nouns_to_tallies

1.2.1

From Nouns to Numbers

A Theory of the Human Development of Number

Introduction

Imagine if you will, a time before numbers, or at least a time before there were any words or symbols for them. We would have to go quite far back in the past for this, perhaps at least 50,000 years ago or even more. Still we might imagine that prehistoric man might have been able to get along without any way to communicate numbers and with only his basic intuition about them. How then did man go from having nothing in the way of numbers to what we have today? In this article I will present a theory of how even primitive humans could easily have gone from only a simple understanding of the world to a way to write out and think about numbers.

Primitive Languages

One thing that is usually pointed out as distinct about humans, or at least used to be, is our ability to use language. However communication is not limited to us and many animals have ways of communicating information to each other. One thing to notice about animals "languages" is that they have very primitive grammar. In fact, they arguably have no grammar. "Grammar" is the rules for how to combine words in a sentence and how to translate a grammatically correct sentence. Animals don't have grammar because their "sentences" are usually just single words. These words communicate very simple ideas of things in the immediate environment. For example, a special call might alert the group that a predator is in the area. This "special call" is essentially a single word that means "predator", and by implication "danger". Essentially all the words in these primitive animal languages are nouns: simple objects or ideas. I believe that nouns are the simplest components of language. They inherently have meaning even in isolation: chair, table, cat, time, space, etc. Other words have meaning only when in combination with other words, for example "of" , "and", "the", "is", etc. If you disagree just try to think about what "the" means. It doesn't really have a meaning, but rather it has a grammatical function. Nouns are therefore very different than other kinds of words in that they are not purely grammatical abstractions but act as stand alone ideas. It is probably true that the first words humans uttered were nouns, and eventually grammatical words were developed to relate nouns to each other.

Thinking about thoughts

So primitive humans would at least have had a concept of things in the world. These I will generically refer to as nouns. Nouns are technically a broader concept that includes such intangibles as "love", "hate", "good" and "evil" etc., but I will be talking only about very tangible things in this discussion. In fact, I want to try to avoid abstraction as much as possible at the outset. It's important to see how we transition from concretes to abstracts so that we can see that the abstract is not groundless as is often implied.

Humans would have been aware of many "tangibles" around him. First an foremost, himself and the members of his tribe. Then there were all the animals in his surroundings. What could be more concrete and real?

But already we have abstraction. For the man's awareness of his surroundings, is an idea in his mind, and is different than his surroundings.

Consider the following situation. A primitive man he has recently killed a rabbit. When the rabbit was living and breathing it was as real as real could be to him. But now that it is dead he reflects back on when it was alive. Suddenly he becomes aware of the rabbit in his mind. Is that rabbit alive? Perhaps he recognizes it as just a memory, ... and yet, he now has a concept of the rabbit that once lived but lives no more. If the idea can exist in his head, then he can also try and represent it physically. He decides to create an illustration of this rabbit on the wall of a cave. Perhaps he draws something simple but representative of rabbits:

And here we have the beginning of number, ... sort of. At very least we have the beginning of written language. An idea has now become manifest. Perhaps the man forgets about the rabbit, but when he goes back to the cave he is reminded of the rabbit that once lived but now is dead. The rabbit has become metaphysical in a sense, living on only as an idea in the mans mind, and as a painting on a wall. So as you can see, it doesn't take a great leap to go from the concrete to the abstract. It's almost immediate any time we reflect upon anything not directly in our preview. But the abstraction at this point still represents something very concrete: A specific rabbit that once lived. But abstractions have a way of breeding further abstractions...

Let's imagine that some day later the man has killed yet another rabbit. Again he recalls it being once alive, and decides to immortalize it as well by drawing yet another rabbit drawing next to the first one he drew:

Far from being a meaningless abstraction the drawings hold a rather strong link to reality in that the drawings correspond directly to things that once existed in the real world. The man might even know which drawing matched up with which rabbit since he knows he drew the rabbit on top first.

But now let's mix things up a bit. Say that the mans son finds the drawings and immediately recognizes them as rabbits. He doesn't recognize them as specific rabbits but rather, the idea of rabbits. He decides to try his own hand at drawing one, and draws yet another one to the side of the first two:

The man comes back just as the son finishes the third rabbit. What is interesting about this is that unlike the first two rabbits, the third has no representative in the real world. There is no third rabbit! The man might mistakenly think the son has killed a rabbit, but when he realizes there is no bones left of this third rabbit, perhaps he realizes that his son was merely copying his drawing. The third rabbit, doesn't represent a specific rabbit, but rather the idea of a rabbit. The man then comes to regard his drawings as symbols for the idea of a rabbit, rather than a specific individual.

On following days, the man and his tribe adopt the practice of using standard drawings to represent the number of hunted animals. These symbols are basically nouns. There can be a symbol for "rabbit", "bull", "deer", and whatever else they choose to hunt. A days hunt might look something like this:

So they would know they had hunted four rabbits and two deers that day. Implicitly these items are grouped together by proximity. Space could be used to distinguish between hunts on different days.

So far so good. The tribe now has a simple way to deal with numbers, and as you can see this notation leaves little to the imagination about what's being counted. It's rather unencumbered with abstraction. In fact, this notation is essentially equivalent to the notation established in Chapter 1.1. For example, to state the number of rabbits killed on this day we could simply write:

This defines (in no uncertain terms) the number four. Of coarse we could use much more abstract symbols that are easier to write. For example let "R" stand for rabbit, then we can simply write:

( R R R R )

To mean the same thing. There is one key difference however. The parenthesis I defined earlier are used to define number. Therefore ( R R R R ) doesn't mean four rabbits but rather just four. Our primitive tribe isn't quite up to that yet, but in essence they can now define numbers if they want to by the size of these collections.

I argue that these drawings represent the most primitive system of numeration possible. They represent precisely what they look like. The numbers are there implicitly, but are not made explicit. The numbers are there on the drawing just as much as they are "out there" during the hunt. It is possible that our tribe hasn't quite gotten the idea of number yet, but still sees only individual animals. But I think that numbers are actually innate to some degree in humans, so that even now number must be on the minds of the primitive tribe, even if only subconsciously.

Over time however the tribe begins to find the notation cumbersome, and their cave begins to fill up with these drawings. One option for this problem is to further simplify the symbols for various types of animals. One could for example just draw the heads: