20 December 2006

Will the real Mr Zero please step forward?

Is zero a number? Or a concept?

If that seems an odd question for me to be asking, look here (last paragraph) for the aside which sparked it. And I hasten to add that, whatever the answer, the validity of what you read there is unaffected.

It's an odd question for me to be asking because I am (on Mondays, Wednesdays and Fridays, at least) a mathematician - and to a mathematician the answer is self evident: yes, of course, it's a number. It's an element of the set of all integers. No question. But, I'm not only a mathematician (which of us is only one thing, ever?) and the question isn't as easily dealt with as that.

It invites other questions. What is a number? What is a concept? Who is asking?

From a societal point of view, the first numbers were the so called "natural numbers", also known as "counting numbers", which are also the same thing as the "positive integers" (abbreviation: J+). They start from 1 ... we don't normally start counting things unless there is at least one of them. Zero is not included ... when did you last start counting money with the words "zero, one, two, three..."? So, from the most fundamental human intuitive definition, no, zero is not a number. (By the same token, negatives then are not numbers either; nor are fractions, never mind the irrationals.)

In any society it is possible to pin down the point at which zero was added. You cannot define a complete number system without zero, and so it comes eventually to every culture once knowledge and understanding reach a certain point. At that point, the concept of zero occurs to somebody. So, philosophically (since you ask: I am a philosopher on the fifth Saturday of the month, if I wake up in time) yes - zero is a concept. (And the analogy with peace as an enabling concept is a valuable one.)

But isn't any number a concept? There are still preindustrial societies where no words exist for numbers above three, or four, or five - the word for any number above that limit usually being the equivalent of "many" - the concept of (for example) "ten" is as absent from such cultures and their linguistics as "zero". And even when we say "one" or "two", we are applying a conceptual label drawn from an abstract mental idea of singularity or twoness - there is nothing concrete connecting identical twins with one whale and one banana ... only n abstract concept of "two".

The Christian and Hindu ideas of a three in one godhead, however different, have in common an existence as theological concept which defies unambiguous linkage to the numbers involved. But as a proximate atheist I'm on shaky ground there. Back to safer territory.

One area where pure mathematics and pure philosophy interpenetrate is number theory. In particular, the definition of our number system from scratch. It goes like this; I leave it to you to decide what it has to say about zero, numbers, and concepts.

For our number system to exist, we must have three minimum irreducible elements: the number "1", the operation of addition (+), and the operation of multiplication (*).

Start with 1. We have a single number; everything else is yet to be imagined.

1

Take that one, and take it again, and apply addition:

1 + 1

We now have a new entity - let's call it "2". We can now apply addition to situations involving our original "1" and the new fangled "2":

1 + 2 = 3 ...or... 2 + 2 = 4

In that way, we can populate the whole space "upwards" from 1, using addition to recombine an ever increasing range of new numbers using the general method a+b=c. These numbers, inevitably generated from "1" by repeated addition, are known as the se of positive integers or { J+} for short.

But that general system does more than we expected, and leads us to unexpected conclusions.

What we are saying is that when we add one known number to a second known number, the result is equal to a third as yet unknown number. Like this:

7 + 1 = n

No difficulty there: the unknown number n is given the name "8" and becomes a known number, catalogued in the ever growing sequence. We can do it the other way round, as well:

n + 1 = 3

Once again, there is no difficulty. Obviously the unknown number n is actually the already known number "2". Similarly:

3 + n = 7

Again, no sweat: this time, n is "4". But now try this one:

9 + n = 3

What's going on here? The system has thrown up a new type of entity: we have invented negative numbers. This time, n is "-6". The definition of our number system has forced us to incorporate not just a new number but a new type of number. We called the numbers marching upward from "1" { J+} in short hand; now we have another set, marching backwards: call them the set of negative integers or, in the same short hand, {J-}.

But the same situation which forced us to recognise negative integers presents us with another new entity which belongs to neither { J+} nor {J-}. Look at this case:

9 + n = 9

This time, the unknown number n cannot be a positive integer (because the answer would then be bigger than nine) and nor can it be a negative integer (because the answer would then be smaller than nine). It belongs to none of the sets of numbers we have discovered so far. This unknown number n is something new ... and so we have discovered, as part of our number system, "0".

Our number line now runs all the way up through the positive integers as far as the eye can see ... and all the way backwards through the negative integers to the opposite horizon ... and in between those sets it necessarily includes (as part of itself) a number which is in neither of them. Taken together, the sets {J-}, 0, { J+} make up a superset {J} which we call the set of all integers.

There is still much business to be done, before we complete the task of defining the full extent of our number system. The next step would involve applying multiplication as we have already applied addition. Out along that long road would lie fractions, irrationals, eventually complex numbers and quaternions ... but it's getting late, and we've come far enough to decide whether or not we are going to count zero as a number.

For a mathematician, the answer is clear cut: the number system has, by its own rules of existence, defined zero as a member of itself. That sort of black and white killjoy attitude to life is why mathematicians are generally lonely and friendless outcasts, never invited to parties, slightly lower on the social scale than tax inspectors. Philosophers are no better; real people are much sexier and have much more fun.

The final, definitive, no messing conclusion, then: zero is a shifty trickster of a character in a coat of many colours, and you'll have to make up your own mind.

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