Assumptions required to develop a Number System

Nature is not deliberately mathematical or even numerate :   if certain numbers keep coming up — and few do systematically — there is generally some physical or biological reason for them ¹.
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. One might say, to paraphrase Guy Debord, “Number has always existed but not always in its  its numerical form” ².
So how do we develop a number system? What are the minimal requirements?
Two, and as far as I can see, only two abilities are necessary to develop a number system :

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 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.
Cardinal number — and unless stated otherwise I shall be referring to cardinal (how many?) number  — cardinal number may be legitimately viewed as some sort of an abstract quality pertaining to all finite sets. Or can it? The best one can do to define ‘number’ is to say it is what results when we have done away with all other distinctions between sets such as colour, weight, position, shape and so on. This is not much of a definition though it does emphasize the curious fact that number is a negative rather than a positive thing (in the usual sense) since it results, as Piaget says, “from an ignoring of differential qualities”.
But, notwithstanding the difficulty of saying what exactly number is, practically speaking there is a perfectly simple and universally applicable test which can decide whether two sets of discrete objects are numerically equivalent or not. If I can pair them off with the same standard set of objects or marks, the two sets are numerically equivalent, if I can’t they are not. Of course, today if I want to assess the ‘number’ of chairs in a room, say, I associate the collection with a number word, seven or four or six as the case may be, but underlying this is a pairing with a standard set. As a matter of fact I find that, though I use the number words one, two, three…..  when counting objects, I still find it necessary to use my fingers, either by pointing my finger at the object or pressing it against my side, one press, one object. And the umpire in a cricket match still uses stones or pebbles : one ball bowled, one stone shifted from the right hand to the left. It is not that the finger or stone pairing off is valid because of our ciphered numerals but the reverse : our written or spoken numerals ‘work’ because underlying them is this pairing off of items with those of a standard set.
Now, one could actually derive the Cantor definition of cardinal number — roughly, “what results from abstracting from a set the order of appearance of the elements and their specific character” —  from what happens when I apply my test. If I rearrange the objects I am supposed to be counting, does that make any difference? No. Because if I could pair off the original collection with items from a standard set, such as so many pebbles or marks, I can do the same after rearrangement. Does the actual identity of the objects matter? Apparently not, since if I replace each original item by a completely different item, I can still pair off the resulting set with my standard set (or subset).

We thus arrive, either by reflection or simply by applying the test, at the two basic numerical principles, the Disordering Principle and the Principle of Replacement

Disordering Principle

The numerical status/cardinal number of a collection is not changed by rearrangement so long as no object is created or destroyed.

Principle of Replacement

The numerical status/cardinal number of a collection is not changed if each individual object is replaced by a different individual object.

           Together these two principles make up a sort of Number Conservation Principle since whatever ‘cardinal number’ is, this ‘something’ persists throughout all the drastic changes the set undergoes exactly as, allegedly, a given amount of mass/energy persists throughout the interactions between molecules within a closed system.

These two principles may either be viewed as Definitions i.e. they tell you what we mean by cardinal number, or as Postulates  since they are the generalisation of actual experiments (pairing off sets with a chosen standard set). They are not, I think, ‘logical truths’ and not strictly speaking axioms.

The  Principle of Correspondence  has a somewhat different status and is more like a true Axiom, i.e. something which we have to take for granted to get started at all but which is not directly culled from experience.

The Principle of Correspondence  

Whatever is found to be numerically the case with respect to a particular set A, will also be numerically the case for any set B that can be put in one-one correspondence with it.

By ‘numerical’ features I mean such features as divisibility which has nothing to do with colour, size and so forth. We certainly do assume the Principle of Correspondence all the time, since otherwise we would not gaily use the same rules of arithmetic when dealing with apples, baboons or stars : indeed, without it there would not be a proper science of arithmetic at all, merely ad hoc rules of thumb. But, though the Principle of Correspondence is justified by experience, I am not so sure that it originates there : it is such a basic and sweeping assertion than it is more appropriate to call it an Axiom than anything else. It is certainly not a theorem since it is quite untestable and not really a definition either.
This does not by any means exhaust the assumptions we implicitly make when we use or apply a Number System : indeed, if we listed all of them we could probably fill a sizeable volume. For example, we continually assume that there is ‘something out there’ to number in the first place (which solipsists and some Buddhists deny), that there are such things as discrete objects (which philosophic monists and in some of his writings even Einstein seems to deny) and so on and so forth. But these ‘axioms’ are best left out of the picture : they underlie most of what we believe and are not specific to numbering and mathematics.

Notes

¹ This is (perhaps) not true of the basic constants such as the gravitational constant or the fine structure constant : they seem to be ‘hard-wired’ into the universe as it were. There seems to be no special reason why they should have the values they actually do have, unless one accepts the Strong Anthropic Principle which claims  that only such values (or very similar ones) would permit the development of human life in the universe. In theory it should be possible to deduce the values of basic constants from a priori principles but to date attempts to do this, such as Eddington’s derivation of the number N, the number of elementary particles in the universe, have not been very successful to say the least. One could argue from ‘logical’ considerations that there must be a limiting value to the transmission of electro-magnetic signals but there is no apparent reason why it should be 3 × 10⁸ m/sec However, proponents of loop quantum gravity do claim to be able to deduce many physical laws from the equations of General relativity and nothing else, though I am not sure if they actually derive the precise values of  constants.
I am personally coming round to the view that there are strictly no numbers as such built into this universe or any other. What we do seem to have is certain ‘constraints’ which drive physical processes towards certain limits : these constraints must seemingly be independent of any actual universe, otherwise they would be part of it and thus vary along with everything else. By ‘constraints’ I have in mind principles like that controlling pressure in fluids, making air or water move from a region of relatively high pressure to one of relatively low pressure (2^nd Law of Thermodynamics if you like). Newton’s Third Law of the equivalence of action and reaction would seem also to be a basic ‘constraint’ though it is not necessary to believe that such forces are ever exactly equivalent : there is a perpetual oscillation of pressure/reaction to pressure which tends towards a limit. Just recently, Adrian Bejan has come up with what seems to be a new ‘basic law’, one of a more positive type, namely his Constructal Law (his term) which states that “If a finite-size flow system is to persist in time, its configuration must evolve in such a way that provides easier access to the currents that flow through it” (see Design in Nature  by Adrian Bejart & J. Peder Zane).  
² The quotation I have in mind is, “L’histoire a toujours existé mais pas toujours sous sa forme historique” (‘History has always existed but not always in its historical form’) from La Société du Spectacle by Guy Debord. The phrase sounds wonderful but means very little.
S.H.     14/3/12