1.2.2 - Carving The Notch: Dawn of Mathematics

1.2.2

Carving The Notch : Dawn of Mathematics

Origin of The Mark

It is said that the origins of mathematical thought lie in the concepts of number, magnitude, and form. As we learned in the previous chapter, animals possess a rudimentary awareness of number and so it is reasonable to assume that prehistoric man also possessed an innate understanding of quantity. Man transcended the animals however, once he made the first leap of abstraction. The ability of man to associate one thing with another, to let one stand for another, this is the leap of genius which propelled him forward in all his endeavors. It can be argued that mathematics really began on the day that the first man carved out the first notch unto a bone to represent the count of one.

The earliest record we have for such man made marks is an animal bone that is dated to approximately 35,000 B.C. Known as the Lebombo Bone, it is the oldest known mathematical artifact.

The bone is actually the fabula of a baboon. On it's surface a series of 29 distinct and deliberate notches can be delineated. Although there is no way for us to know what this was meant to count or record or measure, it seems clear there are really only a few reasonable possibilities. It might be a count of some sort, perhaps animals killed, or the number of members in the tribe. It might be a primitive calendar, perhaps a count of days. It might have a ritualistic purpose, in which each notch was like a prayer bead. Regardless, each of these examples involves some form of counting. It also involves some form of representation of the notch for something else.

There are other well known examples of these notches on bones. Another series of bones known as the Ishango bones, also contain similar notches. These bones are said to date back perhaps as much as 20,000 years ago! That's still well before the beginning of recorded history.

What these bones perhaps represent is the earliest recording of number. In this earliest of "notations", if we may call it that, a single vertical line was used to represent a single item/individual. A group of such marks stood for a group of such items/individuals. As we learned in the last chapter, the fundamental notion of numbers comes from one-to-one correspondence. There was implicitly a one-to-one correspondence between the number of notches and the number the notches represented. We may call this "direct representation". Direct representation represents the lowest level of numerical abstraction. While this leap of intuition is a huge one, it is really only the beginning. Direct representation has all the basic advantages which the concept of number provides, but it also has a lot of the limitations inherent to our raw number-sense.

The Tally Mark

It is interesting to note that the dawn of number did not begin with a word for "oneness", "twoness", "threeness", etc. The real breakthrough was the idea of one-to-one correspondence: The idea that one thing could stand in for another.

A number of methods can be used to record a counting number in this manner. For example, one could place a stone in a cloth bag for every sheep in their flock. I've heard that in ancient times, when men went to war, a pile of stones was made, with each stone representing a fallen soldier. One can of coarse use any object to represent the count of one, small stones being practical simply because they are plentiful. One could of coarse use sea shells, small bones, or chips of wood just as easily.

Imagine a pre-civilization human who owns a flock of sheep; a largish one. How can he be sure that he has not lost any of his animals?

Lets assume the man had a small bag of pebbles. By Pairing each pebble with each sheep in his flock, their would be a one-to-one correspondence between the number of pebbles and the number of sheep.

In modern language we would say that the size of the two sets is "equal". Afterwards if the man wanted to account for all his animals, he would simply pair them off with the pebbles. He could do this by emptying the bag of pebbles and lumping the flock somewhere. He could then let the sheep enter a fencing area one-by-one and for each animal entering the fence take one pebble and place it in the bag. If at the end all the pebbles had been returned, he would know that all the animals were accounted for. But if some pebbles were still out of the bag, some of the sheep were missing. For just as there would be pebbles out of the bag, there would have to be just as many sheep outside of the fence. The bag and pebbles therefore act as a kind of model for the fence and sheep respectively.

What's interesting is that the man would not necessarily have a name for the number of sheep, and would not instantly be able to compare it to someone else's flock. Yet this was a perfect way to measure set size.

Another more practical idea is to create some kind of mark or indication on a single object, such as a bone or stone tablet. One method involves tying knots into a cord, each knot representing a count of one. The most common method however is to simply create a vertical groove into some material. This is what is commonly referred to as a "tally mark". A collection of such marks is a "tally".

We can say that a tally is defined as the number equal to exact count of tally marks it contains. Let's say a shepherd wants to represent his flock. Using ":=" to mean "is defined as" we can say that:

The "tally" mainly serves as a form of primitive data storage. As long as the tally marks remain intact, we can read off the tallies to get a feel for the numbers size. We can also duplicate the number through one-to-one correspondence. Tallies give us a direct sense of a numbers size. However they have all the limitations of our ordinary old number-sense when it comes to quantifying exactly and comparatively. While it may be a simple matter to say:

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This can only be done easily because our inherent number sense allows us to distinguish three from five. For a more difficult example, consider whether there are as many tallies as apples in the next image:

As it turns out there are both 19 tallies and apples in the above image. Without counting however you can not verify this.

The main difficulty is that it is difficult to keep your place in a set of identical lines and errors are likely to occur. To remedy this a practice eventually arose of placing the fifth mark across a set of four. This forms what is called the "5-bar gate", or simply "5-bar":

We can also come up with an ascii form of this trick, by letting the fifth tally mark be slanted to the left:

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While this alleviates some of the problem by tapping into our natural grouping abilities, it does alleviate it by much. With a significant enough count, there will be enough 5-bars that we won't be able to easily count them all:

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Just like with our ordinary number-sense, once we encounter the count of jelly beans in a jar, people in a stadium, or stars in the heavens our tally system will not be of much greater help than our raw senses. Also note that this is no different than the notation established in the first chapter where a set of object symbols was enclosed between parenthesis, as in (ooooo), where "o" stands for an arbitrary object.

Some Alternative Tally Systems

There are also some ancient alternative tally systems of interest. Tally marks used in Spain, South America, Argentina, Brazil, Chile, Venezuela, among many others, follow the following pattern:

Beginning with the usual vertical tally mark, the tallies then continue clockwise to form a box. The fifth tally goes diagonally across.

Another tally system, used in China, Japan, Korea, and Taiwan has the following pattern:

All these tally systems retain this one similar property. However the marks are arranged, the count still remains the number of lines drawn.

Another common shape to use for tallying is the dot. On common six-sided die these are known as pips. The number of pips on a side of a die represents the number of that side:

The arrangement of pips seems designed to tap into our grouping abilities, making them easier to perceive the number without counting. The same pattern of pips can be seen on domino sets. An extended system of pips can be seen in some of the larger sets of dominoes which may go up to as many as 9, 12, or even 15 pips. Here is a typical way of extending up to 15 pips:

No row or column contains more than four pips, so we are able to use grouping to quickly determine the number. Eventually however the pips would be numerous enough that even grouping would not make the total number easily apparent.

In Conclusion

Tally and pip systems are no better at describing the number of jelly beans in a jar, people in a stadium, or stars in the heaven than seeing such multitudes directly. If we are going to move beyond our innate number capabilities we are going to need a much more compact numeric language. In the next articles we will consider the development of numerals from simple tally systems, to our modern decimal system.

Next Article: 1.2.2 - Early Numeration