What do you need to create a Number System?

“He who examines things in their growth and first origins will obtain the clearest view of them”  Aristotle

TODAY the natural numbers are treated as entirely abstract entities and as such are evolved either from the Peano Axioms (Note 1) or, if you want to be really stylish, from the Axioms of Zermelo-Frankel Set Theory, especially ZF6, the so-called Axiom of Infinity. Few people  apart from professional pure mathematicians find such ‘derivations’ convincing and those who are unmathematical find this whole way of proceeding both mystifying and repellent. One does not in fact recognize what we think we know as numbers in these abstruse procedures.
I propose to approach the question in a completely different, pragmatic spirit. What do we need if we are going to make — yes, actually make in the sense of ‘put together’, ‘construct’ — a number system? Well, to start off with you need a lot of ‘ones’, more or less similarly sized distinct objects that are readily manageable, like shells, beans, twigs. Such a set of ‘concrete numbers’, once officially consecrated , serves as a standard set against which you can pair off actual or imaginary collections of ‘ones’. Ideally, you want to be able to make as many new ‘ones’ as you are likely to need : but even in a relatively advanced society you are not likely to need any more shells or beads that you can either find or make.
Secondly, you need to be able to play around with this standard set, dividing it up, bringing little groups together and so on. This is not so easy to achieve since small objects get lost or otherwise ‘disappear’. We thus require objects that can be neatly piled up into stable structures like cowrie shells, or we require hollow objects like beads that can be imprisoned on the threads or wires of an abacus frame and slid to and fro. The Romans used marbles moving in grooves on hand-held calculi, the distant ancestor of our electronic calculator. There are other methods of a rather different nature, such as making knots in cords or marks on the surfaces of larger objects like walls. The important thing is that it should be relatively easy to group and regroup the concrete numbers. Manouevres with groups of vasrying size constitute the three basic operations of arithmetic, division, addition and subtraction. (Multiplication is a rather more comkplex procedure and is not, to my mind, a fundamental arithemtic operation.)
Thirdly, you will find, or might find, that you need to define standard subgroups of increasing size. Human beings are surprisingly inapt at assessing at a glance (subitizing) even quite small collections of objects accurately : indeed, it is in part because we are so bad at doing this that mathematics evolved in the first place. Take a guess at how many shrubs there are in a garden in front of you, how many gulls there are sitting on the grass in a park or on the beach, even how many different objects there are on the table in front of you. Unless you are something of a prodigy, you will be amazed to find how far out you usually are. Most people can only subitize reliably collections of seven or eight objects : any larger collections we have to count  one by one. For accurate assessment, it soon becomes necessary to break up larger collections into easily recognizable smaller ones, and, fairly early on, societies realized that it would be extremely convenient to introduce this feature into the concrete number system. The idea of ‘bases’ was born : one of humanity’s most useful inventions.
It is not absolutely necessary that the various subgroups should have a single ‘base’ like our 1, 10, 100, 1000… The Babylonians had 60 as their principal base — which is why we have sixty seconds to the minute and three hunded and sixty degrees — but, since 60 is relatively large, they found it necessary to have a ‘sub-base’ of 12.  The Yoruba used a mixed five-twenty system : “In the northern parts of the region, the cowrie shells were counted out in groups of five, while along the coast they were pierced and threaded,m generally in strings of forty” (Zaslavsky, Africa Counts p. 225). The same author states that the ‘Lagos system’ went like this :
     “40 = 1 string
2000 =  1 head = 50 strings
20,000 = 1 bag = 10 heads”

There is, however, such an obvious advantage in keeping to a single base that the vast majority of advanced societies have done so.  There is only one ‘transformation rule’ to learn and one can conceptualize the entire process as a matter of collectimg together so many units where the value of the ‘new unit’ changes by the same amount each time, one becomes ten becomes ten tens or a hundred and so on. As long ago as 2000 – 3000 BC  the Egyptians evolved an excellent written system which went up to one million and is very clear and easy to use — I have tried it out myself. Usually the base chosen has been what we call ten and this is certainly no accident. 10 is only slightly larger than the limit of our subitizing capacity (8) so there is not too much of a gap and, more important still, we have ten fingers and thumbs. The first computer was the human hand and had we been born with twelve fingers and thumbs, mathematics would have got off to an even better start since 12 is obviously superior for calculation because it has more factors (we cannot even get a proper third  or quarter in our decimal system). 
  Nature is not consciously mathematical, or even numerate : if certain specific numbers keep coming up — and few do systematically — there is generally some physical or biological reason for their appearance and reappearance.  In this sense it is perfectly true that numbers, or at any rate number systems, are human creations but they are firmly based on features of the natural world that really exist objectively. Even the most recondite properties of the natural numbers are, in effect, ‘brought into existence’ just so soon as we have a universe where there are so many ‘different bits’ as opposed to one single, unified ‘bit. One might say, to paraphrase Guy Debord, “Number has always existed but not always in its numerical form” (Note 2) .
The axiomatic method whereby all sorts of properties of a system are deduced from an initial small number of key assumptions has had a long and glorious history but there is no reason to suppose that this has anything to do with how the universe became what it is today. Certain numerical features can be singled out and thrown into a deductive mould, but all this is done for our convenience, that is all. Practically speaking, there are two only two abilities are necessary to develop and use with confidence a number system. They are  :
1. The ability to distinguish between what is singular and plural, i.e. recognize a ‘one’ when you see it;
2. The ability to carry out a one-one correspondence (pairing off).
All the mathematicians who have developed abstract number systems, for example  Zermelo and von Neumann, had these two
perceptual/cognitive abilities — otherwise they would have been denied access to higher education and would not even have been able to read a maths book. Animals, with one or two possible exceptions, seem to have 1.) but not 2.) which is perhaps the reason why they have not developed symbolic number systems (though a more important reason is that they did not feel the need to). Computers are capable of 1.) and 2.) but only because they have been programmed by human beings.
Since numbering has been such a successful activity, there must seeingly be features shared by our man-made numerical systems and the inanimate world : in other words, there must be an objective, physical basis to numbering and calculation. In a future post I will consider what natural ‘principles’ are involved, i.e. I will look for basic facts of existence which are the equivalent of the Formalists’ and Platonists’ Axioms.     S.H.

Note (1)  The Peano Axioms are

P1. 0 is a natural number.
P2. If n  is a natural number, then n’ is a uniquely determined natural number.
P3. For all natural numbers m and n, if m’ = n’, then m = n.
P4. For each natural number n, n’ ≠ 0.
P5. If M is a subset of N such that 0 Î M  and m’ Î M, then M = N.

Note (2) :  Guy Debord wrote, “L’histoire a toujours existé, mais pas toujours sous sa forme historique” (Debord, La Société du Spectacle)