Article: Mathematics of the universe

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Jain cosmology observes mathematical principles found in calculations and geometry formulae. Mathematics is an intellectual field to which Jain theorists have made valuable contributions. More importantly, it is a way of thinking about and mentally organising details of the universe and time in a coherent manner. Classifying space and time into measurable units and seeing recurring patterns in these units, categories and structure of units is a feature of the Jain approach to cosmology.

The unique Jain notion of the universe has deep religious significance. Understanding the whole concept and the different elements that form it, including the structure, cycles of time and actions over time, is vital to spiritual progress. Crucial religious ideas – such as the physical soul going through the cycle of birth, being reborn in different parts of the three worlds, in order to gain omniscience and then liberation – are tied into understanding traditional cosmology. For this reason, passing on accurate knowledge of Jain cosmology has been an important part of scholarly work for centuries.

Communicating information about the universe is based on texts, including scriptures and popular stories, but images of the universe have also been important teaching aids since the earliest times. Since Jain cosmology is remarkably intricate, with recurring patterns, such descriptions and artwork can seem confusingly exhaustive and repetitive.

Units of time and space

The Jain universe is thought of in terms of dimensions and quantities of units. These ideas are discussed at length in philosophical and sacred literature, starting with definitions of time and space units. Jain thinkers have produced a vast vocabulary to describe and understand units of time and space, going from the smallest to the largest, beyond what can be imagined.

Components of Jain cosmology are classified in one of the following ways:

numerable – saṃkhyeya
innumerable – asaṃkhyeya
infinite – ananta.

The smallest unit of physical matter is the atom. Infinite combinations of atoms make up the smallest unit of measurement that can be counted. This is called the ‘extremely fine’.

Similarly, the Jains have gone into great detail in analysing the extremely large or highly numerous (Plofker in Granoff 2009: 65ff.).

Dimensions and quantities

This detail of a manuscript page gives information about the distances between the suns and moons in the Jain triple world. Numbers and mathematics underlie the symmetry and repetition that are noticeable in traditional Jain cosmology.

Distances separating the suns and moons
Image by British Library © CC0 1.0 (Creative Commons Public Domain)

Cosmological texts detail the quantities and proportions of parts of the universe. There are many examples of the vital part mathematics plays in traditional Jain concepts of the universe.

One is the location and number of certain items. For instance, in the middle world of humans, the Jyotiṣka gods – the astral bodies such as planets – are distributed as follows:

Jambū-dvīpa – 2 suns, 2 moons

Lavaṇa-samudra – 4 suns, 4 moons

Dhātakīkhaṇḍa – 12 suns, 12 moons

Kālodadhi – 42 suns, 42 moons

Half-Puṣkara – 72 suns, 72 moons.

Each moon has the following retinue of:

planets – 88

constellations – 28

stars – 6,697,500,000,000,000,000,000.

A second example is the dimensions of the various continents and oceans that make up the Two and A Half Continents.

Sizes of the parts of the Two and A Half Continents
Name	Size in yojanas
Jambū-dvīpa	100,000
Lavaṇa-samudra	200,000 x 2 (east and west) = 400,000
Dhātakīkhaṇḍa	400,000 x 2 = 800,000
Kālodadhi	800,000 x 2 = 1,600,000
Half-Puṣkara	800,000 x 2 = 1,600,000
Aḍhāī-dvīpa	4,500,000 yojanas in total

Repetition and symmetry

This manuscript painting of the temples and trees on the three terraces of Mount Meru emphasises the symmetry and repetition that are hallmarks of Jain cosmology. Mount Meru is the cosmic axis, centre of the three worlds of the Jain universe

Temples and trees on Mount Meru
Image by British Library © CC0 1.0 (Creative Commons Public Domain)

The mathematical characteristics of recurrence and symmetry are very noticeable features of Jain cosmology. The universe is ‘a self-replicating composite’ (Granoff 2009: 56). These qualities are especially obvious in the intricate pictures of the universe, which may make them puzzlingly complex.

There are plenty of illustrations of the universe’s repetitive, symmetrical nature, chiefly in the three worlds. Examples abound in the middle world, particularly in the Two and A Half Continents, where humans live.

Firstly, the 45 continents of the middle world are all modelled on the central continent of Jambū-dvīpa.

Next, each of the 90 continents and oceans is bigger by a factor of two than the one inside it.

Thirdly, in the Two and a Half Continents there are a total of five Mount Merus, the cosmic axis, distributed as follows:

one at the centre of Jambū-dvīpa
two in the second continent of Dhātakīkhaṇḍa, one per half
two in the inner part of the third continent of Puṣkara, one in each half.

In addition, Jambū-dvīpa has many features that are noticeably identical, symmetrical or both. For instance, the landmass itself demonstrates recurring, often symmetrical, proportions of mountains, regions, lakes, rivers and so on.

Prominent examples include the regions of Airāvata, Bharata and Mahā-videha. The northern region of Airāvata is a mirror image of Bharata, the southern region. Jambū-dvīpa’s ten capital cities are equally divided between Bharata and Airāvata. They sit between identical branches of the river that flows across each region and are all called Ayodhyā. The central region of Mahā-videha is comprised of 32 provinces laid out in two matching sets of eight rectangles. These are either side of each of the twin rivers that flow east and west across the region. Each pair of sets therefore makes up 16 rectangles in the east and 16 in the west.

The next example is the social organisation of the deities in the three worlds. The gods and goddesses have a hierarchical society very similar to traditional Indian cultures, in which the ideal form of government is monarchy.

Finally, in the Lands of Actions – karma-bhūmi – around half of Mount Meru, time moves in cycles. These cycles of descending and ascending periods of time are endless. Although not of equal length, they are characterised by repetitiveness and predictability.

Reading

Commentary on Tattvārtha Sūtra of Vācaka Umāsvāti Pandit Sukhlalji: translated by K. K. Dixit
L. D. series; volume 44
L. D. Institute of Indology; Ahmedabad, Gujarat, India; 1974