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Two Approaches to Number : Standard set and Set of All Sets

November 15, 2011

Russell and Frege viewed number — and in what follows ‘number’ should be taken to mean cardinal number, the answer to How many? — as the Set of All Sets that is ‘similar’ to a particular set M. Thus ‘seven’ is a label that English speakers agree to attach to certain collections of objects that are ‘similar’. How do we know the collections are similar ? Because we can pair them off member for member. We can do this for the two sets  ∏ ∏ ∏ ∏ ∏ ∏ ∏  and  Δ Δ Δ Δ Δ Δ Δ because we can form the pairs  ∏ Δ   ∏ Δ    ∏ Δ   ∏ Δ   ∏ Δ   ∏ Δ    ∏ Δ
No member of either set has been left out so if ‘seven’ or sept’  or 7 or VII is to be attached as an identifying mark to the first set, to be consistent, we must also apply it to the seond. As a further step we can form sets with different objects or exchange nembers of one set with members of another. Provided the resulting sets can be paired off with an original ‘seven’ set, we can legitimately apply the same mark or label.
This is a very useful approach provided we do not follow Russell into the quagmires of infinite sets, the Axiom of Choice and so on  : we may at this stage of the game assume  that all sets of objects we wish to number can be exhibited or listed.  It is not expected that we actually exhibit or group together all the members of a particular set since the objects in question may be far away like stars or very numerous like pebbles on a beach,  but it is essential to  believe  that in principle  this could be done.
A rather different approach to number is taken by von Neumann who identifies the Cardinal Number of a Set with a standard set which is selected from all the various sets similar to it. As it happens, everyone in society barring certain disabled or deformed persons is provided with a small portable standard set of numbers, namely their thumbs and fingers. For larger quantities the feet and other parts of the body can be brought into play. For more complex early societies the same principle applied : stacks of cowrie shells were made the ‘standard set’  amongst the Yoruba in West Africa, knots in cords amongst  the Inca. This ‘standard set’ approach, supplemented by the introduction of bases, even applies to the earliest Egyptian written numerals, the so-called hieroglyphic numerals. Here all quantities less than ten were represented by so many upright strokes   | | |  or | | | | |   and the same applied to groups of tens .
The beauty of such a system is that the label or marker for the entire set of all ‘similar’ groups of objects is itself a member of the set and can be paired off with anything you like to mention. One can, using one’s fingers pair them off with a small group of beans, a distant but clearly recognizable clump of trees or even with completely inaccessible groups of stars.  The crippling disadvantage of the system is
that we soon run out of bodily parts and, if we use beads or notches as numbers, we find we require a lot of space to represent even quite modest collections. Also, in the case of recorded numerals, it soon becomes difficult to tell at a glance what quantity is being represented as | | | | | |  looks more or less the same as | | | | |   Some Egyptian scribes got round this by grouping the numerals in clumps with no more than five identical signs in one clump. This makes it a little easier on the eye, but it was not long before the scribes moved on to a
ciphered system so-called where the ‘one symbol’ is not repeated endlessly but different single marks represent quite diverse quantities.
On the other hand, what is infuriating about our ciphered system of numerals is that the written or spoken words cannot be paired off with the groups they represent. Seven is a single word, and even if we take it letter  by letter we cannot pair it off with  Δ Δ Δ Δ Δ Δ Δ
Worse still, each of the following numerical labels 7, 8, 5  or again 785, 462 can be paired off with each other. This being so, it is something of a miracle if children finally catch on to numbering at all. Number books for young children generally present several pictures of sets of well known objects, apples, trees, goldfish and so on with the word SEVEN or SIX in bold at the head of the page. The hope is that by dint of varying the exhibits the child will eventually associate the word with the numerically similar groups portrayed and with no others.
Actually, this conflict between the Russellian and von Neumann approaches to number existed from the very earliest days of humanity. One can, I think, assume that number words came a long time before recorded systems of numerals. Many societies made do with no more than a handful of number words, sometimes just three or two. Thus the Bacairi of Brazil use only the words tokale (one) and ahage (two). Three is ahage tokale (though ahewao also exists), ‘four’ is ahage ahage  and so on (Closs, 1986). This means that even quite small quantities require a lot of words. But inhabitants of such societies often supplement spoken numerals with signs : “They [the Waica] show
exact numbers higher than two by raising their fingers. I have seen them crossing the dwelling to see if a person is holding up three or four fingers in the semidarkness. I have asked for as many as 12 objects and received the exact quantity by showing them four fingers of my hand three consecutive times” (James Barker, 1953 quoted Closs).
Today, we no longer have a standard set of objects that are universally recpognized as numbers : our written numerals are ciphered and calculations are carried out by arrays of minute lights that we do not even see. Numbers have been handed over to the care of professionals, electrical engineers on the one hand, pure mathematicians on the other. But old habits die hard : we still use our fingers
to point to objects on the rare occasions we need to count them and umpires in cricket matches transfer a marble or other small object from one hand to another as each ball is bowled. Curiously, the species which invented mathematics has a very poor innate sense of number, and the more advanced the society becomes the poorer the individual’s perception of quantity. Whereas many missionaries and traders in remote parts of Africa and Latin America were astounded to find that illiterate and innumerate tribesmen could tell at a glance whether a single member of their large herds of livestock and dogs was missing, we are perpetually losing things and making ludicrously inaccurate guesses at the quantities all around us. (Note 2) .   S. H.

Note 1  :     Closs (Editor), Native American Mathematics, 1986  University of Texas Press

Note 2 :   “The long train of mounted women was surrounded in front, in the rear, and on  both sides by countless numbers of dogs. From their saddles the Indians would look around and inspect them. If so much as a single dog was missing from the huge pack, they would keep calling until all were collected again. I have often wondered since how they, without knowing how to count, would tell at once, in spite of the confused throng, that one dog was missing.”
This is taken from an account written by made by of a missionary amongst the Abipones, a tribe of South American Indians and is quoted in Menninger, Number Words and Number Symbols, Dover 1992, p. 10-11.

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