This is my fourth article in the series on History of Mathematics.  If the approach of Vedas to knowledge is different from the western concept of knowledge ( I have explained this in my last article), the approach for gaining such knowledge ,Apara , relating to mundane lifetime activities, which include Mathematics,in Indian system’ basic approach to basic axiom of Knowledge is totally different from the western concept.

Western system of thought has the basic concept as ‘ Ex Nihilo Nihil Fit’ – Out of Nothing, Nothing Comes’.

But Indian system’s axiom is

This is full,That is Full,Having Taken Full out of Full, Full remains Full.

ॐ पूर्णमदः पूर्णमिदं पूर्णात्पूर्णमुदच्यते ।
पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ॥
ॐ शान्तिः शान्तिः शान्तिः ॥
Om Puurnnam-Adah Puurnnam-Idam Puurnnaat-Puurnnam-Udacyate |
Puurnnasya Puurnnam-Aadaaya Puurnnam-Eva-Avashissyate ||
Om Shaantih Shaantih Shaantih ||

Meaning:
1: Om, That (Outer World) is Purna (Full with Divine Consciousness); This (Inner World) is also Purna (Full with Divine Consciousness); From Purna is manifested Purna (From the Fullness of Divine Consciousness the World is manifested),
2: Taking Purna from Purna, Purna indeed remains (Because Divine Consciousness is Non-Dual and Infinite),
3: Om, Peace, Peace, Peace. Source.Green Message Isa Upanishad, Yajurveda (400 BC)

What is applicable to Metaphysics is applicable to Physics in Indian Thought.So, we have totally varied approaches to knowledge,but the essence is the same, except that in the western axiom it is Nihilism and in Indian Thought ,it is Positivism.

Hence difference in paths traveled by Indian mathematics and other sciences is different from the West.

In Indian Thought, Darkness is a positive concept,absence of darkness is Light and not the other way; Acquisition of something from outside sources is Knowledge in the West and is a positive concept.

We find two Concepts in Vedic Thought. That of Infinity and Sunya. Infinity is limitless, beyond description and comprehension for it is beyond our mind which is shackled by Space and Time.Indian thought considers Infinity as a positive concept and ‘Finite’ is a negative concept in that What is Infinite is Real and what is Finite is because of our Limitations in Perception. On Infinity and what is Finite is a topic that needs a detailed analysis and it would be out of place here.

Now onto Nothingness.

Nadadiya Suktha is the 129th hymn of the 10th mandala of the Rigveda (10:129). It is concerned with cosmology and the origin of the universe. Source.Nasadiya Sukta

Nasadiya Sukta (Hymn of non-Eternity, origin of universe):

There was neither non-existence nor existence then;
Neither the realm of space, nor the sky which is beyond;
What stirred? Where? In whose protection?

There was neither death nor immortality then;
No distinguishing sign of night nor of day;
That One breathed, windless, by its own impulse;
Other than that there was nothing beyond.

Darkness there was at first, by darkness hidden;
Without distinctive marks, this all was water;
That which, becoming, by the void was covered;
That One by force of heat came into being;

Who really knows? Who will here proclaim it?
Whence was it produced? Whence is this creation?
Gods came afterwards, with the creation of this universe.
Who then knows whence it has arisen?

Whether God's will created it, or whether He was mute;
Perhaps it formed itself, or perhaps it did not;
Only He who is its overseer in highest heaven knows,
Only He knows, or perhaps He does not know.

—Rigveda 10.129 (Abridged, Tr: Kramer / Christian source.

Nothingness – Is it a positive entity? Absence og Attributes is an Attribute.

And Infinity has directions.

Surya Prajnapti which is thought to be around the 4th century BC and the Jambudvipa Prajnapti from around the same period, have recently received attention through the study of later commentaries. The Bhagabati Sutra dates from around 300 BC and contains interesting information on combinations. From about the second century BC is the Sthananga Sutra which is particularly interesting in that it lists the topics which made up the mathematics studied at the time. In fact this list of topics sets the scene for the areas of study for a long time to come in the Indian subcontinent. The topics are listed in [2] as:-

… the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

The ideas of the mathematical infinite in Jaina mathematics is very interesting indeed and they evolve largely due to the Jaina’s cosmological ideas. In Jaina cosmology time is thought of as eternal and without form. The world is infinite, it was never created and has always existed. Space pervades everything and is without form. All the objects of the universe exist in space which is divided into the space of the universe and the space of the non-universe. There is a central region of the universe in which all living beings, including men, animals, gods and devils, live. Above this central region is the upper world which is itself divided into two parts. Below the central region is the lower world which is divided into seven tiers. This led to the work described in on a mathematical topic in the Jaina work, Tiloyapannatti by Yativrsabha. A circle is divided by parallel lines into regions of prescribed widths. The lengths of the boundary chords and the areas of the regions are given, based on stated rules.

The Jaina cosmology contained a time period of 2^{588}2588 years. Note that 2^{588}2588 is a very large number!

2^{588}2588 = 1013 065324 433836 171511 818326 096474 890383 898005 918563 696288 002277 756507 034036 354527 929615 978746 851512 277392 062160 962106 733983 191180 520452 956027 069051 297354 415786 421338 721071 661056…….

Infinity Jainism

Jaina mathematics recognised five different types of infinity [2]:-

… infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite.

The Anuyoga Dwara Sutra contains other remarkable numerical speculations by the Jainas.

For example several times in the work the number of human beings that ever existed is given as 2^{96}296. Source. Jaina Mathematics ‘ Accessibility Statement.’https://mathshistory.st-andrews.ac.uk/Miscellaneous/accessibility/

These Jain texts date from 4th Century BC.

Jainism is a part of Indian philosophical system ,though it does not recognise Vedas as a source of Knowledge,much like Buddhism. These are called Nastika systems of Indian philosophy along with Carvaka,Ajivaka,Mimamsa, Vaiseshika and Nyaya systems.

More articles on Vedic Mathematical Concepts follow.