In my last post, I argued that a distinction needs to be made between calculation and math. But certainly these are related, because calculation is about numbers, and math uses a lot of numbers. So what exactly is the connection?
Working from my own experience being trained to be a mathematician and educator, as well as from all I have learned about the history of math through the years, it seems fair to say that math arose primarily out of calculation. But these are not identical – it could equally well be said that poetry arose from the most primitive forms of communication, but that in no way means that literature can be reduced to the musings of a civilization that only just begun to develop language. So it is with mathematics.
A particular paradigm shift within the development of language will be useful to talk about. Language most naturally would have began as a tool for describing things around us and visible to us. Whenever words capable of going outside these limits began to emerge, the power of language would greatly increase. In particular, the ability to tell a story is revolutionary. Stories necessarily require some degree of abstract language. You need to be able to talk about times other than right now, places other than right here, things and people that you cannot see, or have never seen, or that might even not exist at all. Development from specific, immediately present events and objects to more abstract, broader concepts would greatly increase the power of language. When you can tell a story, you can also explain why things are the way they are, not just what they are.
Mathematics has a nearly identical progression. Naturally, counting physical objects is where math started. The very most ancient form of counting would not have looked much like we think of it today, either. What we do is more abstract. Say, for instance, that you have some tools that you keep track of, and that you nor anyone around you has yet developed any idea of counting things or of numbers. You still want to keep track of your tools, so you probably want to come up with a way to check whether you have them all. One way to resolve this dilemma would be to learn some verbal pattern that you’d rattle off as you pick up your tools, some pattern of sounds or movements you remember well. These rituals would most likely have been be something rhythmic, like a chant at a sporting event or musical lyrics. Perhaps you pick up one tool for every line of your jingle, or in some way you can remember. Every day, you finish picking up your tools as soon as your memorized ritual is over. If one day, you finished gathering all your tools but you hadn’t finished yet, you’d conclude that someone took something from you!
That is likely something like how things began. Notice that nowhere is the idea of a number mentioned. You don’t need number to do this. All you need is some vague notion of “more” and of “less,” and for day-to-day life that would serve you very well. In fact, this is also how we count today. We all memorize the same ritual. Our ritual sounds like “one, two, three, four, …” and so on. This is the essence of counting at its beginnings. With this comparison, we can already begin to see how things might develop. Perhaps disputes that involve what we now call counting led people to realize that having a shared ritual for counting objects would be beneficial for solving disputes. I certainly don’t claim to be an expert on these things, it is quite possible some of what I said is not quite right. But there is some reading behind this, and it does seem like this is a quite plausible account of how humans could gradually develop a counting system.
Like with language, the paradigm shift here occurs when things get more abstract. Today, we know what “two” means, even if we are not told what it is there are two of. In terms of the previous discussion, this is because we understand where “two” belongs in our ritual. But over time, this understanding develops beyond the ritual, and we start to think of the number 2 as a concept in and of itself. Concepts like number and shape, just slightly more abstract than what we see with out senses, radically change how humans were able to socialize.
With this background, it is now easier for me to explain what math is. One possible attempt at defining mathematics is the effort towards understanding concepts like number, shape, and measurement. It is not really about any particular number of trees or cows, or the shape of any particular cave or mountain, but about the ideas of number and shape themselves. For example, there are many round things, and when we do mathematics, we want to understand what roundness is. When we study number, we want to discover rules that we can use to count things.
Hopefully, this has provided something of an introduction to what math is and its history. Later on, I hope to go more in depth into some of the mathematical techniques of ancient civilizations like Egypt, Babylon, and China, because they truly did develop remarkable insights for their time. The next big transition in the history of mathematics came with the ancient Greeks. Even though Greek mathematics is roughly 2000 years old, the work done in this marvelous ancient civilization is in many important ways very similar to what is done in universities today. That key development will be the topic the next post.