult to reconcile Jain Universe with the Big Bang Universe. Some crude agreement between modern and Jain concepts can be obtained if the universe is assumed to be a triplet with density greater, equal to and less than the critical density, superimposed on each other as proposed by N.L. Kachhara. A spherical, close Universe with positive curvature, a flat universe with no curvature and an open Universe with negative curvature can co-exist side by side or even overlapping each other. The Loka, which is our Universe, is a flat Universe acting as an interface between the Universe with negative curvature (called Hell by Jains) and the closed spherical Universe is the sphere on top (called Devaloka by Jains) as shown in Fig. 8.1a.

Alternatively, the shape of the Jain Universe looks more like a projection of a four (or more) dimensional object on our 3 dimensional space (see, e.g. P.D. Ouspensky). Some of the modern theories suggest that our Universe may have 11 dimensions and it may be difficult to draw it on the 3D Euclidean space.

The Jain universe basically goes through a cyclic change called the Jain wheel of time (kālchakra). This cycle repeats itself over and over again for ever. One complete cycle is divided in to two parts, Utsarpini and Avsarpini. Each of these have a period of 10 Kodā Kodi Sagaropams, the exact equivalent in years is controversial, but it is very large, and some estimates indicate that 1 KK Sagaropams is equal to 10¹⁴ years, which is difficult to rationalise because the age of the Universe, determined based on Big Bang theory, is only 14x10⁹ years. Alternatively, this would imply that Jain cosmology goes much beyond the Big Bang, and Big Bang Universe is only the current state within a larger cycle in the eternal Steady State Universe. Each of these Utsarpini and Avsarpini are further divided in to 6 epochs or eras each, called Ārā. The period of 1^st Ārā is 4 KK Sagaropams, 2^nd is 3 KK Sagaropams, 3^rd is 2 KK Sagaropams, 4^th, 5^th and 6^th together is 1 KK Sagaropams. The 5^th and 6^th Ārās are equal to 21000 years each. The Utsarpini likewise have six Ārās with periods in reverse order, beginning with the smallest period of21000 years. The names of various Ārās have been styled after "degree of happiness", which may imply favourable or unfavourable climatic conditions. The Ārās are not cycles in the sense that they are not repetative in contrast to climatic, geologic or astronomical cycles which go on repeating over and over again.

The period of the smallest Ārā of 21000 years agrees with one of the Milankovitch solar insolation climatic cycles, as discussed below. Modern climatic changes occur cyclically on several time scales. If we consider cycles which affect life on the earth, then there are cycles which can be classified as climatic cycles, geological cycles and astronomical cycles with increasing periods. Considering only the important ones, there is diurnal cycle, monthly lunar cycle and annual solar cycle. The climate of the earth depends on solar irradiance received by the Earth. Milankovitch found that the solar irradiance depends on three cycles, the precession of equinoxes (which changes inclination of the spin axis of the earth) with a period of 21000 years (19 and 23 thousand years respectively when, at aphelion, the northern hemisphere is tilted away from the sun and towards the sun), obliquity of the earth, which again depends on the inclination of earth's axis to the ecliptic (the plane in which earth moves around the sun in its orbit) which changes with 41000 years period and change of eccentricity of the earth's orbit (which changes the distance of earth from the sun) which varies with a period of 100,000 years. All these cycles have been experimentally confirmed by climate markers (isotopic fractionation records) preserved in the deep sea sediments. It is difficult to say if the jain cycles are climatic cycles, because their names are related to "happiness and unhappiness" and it may just be a coincidence that the two of the Milankovitch cycles agree with the period of 5^th and 6^th Ārā. The 100,000 year climatic cycle, however, does not match with the period of 4^th Ārā, which has a large period, again raising doubt about the jain units of time^[1]. None the less this can be a subject of further investigation.

An important geological cycle (probably related to volcanic periodicity) is 33 million years. The sun moves in the galaxy like a carousel. Astronomical cycles include motion of sun in and out of the galactic plane which is about 60 million years and rotation of sun around the galaxy which is about 250 million years. The sun itself has an expected life of about 10 billion years, before all the nuclear fuel will burn out, and presently we are about half way through its life cycle. The Universe was formed about 14 billion years ago and we know reasonably well how it evolved since then. However not much is known about what happened before the Big Bang and what will happen in the future. The future of the universe however depends on the model used.

We may summarise the discussion in this chapter by stating that modern cosmology is firmly based on observations and theories and Jain cosmology has many appealing concepts. There is scope of reconciling some aspects of Jain cosmology with modern theories. No doubt there are many points of debate, disagreements and dispute but this itself is a reason enough for further investigations.