Human Number Perception

We discussed the reasoning behind using URLs (which are names) instead of IP addresses (which are numbers) for the location of websites. One of the reasons for doing things in this manner is the concept that humankind can intuitively recognize no more than four objects without having to resort to counting them.

I found this in The Universal History of Numbers by Georges Ifrah (John Wiley & Sons, 2000). I have included below an excerpt from pages 3-10 of Chapter 1, "Explaining the Origins: Ethnological and Psychological Approaches to the Sources of Numbers." The bolding in the extract below is something I added to bring out what I felt were key points.

If you don't want to read all of the item, you can skip directly to the key point: THE LIMITS OF PERCEPTION

WHEN THE SLATE WAS CLEAN

There must have been a time when nobody knew how to count. All we can surmise is that the concept of number must then have been indissociable from actual objects -- nothing very much more than a direct apperception of the plurality of things. In this picture of early humanity, no one would have been able to conceive of a number as such, that is to say as an abstraction, nor to grasp the fact that sets such as "day-and-night", a brace of hares, the wings of a bird, or the eyes, ears, arms and legs of a human being had a common property, that of "being two."

Mathematics has made such rapid and spectacular progress in what are still relatively recent periods that we may find it hard to credit the existence of a time without number. However, research into behaviour in early infancy and ethnographic studies of contemporary so-called primitive populations support such a hypothesis.

CAN ANIMALS COUNT?

Some animal species possess some kind of notion of number. At a rudimentary level, they can distinguish concrete quantities (an ability that must be differentiated from the ability to count numbers in abstract). For want of a better term we will call animals' basic number-recognition the sense of number. It is a sense which human infants do not possess at birth.

[The selection goes on to discuss the notion of numbers among animals with some discussion of various experiments. One of the findings was that animals seem to be able to recognize when a small set of objects has undergone a numerical change when seen for a second time. However, that sense of change seems to be limited to big changes. They can recognize when the number has been halved or increased by a third, but have a hard time noticing that what once was six is now seven.]

NUMBERS AND SMALL CHILDREN

... Infancy is ... a long drawn-out phase of preparation, in which the various stages in the development of human intelligence are re-enacted and reconstitute the successive steps through which our ancestors must have gone since the dawn of time ... Oddly enough, when a child has acquired the use of speech and learned to name the first few numbers, he or she often has great difficulty in symbolising the number three. Children often count from one to two and then miss three, jumping straight to four. Although the child can recognise, visually and intuitively, the concrete quantities from one to four, at this stage of development he or she is still at the very doorstep of abstract numbering, which corresponds to one, two, many.

However, once this stage is passed ..., the child quickly becomes able to count properly. From then on, progress is made by virtue of the fact that the abstract concept of number progressively takes over from the purely perceptual aspect of a collection of objects. The road then lies open which leads on to the acquisition of a true grasp of abstract calculation ...

NUMBERS AND THE PRIMITIVE MIND

A good number of so-called primitive people in the world today seem similarly unable to grasp number as an abstract concept. Amongst these populations, number is "felt" and "registered", but it is perceived as a quality, rather as we perceive smell, colour, noise, or the presence of a person or thing outside of ourselves. In other words, "primitive" peoples are affected only by changes in their visual field, in a direct subject-object relationship. Their grasp of number is thus limited to what their predispositions allow them to see in a single visual glance.

However, that does not mean that they have no perception of quantity. It is just that the plurality of beings and things is measured by them not in a quantitative but in a qualitative way, without differentiating individual items. Cardinal reckoning of this sort is never fixed in the abstract, but always related to concrete sets, varying naturally according to the type of set considered ...

[The selection goes on to talk about studies of peoples in various areas as having terms for numbers in their languages which were, in essence, limited to one, two, sometimes three, and a lot. But even without the concepts of numbers beyond this small set, these people could have manipulated numbers in excess of four by using base 2.]

... The rule is what we call the principle of base 2 (or binary principle). In this kind of numbering, five is "two, two, one", six is "two-two-two", seven is "two-two-two-one", and so on. But primitive societies did not develop binary numbering because ... they possessed only the most basic degree of numeracy, that which distinguishes between the singular and the dual.

THE LIMITS OF PERCEPTION

The limited arithmetic of "primitive" societies does not mean that their members were unintelligent, nor that their innate abilities were or are lesser than ours ... In practice, when we want to distinguish a quantity we have recourse to our memories and/or to acquired techniques such as comparison, splitting, mental grouping, or, best of all actual counting. For that reason it is rather difficult to get to our natural sense of number ...

[The selection then goes on to discuss some experiments that illustrate the difficulty of recognizing quantities without actually counting them]

... Everyone can see the sets of one, of two, and of three objects in the figure, and most people can see the set of four. But that's about the limit of our natural ability to numerate. Beyond four, quantities are vague, and our eyes alone cannot tell us how many things there are.

Are there fifteen or twenty plates in that pile?

Thirteen or fourteen cars parked along the street?

Eleven or twelve bushes in that garden?

Ten or fifteen steps on this staircase?

Nine, eight or six windows in the facade of that house?

The correct answers cannot be just seen. We have to count to find out!

The eye is simply not a sufficiently precise measuring tool: its natural number ability virtually never exceeds four.

... Perhaps the most obvious confirmation of the basic psychological rule of the "limit of four" can be found in the almost universal counting-device called (in England) the "five-barred gate". It is used by innkeepers keeping a tally or "slate" of drinks ordered, by card-players totting up scores, by prisoners keeping count of their days in jail, even by examiners working out the mark distribution of a cohort of students.

  1. I
  2. II
  3. III
  4. IIII
  5. five - four lines with one crossed line
  6. five - four lines with one crossed line I
  7. five - four lines with one crossed line II
  8. five - four lines with one crossed line III
  9. five - four lines with one crossed line IIII
  10. five - four lines with one crossed line five - four lines with one crossed line
  11. five - four lines with one crossed line five - four lines with one crossed line I
  12. five - four lines with one crossed line five - four lines with one crossed line II
  13. five - four lines with one crossed line five - four lines with one crossed line III
  14. five - four lines with one crossed line five - four lines with one crossed line IIII
  15. five - four lines with one crossed line five - four lines with one crossed line five - four lines with one crossed line

Most human societies the world has known have used this kind of number-notation at some stage in their development and all have tried to find ways of coping with the unavoidable fact that beyond four (IIII) nobody can "read" intuitively a sequence of five strokes (IIIII) or more.

[The selection then compared the stroke numbering system of 22 different ancient systems in Asia, Europe, and the Americas.]

To recapitulate: at the start of this story, people began by counting the first nine numbers by placing in sequence the corresponding number of strokes, circles, dots or other similar signs representing "one", more or less as follows:

  1. I
  2. II
  3. III
  4. IIII
  5. IIIII
  6. IIIIII
  7. IIIIIII
  8. IIIIIIII
  9. IIIIIIIII

But because series of identical signs are not easy to read quickly for numbers above four, the system was rapidly abandoned. Some civilisations ... got round the difficulty by grouping the signs for numbers from five to nine according to a principle that we might call dyadic representation:

  1. I
  2. II
  3. III
  4. IIII
  5. III II (3+2)
  6. III III (3+3)
  7. IIII III (4+3)
  8. IIII IIII (4+4)
  9. IIIII IIII (5+4)

Other civilisations solved the problem by recourse to a rule of three:

  1. I
  2. II
  3. III
  4. III I (3+1)
  5. III II (3+2)
  6. III III (3+3)
  7. III III I (3+3+1)
  8. III III II (3+3+2)
  9. III III III (3+3+3)

And yet others ... came up with an idea (probably based on finger counting) for a special sign for the number five, proceeding thereafter on a rule of five or quinary system (6=5+1, 7=5+2, and so on).

There can be no debate about it now: natural human ability to perceive number does not exceed four!

So the basic root of arithmetic as we know it today is a very rudimentary numerical capacity indeed, a capacity barely greater than that of some animals. There's no doubt that the human mind could no more accede by innate aptitude alone to the abstraction of counting than could crows or goldfinches. But human societies have enlarged the potential of these very limited abilities by inventing a number of mental procedures of enormous fertility, procedures which opened up a pathway into the universe of numbers and mathematics ...

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