Zero: A philosophical history of an Indian Idea – II
The Men Who Knew Infinity
Zero and infinity are curious beasts; they defy our logical understanding of the world. Try this: could you name an infinitely large number? Any number you think of, however large, will be a finite number. See how a similar problem creeps in when we deal with time related notions of ‘the present’ and ‘eternity’. We tend to think of eternity as an ‘endless’ duration of time. But if it is some duration of time, however long, it cannot be eternity- just the same way as we cannot think of an infinitely large number. What is eternity then? Ludwig Wittgenstein cautioned us against the trap of treating the present and eternity as moments in time. Rather he said they symbolize timelessness: the absence of time. (more on eternity and Wittgenstein some other day)
The properties of Zero are just as logically absurd as that of eternity. It is common knowledge now that one divided by Zero is infinity and one multiplied by Zero is Zero. I fear however most of us don’t actually grasp the profundity of these simple formulae, perhaps because we are so familiar with them. To understand Zero better, we first need to shrug off the anaesthetic soporific of familiarity with this thought experiment suggested by Charles Siefe. Imagine an elastic band as the number line. When you multiply you stretch the band. When you divide you relax the band. This is what happens with routine divisions and multiplications with numbers other than zero. However, the bizarre happens when you operate with zero. Now when you multiply by Zero the band is so stretched that it explodes: and nothing remains as Zero sucks any number into itself. Similarly when you divide by zero the band so contracts that it implodes; nothing of it remains. Such are the properties of the Zero. These operations represent the end of logic as we know it: any equation in the world can be proved by multiplying both sides by Zero. In what philosophical world view could any of these make sense?
All of this absurdity made perfect sense in the Indian world view. The Indians not only embraced the idea of Zero, they actually celebrated it. What follows is the gist – oversimplification even- of the idea of emptiness at the heart of Indian philosophy. The central idea of Indian philosophy was that the infinite phenomenal world (apparently) emerges from emptiness. And when the appearance of the infinite phenomenal world – which is illusionary- is realized to be nothing but mere appearance, the individual consciousness is restored to its real state of attributeless and formless emptiness: Brahman (not to be confused with the caste Brahmin). By a perceptual illusion (maya) the apparently individual consciousness (atman) perceives the phenomenal world of names and forms- with the individual being the subject and the phenomenal world being the object of perception. Realization or enlightenment is end of this illusion – with there being no differentiation between the subject and the object- and state where this obtains is that of emptiness (Brahman).
Different Indian schools of philosophy give different names to this ineffable and nameless emptiness. Some called it Sunya (Zero or emptiness) others called it Nirguna Brahman (attributeless truth) some others, Sunya Brahman. This idea forms the philosophical core of Buddhism and the many branches of Vedanta. Much has been written about whether the Buddhist Sunya and the Vedantic Nirguna Brahmana are metaphysically identical. But this is not the occasion to get into such logomachical and ultimately -I dare say- futile debates. Truth according to both Buddhism and Vedanta is ineffable, being beyond illusionary sensory perception, beyond thought, and eventually beyond language. Both assert that the truth or emptiness is neither equivalent to existence nor to non existence. And finally both assert that enlightenment is the restoration to that very truth; and the path to that involves thoughtlessness and surrender of the false sense of ego.
In short, emptiness or zero was thought to be the truth of human existence and indeed the meaning of life. In Douglas Adams’ Hitchhiker’s Guide to the Galaxy, the supercomputer, Deep Thought, after seven and a half million years of computation, computes the meaning of life to be a meaningless number: 42. This may sound amusing in a humorous fictional work. But here was a philosophy that asserted seriously that the meaning of life was a number: and the number was Zero (Sunya). Sunya is one of the many words for Zero in Sanskrit. Another Sanskrit word for Zero was Poojyam. In Tamil, Zero is still referred to as Poojyam. Poojyam in Sanskrit also means the object of worship. The goal of human life was indeed worship (surrendering the illusionary sense of individual ego) for the realization of the ultimate emptiness (Sunya ). Thus in Zero was to be found the whole meaning and goal of existence.
When Indian Mathematicians introduced Zero they made it clear that they were applying this very philosophical idea in the realm of mathematics. They asserted that Zero ( Sunya or Poojyam) was Nirguna Brahman (attributeless truth) and infinity (Poornam or ananta) was Saguna Brahman ( truth with attributes). Infinity was a manifestation of Zero. No doubt all this sounds too mystical and far removed from modern mathematics. Yet no less a genius than Srinivasa Ramanujam( 20th Century) also held this view, and expressed it in precisely the same Vedantic terminology. After Ramanujam died, amongst his papers at Trinity College, Cambridge was found a note where Ramanujam equated Zero and Infinity with Saguna and Nirguna Brahman. That note provided the inspiration for Robert Kanigel to name his biography of Ramanujam as The Man Who Knew Infinity. But Ramanujam was just one in a line of Indian mathematicians who for over 1300 years understood Zero and infinity in this way. Brahmagupta ( 7th Century AD) who formalized most of the properties of Zero we visited earlier, understood these concepts in just this way. Bhaskara (12th Century AD) who improved upon Brahmagupta’s formalization by introducing the idea that any number divided by zero was infinity also relied on the same Vedantic equation.