The Whole in Zero
Rupert Wegerif
Rupert Wegerif
A review of The Nothing That Is: a Natural History of Zero, by Robert Kaplan.
(Hardback published in 1999 at £11.99 by The Penguin Press, Harmondsworth.
Paperback due out October 2000 at £6.99)
Resurgence. Spring 2001
We normally take the use of the zero for granted but where does it come from and what
does it really represent? In this fascinating book Robert Kaplan explores the history and
implications of using a sign for ‘nothing’. His account weaves together strands of history,
philosophy and mathematics to show that thinking about nothing provides a way to think
about almost everything.
Perhaps only children are in a good position to question the zero since its use has become
so apparently obvious and so ubiquitous. I once observed a young girl being taught about
numbers.
‘If there are three apples in a basket and I take one out and eat it, how many apples are
left?’ the teacher asked.
‘Two apples’ the girl replied. The teacher was pleased and asked her to write down the
number 2.
‘Now if two of your friends come along and each take one apple and eat it, how many
apples are left in the basket?'
‘There aren’t any more apples’ the girl said. ‘Good’ said the teacher, ‘so how many is
that? Can you write it down?’
But however hard the teacher tried the girl would not go beyond the fact that there were
no apples in the basket. While she could see that three apples had a number and that two
apples had a number she did not accept that ‘there aren’t any more apples’ could also be
expressed as a number. To give the absence of things a visible form as "0", and treat it as
if it was a number looked to this child like slippery dealing.
The first recorded use of the zero was in ancient Sumeria as a punctuation mark or
comma indicating that there were no numbers in one position. This early sign for zero
was written as two wedges pressed into clay. Robert Kaplan claims that the circular shape
it later assumed reflects the indentation made in sand when one of the round pebbles used
for counting was removed. In other words "0" began as a visible trace of something that
was no longer there. For centuries this sign was treated as a handy trick rather than as a
number. This is indicated by the fact that the common word for zero "null" is short for the
Latin "nulla figura" meaning "no number".
That "0" is not a number just like any other number can be easily seen. You can divide
real numbers by other numbers and get a third number, but what happens when we try to
divide by "0"? What is something divided by nothing? The rules of mathematics say that
this move is not allowed but this sort of patching up of problems with extra rules looks
suspicious. Nietzsche often claims that all the great 'truths' of our civilisation were
originally just useful lies. Is it possible that children who refuse the idea of zero are right
and their teachers wrong? Perhaps the very idea of making nothing visible in the form of
"0" is a kind of lie that we should not be tricked into accepting? Mathematicians might
reply to this that mathematics is not about truth and falsehood but is more like a game
that is played by the rules on the basis of axioms. One of the key axioms behind the
current number system is that, 'there exists such a thing as an empty set'. This amounts to
the self-contradictory sounding assertion that 'a nothing exists'. Should we accept this?
Does not the whole point of nothing lie in the fact that it does not exist?
In his big book about nothing, Being and Nothingness, Sartre describes how, when he
goes to the café cafe looking for his friend Pierre, who is not there, the whole café is
permeated by the absence of Pierre. The other people in the café are not really noticed
anymore, they recede into the background in his search for Pierre. Sartre's point is that we
do not find nothing in the world but we bring it to the world. You cannot really
experience nothing but you can make nothing of what you experience by looking for
something else.
If we assert that 'there exists such a thing as an empty set’ an obvious response could be
'empty of what exactly?' or 'What were you looking for that was not there so that you say
it is empty?' If you were out looking for a useful bit of nothing to play mathematics
games with you might want to say that the set is full – it is full of nothing. As
Wittgenstein would say the problem here lies with our grammar. The word nothing is a
noun that we can act on as if it was some kind of thing but in fact it is not. It does not
correspond to anything in the world that you could possibly have a set of.
Heidegger wrote that the fundamental question of philosophy is not why this or that but
why is there anything at all? Or 'why is there something and not nothing'? In asking this
question he was positing a profound nothing that is not some kind of thing, like an empty
set, but the context of the possibility of things. I think his point was that in order to see a
thing as a thing and to define it, it is necessary to contrast it to what it is not, so in order
to see the property of Being we need to contrast this to not-being or the possibility of
nothingness. We can see an apple in a basket or a tree in a field but we cannot normally
see the property of ‘Being’ precisely because everything that is has this property. Being
only becomes visible or tangible for us when it is framed by the possibility of not-being
or Heidegger’s profound nothing. His argument is a bit like the idea in Tolstoy's short
story, The Death of Ivan Illich, that you can only discover what is of real value in life
when you face death. Heidegger seems to argue that the encounter with the possibility of
not-being, is a way to discover the truth of Being. Heidegger was careful to distance
himself from theology but despite this his philosophy has often been compared to the
teachings of the 16th century German mystic Meister Eckhart who claimed that if we
make ourselves and the world nothing we become one with God.
Zero was invented in ancient Sumeria but according to Robert Kaplan its first use as
more than just a punctuation mark is found in Indian mathematical treatises in the 7th
century AD where the new sign of “0” is referred to by the Sanskrit term sünya. Sünya is
also a term in Hindu and Buddhist thinking referring to the full void. This void is ‘empty’
because it has no things in it – you have to strip away all sense of thinglikeness through
meditation in order to experience the void – but at the same time it is full because it is the
pregnant ground from which all things emerge and to which all things return. In other
words sünya is both nothing and everything. It is only perhaps because of this tradition in
India of nothing (sünya) not just being an absence but also a positive force that "0" was
promoted from a kind of comma to an active number and even the ruling divinity of all
numbers. This promotion to number status was seen in attempting to do all the normal
calculations with the “0”. Kaplan describes how the Indian mathematician Bhaskara, for
example, explored adding with zero, subtracting with zero, multiplying with zero and
dividing by zero. But we have already seen that dividing by zero is a sum that we are not
supposed to do because it threatens to undermine the whole number system. The problem
is that if we divide by zero we can easily show that all numbers are equal because any
number divided by zero equals any other number divided by zero. If all numbers are
equal the game of mathematics comes to an end. Despite this Bhaskara referred to the
sum of dividing any real number by 0 as 'the infinite and immutable', terms normally
reserved for the highest divinity also called Brahman. Kaplan expresses surprise that
Bhaskara would allow a move that threatens to undermine the whole of mathematics. But
perhaps Bhaskara was trying to say something beyond mathematics. If we think of “0” as
the smallest possible number, then dividing any real number by 0 must produce an
infinite result. So we get the sum, a/0 = ∞. To describe this sum as Brahman is like
saying that dividing something by nothing equals divinity. Perhaps this is nonsense as
Kaplan seems to think but another possible interpretation is that it is mathematical poetry
pointing to a spiritual truth. Is it not possible that for Bhaskara the formula a/0 = ∞ was a
way to say what Heidegger and Eckhart also appear to say, that the encounter between a
limited being on the one hand and profound nothingness (sünya) on the other has a
transforming effect leading to the realisation of universality?
Robert Kaplan has obviously set out to write an entertaining book as much as a scholarly
book. His friendly style of writing mean that even the more mathematical sections of this
book are accessible and interesting. On the other hand I became a little tired of his
insistence on being amusing at all times and felt that this sometimes obscured the more
interesting depths of the story he tells. Despite this slight criticism I came away feeling
enriched by the fascinating details and images to think with that he provides in the many
asides and anecdotes that are very much a part of this style of writing. For example, the
ritual conflict between the Mayan god of the underworld, ‘zero’, and a god representing
what could be called the ‘overworld’ of numbers and dates and Indian traders counting
with pebbles on trays lightly dusted with sand so that they could see not only where the
pebbles were at any time but also where they had been. Robert Kaplan should be praised
for drawing attention to this extraordinary story and for illustrating that it is possible to
learn something about the whole of reality through a close examination of just one small
part.
Rupert Wegerif is an educational researcher at the OpenUniversity.http://elac.ex.ac.uk/dialogiceducation/userfiles/nothResurge
nce.pdf
再生核研究所声明297(2016.05.19) 豊かなゼロ、空の世界、隠れた未知の世界
ゼロ除算の研究を進めているが、微分方程式のある項を落とした場合の解と落とす前の解を結び付ける具体的な方法として、ゼロ除算の解析の具体的な応用がある事が分かった。この事実は、広く世の現象として、面白い視点に気づかせたので、普遍的な現象として、生きた形で表現したい。
ある項を落とした微分方程式とは、逆に言えば、与えられた微分方程式はさらに 複雑な微分方程式において、沢山の項を落として考えられた簡略の微分方程式であると考えられる。どのくらいの項を落としたかと考えれば、限りない項が存在して、殆どがゼロとして消された微分方程式であると見なせる。この意味で、ゼロの世界は限りなく広がっていると考えられる。
消された見えない世界は ゼロの世界、空、ある隠された世界として、無限に存在していると考えられる。たまたま、現れた項が 表現する物理現象を記述していると言える。
これは、地球に繁茂する動植物が、大海や大地から、生まれては、それらに回帰する現象と同様と言える。大量に発生した卵の極一部がそれぞれの生物に成長して、やがて元の世界に戻り、豊かな大海や大地は生命の存在の元、隠れた存在の大いなる世界であると考えられる。無数の生命の発生と回帰した世界の様は 生物、生体の様の変化は捉えられても、人間の精神活用や生命の生命活動の様の精しい様などは 殆ど何も分からない存在であると言える。我々の認知した世界と発生して来た世界と消えて行った認知できない世界である。
このような視点で、人間にとって最も大事なことは 何だろうか。それは、個々の人間も、人類も 大きな存在の中の小さな存在であることを先ず自覚して、背後に存在する大いなる基礎、環境に畏敬の念を抱き、謙虚さを保つことではないだろうか。この視点では日本古来の神道の精神こそ、宗教の原点として大事では ないだろうか。未知なる自然に対する畏敬の念である。実際、日本でも、世界各地でも人工物を建設するとき、神事を行い、神の許しを求めてきたものである。その心は大いなる存在と人間の調和を志向する意味で人間存在の原理ではないだろうか。それはそもそも 原罪の概念そのものであると言える。
しかしながら、人類が好きなように生きたいも道理であり、巨大都市を建設して、環境を汚染して生存を享受したいも道理であるから、それらの一面も否定できず、それは結局全体的な有り様の中でのバランスの問題ではないだろうか。人類の進化の面には必然的に人類絶滅の要素が内在していると考えられる：
再生核研究所声明 144（2013.12.12） 人類滅亡の概念 － 進化とは 滅亡への過程である
そこで、結局は全体的な調和、バランスの問題である：
再生核研究所声明 56： アースデイ の理念
発想における最も大事なことに触れたが、表現したかった元を回想したい。― それは存在と非存在の間の微妙な有り様と非存在の認知できない限りない世界に想いを致す心情そのものであった。無数とも言える人間の想いはどこに消えて行ったのだろうか。先も分からず、由来も分からない。世の中は雲のような存在であると言える。
以 上
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