Thanks, everyone! I'm stoked to see that I've got 600 page views as of today. In appreciation of that, I'll be writing a post about numbers today. We're going to try to figure out what a number actually is, and how we might have arrived with a concept like that.
Let's jump right in. What is a number, and where does it come from? Functionally, it's a concept that allows us to keep track of quantity. That doesn't tell us very much, though - it only pushes the question back so that it becomes "What is a quantity?" We could of course define quantity in some other convenient term, but we'd end up with an infinite regress of questions that don't quite touch at the root of it. Number is very fundamentally rooted into our ways of thinking, making it harder to get at than many concepts. The question I'm trying to look at here is not "How can we show the correctness of numbers once we have them?" but "How can we get to numbers from a place without numbers?"
While studies of the evolution of cognition are only getting started (notably, the recent book Adam's Tongue explores the evolution from animal communication systems to linguistic communication in great detail), I do have a hypothesis to suggest about how numbers arose. First, though, a question: what do you think is more fundamental to the experience of being human: noticing quantity, or noticing resemblance? If we look back at animal communication systems, which is a great way for telling which category has been around longer, it's fairly clear that resemblance is a more fundamental category. Vervet monkeys, for instance, have multiple alarm calls to warn their fellows of danger. They don't use the different alarm calls to indicate different numbers of predators, but they do use them to indicate different types of predators, distinguishing a threat from above from one on the ground by use of two different calls.
It's worth noting, however, that chimpanzees can be taught to use a token economy. With the coaxing of the experimenters in control of their environment, they will compete for monetary units that can be exchanged for food. After some time spent teaching them how it works, they simply understand. It works the same way when teaching them to communicate with hand signs - the first sign takes sometimes months to learn, and the fiftieth takes maybe ten minutes. More data would be required to make a conclusive case, but it seems that the faculty of counting and the capability to learn proto-language are linked.
I suspect that the linearity of language is what assists with the development of counting. Only when we can disentangle the flow of our consciousness out of parallel processing and into a stream with a definite beginning and end can we have any hope of counting. Until then, we surely can't hope to learn anything about the internal relations between 'one' and 'two' - we would simply know that we heard or said both of them. When linear processing comes into play, we can realize the implications of the fact that 'one' was said before 'two'. But that doesn't solve the whole problem. Language gives us counting, but it doesn't give us numbers, at least not solely by virtue of its linearity.
The more fundamental pattern-recognition faculty mentioned earlier, which concerns itself with resemblance rather than quantity, however, does. I suspect that the first math done did not involve equations but analogies. One of the neat things we can do with language and our pattern-recognition faculty combined is to make names for qualities. Suppose we encounter two different organisms in whose presence the cells in our olfactory bulb shrivel and recoil, flooding our system with unpleasant neuropeptides. It's natural for us to refer to each of them as "smelly", due to the similar reaction elicited from their presence. We can create words not only for objects in space-time, but also for classes of objects that resemble each other according to our own pattern recognition capabilities.
Once we have similarity, difference follows quite easily. Organism A might have a bit of malodorousness about him, but we might find that organism B's stench is truly profound to behold. A little linguistic innovation makes it easy to express these differences. The exploration of differences and similarities is one of the chief functions of language even today. Once we have these two, spawned from pattern-recognition, we are close to quantity.
Quantity, in fact, could be referred to as a special subset of quality. Quantity is what two apples have in common with two pears (besides being edible fruit), what three bears have in common with three blades of grass (besides being alive), and what distinguishes five dogs from one (besides the temperament and appearance of each). So just like we coined words for degrees of something overt to the senses like color, we coined words for different quantities. However, it's likely we couldn't do this with any precision until linear thought and language pulled themselves together. Counting, as stated earlier, depends heavily on the ability to maintain a linear train of thought.