Apr 212015

Parmenides‘ trouble with ‘nothing’ was nothing new. The ancient Greeks thought the world started with Chaos, a variably imagined primordial mess, where the principle of all things (Arche) eventually gave rise to an ordered Cosmos. Whatever that was – Anaximander called it Apeiron, the limitless – it was something.

This was common to all ancient creation myths, including the Bible. They all started with something. It was only in the second and third century CE that Christian theologians, eager to affirm God’s absolute omnipotence, reinterpreted Genesis as creation ex nihilo. Why is there something rather than nothing? Because God created it. Leibniz was so keen to demonstrate it that he got into his own muddle.

Take the infinite series S=1+x+x2+x3+…= 1/(1-x). For x=-1 we have S=1-1+1-1+…=1/2. That is S=(1-1)+(1-1)+…=0+0+…=1/2. Amazing but true: an infinite sum of zeros equals 1/2. Nothing=Something. Luigi Guido Grandi, a Camaldolese monk and mathematician, saw this as a marvellous representation of creation ex nihilo. Here is another way to see it: S can be written as (1-1)+(1-1)+… or as 1-(1-1)-(1-1)-… . The first sum, with an even number of terms, equals zero, while the second sum, with an odd number of terms, equals 1. So – just like a bit – S is either 0 or 1, depending on when we stop counting. Since we have an equal probability of stopping at an even or at an odd number, the expected value of S is 1/2.

Leibniz – one of the smartest men on earth and co-inventor of infinitesimal calculus – bought the argument. The wonders of binary arithmetic – he thought: God, the One, created all Being from Nothing. He was so impressed with the idea that, according to Laplace, he wrote a letter to the Jesuit missionary Claudio Filippo Grimaldi, president of the tribunal of Mathematics in China, asking him to show it to the Chinese emperor and convince him to convert to Christianity! “I report this incident only to show to what extent the prejudices of infancy can mislead the greatest men” (A Philosophical Essay on Probabilities, p. 169).

As should have been obvious to Leibniz (and likely it was to Grimaldi and to the emperor, who remained an infidel), S is only convergent for -1<x<1. For x=-1 it does not converge to any number, but bounces aimlessly between 1 and 0 – between something and nothing, if you want to insist on the metaphor. But in that case you don’t need the whole series. The first two terms are enough: 1-1=0. Nothing is the sum of two opposite somethings – hot and cold, wet and dry (as in Anaximander’s apeiron), positive and negative, good and evil, light and darkness, matter and antimatter, Yin and Yang, Laurel and Hardy or whatever opposite pair you may fancy. Why not 42-42=0?

Grandi’s series is not a good answer to Leibniz’s question. No wonder – Parmenides would have said. ‘Why is there something rather than nothing’ is a meaningless question: there is no such thing as nothing – we cannot even speak or think about it. In his Tractatus Logico-Philosophicus Wittgenstein agreed:

6.44    Not how the world is, is the mystical, but that it is.

6.45    The contemplation of the world sub specie aeterni is its contemplation as a limited whole.
The feeling of the world as a limited whole is the mystical feeling.

6.5      For an answer which cannot be expressed the question too cannot be expressed.
The riddle does not exist.
If a question can be put at all, then it can also be answered.

Wittgenstein’s world sub specie aeterni is the Einstein-Weyl block universe, and his world as a limited whole resembles Parmenides’ well-rounded sphere. In his 1929 Lecture on Ethics, Wittgenstein described thaumazein as ‘my experience par excellence‘: ‘when I have it I wonder at the existence of the world‘. Like Parmenides, however, he thought that any ‘verbal expression’ about thaumazein was ‘nonsense! If I say “I wonder at the existence of the world” I am misusing language’. ‘To say “I wonder at such and such being the case” has only sense if I can imagine it not to be the case’. ‘But it is nonsense to say that I wonder at the existence of the world, because I cannot imagine it not existing’ (p. 41). It is clear from this that the question Wittgenstein had in mind in 6.5 was precisely Leibniz’s question, which he regarded as senseless – a question that cannot be put at all and therefore cannot be answered. The riddle does not exist.

I find this very strange. What’s so difficult about imagining that nothing exists? Just imagine the absence of everything – any thing, all the 1080 atoms in the observable universe, or however many there are in the whole universe. And if after that you are left with something – a vacuum space – imagine that away as well, until there is nothing – nothing at all. What’s the big deal? Let’s call this ‘Nall’ – short for Nothing at all – which of course is not a thing but just a name for the absence of any thing. Nall may be impossible, but it is certainly not unimaginable. In fact, ‘Why is there All rather than Nall?’ is a shorter version of Leibniz’s question. Nothing senseless about it.

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