Assumptions and Abilities Necessary for 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 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.

What is number? One could describe ‘number’ as the ‘property’ that 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 but it does emphasize the curious fact that number is more of a negative rather than a positive property 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, i.e. can be validly allocated the same number label. If I can pair each of 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 —  “that which 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 to the ‘number’ representing the sum? 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 just 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 ‘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 things 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. Note that physical science uses a similar principle which is today so familiar that we take it for granted though it is far from ‘obvious’ (and possibly not entirely true), namely that “what is found to be physically the case for a physical body in a particular place and time is the case for a similar body at a completely different place and time”. Newton’s law of gravitation is not just true here on Earth but is assumed to be true everywhere in the universe — a fantastic generalization that many scientists at the time thought unwarranted and arbitrary.

These principles do 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 a physical reality ‘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 and there seems to be no special reason why they should have the values they actually do have, unless one accepts the Strong Anthropic Principle.  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

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.

SH   4/03/2018