On reading my recent article on large numbers in Hindu literature, my friend Shri Guruprasath, a neuroscientist, was curious. If very large numbers find mention in ancient Bhārata, what about the pinnacle of large numbers that defies human logic and flummoxes mathematicians to the extent of retiring from their cherished field of mathematics? What about INFINITY? Did the Ancients of Bhārata have words for it or did they even dwell on such a concept? Anyone acquainted with that “sleeping 8” symbol (∞) would find this an interesting question. Here goes my amateur excursion into understanding the concept of Infinity in Hindu mathematics.
Etymology and psycholinguistics
The word ‘Infinity’ is ultimately derived from the Latin word ‘infinitas’, the accusative feminine plural of ‘infinitus’. The original word, ‘infinitus’, means ‘(n)ot enclosed within boundaries, boundless, unlimited’. (1) The parts “in-” is a negative prefix and “finis” denotes ‘end’ or ‘limit’.
A simple search for the word ‘Infinity’ on the popular ‘Sanskrit Dictionary for Spoken Sanskrit’ returns a small list of results. (2)
Figure 1: List of results for the search query ‘Infinity’
The hyperlink to the word ‘ananta’ takes the user to yet another set of results. (3) Of these, the following are most relevant to the current cause.
Figure 2: A subset of the results for the search query ‘ananta’
As ‘ananta’ seems like the only possible synonymous candidate of ‘infinity’, let us analyse this word. The parts ‘an-’ is a negative prefix and ‘-anta’ means ‘end’. Hence, ‘ananta’ means endless.
It is clear from this that both ‘infinity’ and ‘ananta’ have a similar general linguistic structure, i.e. a negative prefix followed by the word for the concept of ‘end’.
Also, the knowledge of existence precedes that of its absence. Simply put, the knowledge of the existence of an object, say a pot (ghaṭavṛttitva), precedes the knowledge of the absence of the object, i.e. absence of the said pot (ghaṭābhāvatva). (see Figure 3)
Similarly, the knowledge of the concept of ‘end-ness’ (antantva) precedes the knowledge of the concept of ‘endless-ness’ (anantatva). Hence, we may infer that the ancients developed the concept of ‘infiniteness’ independently after being exposed to and exhausted by ‘finiteness’.
Figure 3: Knowledge of existence precedes that of absence – a simple visiualisation
Hypothetically, ‘infinity’ may be said to have been a lucrative philosophical musing during the days of high mortality, low health services, and low resource availability. But this hypothesis may not necessarily hold good in all cultural contexts.
Mentions in Śāstra and historical issues
Amongst others, the word ‘ananta’ has been used to refer to Ādiṣeṣa (one of the Prajāpatis) (4), Sūrya (the Sun god) (4), Śrīkṛṣṇa (4), Viṣṇu (4), Śiva (4) and Balarāma (5) in different Purāṇas and the Itihāsas.
The word ‘ananta’ finds mention several times in the Upaniṣads. (6) Anyone exposed to the fundamentals of Vedānta would remember the fact that one of the major differences between the Viśiṣṭādvaita and Advaita schools of thought is the interpretation of the statement “satyaṃ jñānam anantam bramha” from the Taittirīya Upaniṣad.
The word ‘ananta’ also is mentioned a number of times in the Ṛgveda, Yajurveda and Atharvaveda. (6) Furthermore, it is very well known that while describing the Veda, scholars begin with the description: “anantā vai vedāḥ”, i.e. “(The) Vedas (are) verily endless”.
The Veda is considered to be beginningless by traditional scholars. The Veda, primarily being philosophical treatises, is dated closer to the Sarawati -Indus valley civilization by the colonial era influenced historians. This would mean that the Vedic period is dated to 1500 B.C.E. But the philosophical concept of ‘infinity’ is traditionally attributed to the Greeks of the pre-socratic period. Hence, the colonial estimate of the age of the Veda would help us conclude that the concept of ‘ananta’ or ‘infinity’, as a philosophy, may have existed a full millennium before being spoken of by the early Greeks. To put it in simple words, ‘infinity’,i.e. ‘ananta’ must have existed in the minds of the Hindu Ancients at least since about 3500 years ago.
Further, we must understand, as Swami Vivekananda said, “It is an insult to a starving people to offer them religion; it is an insult to a starving man to teach him metaphysics.” (7) The Ancients of Bhārata must have involved in the musings on ‘infinity’ as a philosophy or a mathematical concept after being well established in terms of material prosperity. This is also scientifically plausible based on Maslow’s triangle of hierarchical needs.
The mathematical clues
Shri Joseph George Gheverghese is well known for his book ‘The Crest of the Peacock’. (8) This is what he says in his book:
Again, when it comes to the mathematical concept of ‘infinity’, the Jains have made a significant contribution. Shri Gheverghese says:
Shri Gheverghese writes many more fascinating aspects of Hindu mathematics in his book.
Shri Avinash Sathaye, in his interesting article titled “Infinity in Classical Indian Mathematics” (9), based on Bhskara’s Bījagaṇita and Līlāvatī, states that a zero-divided number is called a ‘khahara’ (1/0). Shri Sathaye goes on to clearly explain much more in his article.
Unfortunately, elucidating the work of Shri Gheverghese and Shri Sathaye is beyond the scope of this introductory article.
I began my quest to know how the Hindu Ancients called ‘infinity’. But there’s a lot more that meets the eye. The ‘ananta’ of the Hindu Ancients is the ‘infinity’ of today. The philosophy around ‘ananta’ dates back to the Veda. As ‘ananta’ occupies a significant place in philosophy, we may dare to guess that the philosophical basis of the concept of ‘infinity’ finds its roots in ancient Indian resource abundance. Based on the colonial estimates for the age of the Veda, we may also dare a guess that the philosophy of ‘ananta’ predated the Greeks by a full millennium.
Mathematical operations with ‘ananta’ have been significantly dealt with by the Jains, Bhāskara, Bramhagupta, and others. Numbers that are divided by zero have also been termed as ‘khahara’.
However long this article is, what needs to be said is forever incomplete… or should I say ‘endless’? 😉
- Lewis, Charlton T. and Short, Charles. A New Latin Dictionary. New York : Harper and Brothers Publishers, 1891. p. 946.
- Sanskrit Dictionary for Spoken Sanskrit. [Online] [Cited: July 2, 2017.] http://spokensanskrit.de/index.php?tinput=infinity&direction=ES&script=HK&link=yes&beginning=0..
- Sanskrit Dictionary of Spoken Sanskrit. [Online] [Cited: July 2, 2017.] http://spokensanskrit.de/index.php?tinput=ananta&direction=SE&script=HK&link=yes&beginning=0.
- Mani, Vettam. Purāṇic Encyclopaedia. Delhi : Motilal Banarsidass, 1975. pp. 34-35. 0842608222.
- Dikshitar, V. R. Ramachandra, [ed.]. The Purana Index. Madras : University of Madras, 1952. p. 50. Vol. 1.
- Bloomfield, Maurice. A Vedic Concordance. Cambridge, Massachusetts : Harvard University, 1906. p. 60.
- Vivekananda, Swami. Religion not the Crying need of India. The Complete Works of Swami Vivekananda.
- Gheverghese, Joseph George. The Crest of the Peacock. Princeton : Princeton University Press, 2011. 9780691135267.
- Sathaye, Avinash. University of Kentucky. /~sohum/ma330/files. [Online] February 22, 2013. [Cited: July 3, 2017.] http://www.ms.uky.edu/~sohum/ma330/files/Infinity_in_india.pdf.