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Minimal Conceptual Abilities for Development of a Number System

July 29, 2017


With one or two possible rare exceptions that will not be dealt with here, animals (including birds, insects &c.) do not use numbers though some primates and birds can be (with difficulty) taught by humans to use them. The reason is quite simple: all species including our own until quite recently (on an evolutionary scale) got along perfectly well without number systems as such. What most, if not all, ‘advanced’ species do have is the ability to make ‘Rough Quantitative Assessments’  (RQA). Rivals for food or females have to decide rapidly whether it’s safer to fight or flee and herbivores of whatever gender have to decide whether one locality has more, or less, nutritious plants. Such assessments usually, implicitly or explicitly, distinguish a threshold: below the threshold it is considered advantageous to fight, above it not. Experience, the great teacher, aids ― or rather obliges ― the species or tribe to hone their rapid assessment abilities since survival and reproductive success in a competitive world may well depend on it. When faced, say, with a threatening group of hungry rivals after you have made a kill (as often occurs), you don’t need to know the exact number of your assailants or their age and state of health ― and trying to work all this out would waste valuable time. You only need to make a snap decision on the information you’ve got ― but this is easier said than done. Subtle though such abilities are, they must be distinguished from numerical procedures.
A very important extension of this ‘RQA’ sense is the ability to notice at once when something important from a set is missing. Primitive peoples so-called regularly astounded explorers or missionaries by their ability to keep check on their packs of dogs, horses or herds of cattle with staggering accuracy (Note 1) even when the culture barely had a number system at all. We still have this ability up to a point but it has been allowed to atrophy because we don’t practise it enough. A schoolteacher generally recognizes at once if a pupil is absent, a collector surveying a roomful of curios at once spots a missing item in a show case and so on. This ability is impressive and doubtless once again of evolutionary importance but it is a ‘pre-numerate’ ability.
To be able to develop a number sense and be capable of manipulating numbers reliably, two ― and as far as I can see only two ― cognitive abilities are required.

The first is the ability to sharply distinguish between ‘one’ and ‘many’, singular and plural, ‘one’ and ‘more-than-one’ (Note 2). But doesn’t everyone have this ability all the time? This is debatable. Some psychologists and philosophers claim that the newborn baby, though perfectly conscious, exists in a completely unified world where no proper distinction is made between itself and its surroundings ― everything is a “buzzing, blooming confusion” (Piaget) but a unified and coherent kind of confusion. According to this view, the great attraction of mysticism is that the practitioner temporarily regains this blissful unitary consciousness ― “Everything is One”. Certainly, it would seem that there can be no awareness of the ‘I’ without awareness of the ‘non-I’. Interestingly, in at least one ancient language, the word for ‘one’ or ‘single’ is the same as the word for ‘alone’ (Note 3).

The second absolutely essential ability for number development is the ability to ‘pair off’  two collections of objects. This ability does not come naturally and primary schoolteachers often have great difficulty in getting tiny children to develop it ― apart from anything else, it seems a rather pointless thing to do. So what, if we can line up two groups of apples (or boys and girls) so that each apple, or boy or girl, from one group is paired off exactly with an apple, or boy or girl, from the other group? Modern mathematicians call this carrying out a ‘One-One Correspondence’ and it is only since the latter 19th century that mathematicians have realized it is the key to understanding numbering.
It is important to note that the child may perfectly well be able to form correctly two sets of paired similar objects, for example pens taken from a pool of pens, but be unable, or refuse, to pair off apples and pens.
Not only children but whole cultures strongly resisted the idea that any set of ‘ones’ ― i.e. discrete objects that do not fuse when brought close together ― can be ‘paired off’ with any other set of ‘ones’ (provided, of course, that there are enough objects in the second set). Several societies, when they did eventually develop spoken and written numerals, had more than one set depending on what sort of ‘things’ were being compared: the Nootka of British Columbia, for example, went so far as to use different number words for rounded objects and long, thin objects(Note 4). Other cultures, understandably, considered it blasphemous to use the same number words or number signs for humans as for gods, which is one reason why the Mayans had three different sets of numerals. Even in our own books, until recently, the date of publication was always given in Roman numerals ― as if ‘years’ somehow required a different set of numerals to everything else. More generally, even today, there is the persistent feeling that there is something degrading and dehumanizing about humans being numbered in the same way as cattle or pieces of wood. Accepting that two sets of objects, no matter what the objects are, provided  they can be exactly paired off, item for item, are ‘numerically equivalent’, ‘represent the same number’, involves making a giant conceptual leap that we still baulk at.

Note 1 : “It was related by a missionary to the Abipones, a tribe of South American Indians compelled by a shortage of food to migrate (in the 18th century): “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 together again.” (…) Yet they [the Abipones] had only three number words and showed the strongest resistance to learning the number sequence from white men.
(…) We can understand such phenomena if we remember the far closer relationship of these people with the world around them: the keen observation that unhesitatingly notes the absence of a single animal and can say which one is missing, and the translation of a number that cannot be visualized into a clearly perceived spatial form.”        Menninger, Number Words and Number Symbols p. 11

Note 2 : “A few other South American languages are almost equally destitute of pure numerical words. But even here, rudimentary as the number sense undoubtedly is, it is not wholly lacking; and some indirect expression, or some form of circumlocution, shows a conception of the difference between one and two, or at least, between one and many” (Courant, The Number Concept Its Origin and Development p. 5)

Note 3 : “The Tacanas of the same country [Bolivia] have no real numerals whatever, but expressed their idea for “one” by the word etama, meaning alone” (Courant, The Number Concept Its Origin and Development p.5)

Note 4 : See The Calendrical and Numerical Systems of the Nootka by William Folan in Native American Mathematics edited by Michael Closs.




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