Crash course in mathematics: Lesson 4: The genesis of numbers

This lesson is about mathematical thinking. If it is the first lesson of this crash course you encounter, or you missed some lessons, please go back to the beginning and follow the course trough, by using the framework and index, at the end of the introduction

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Long, long time ago, a galaxy far far away existed only in the imagination of a child, called George Lucas. And then, when he grew up, he made the first Star Wars movie and the story that existed only in his mind became an epos.

Epical stories have a life of their own. Even if we can trace them back to a single original author (which in many cases we can't), once this author set his epos free, it is no longer his, as Lucas learned when wanted to re-edit his movies. By the time he did that, the heroes of his Star Wars epos were too real in the minds of millions of others, who knew that Han shoots first!

The reason we take epics so seriously, is that their internal logic, embedded in them by the genius of their creators, helps us solve complex problems in the real world. An epos is a story that generalizes private problems, and once this generalization is established, it may be very upsetting if it changes and thus becomes useless.

Can you see how mathematics is the greatest epos ever told?

If you don't, it's probably because you and mathematics were not properly introduced. If your first Introduction to mathematics was through the standard elementary school arithmetics curriculum, you probably simply had to memorize a lot of meaningless stuff because “that's the way it is”. Imagine a teacher shows pictures of the characters Star Wars and says, “This is Luke”, “This is princess Leia”, “This is Darth Vader”... Without the story or any other context, wouldn't​ you just hate it?

But mathematics is a great epos, and realizing that is the key for understanding both its beauty and its usefulness. “Why learn math?”, is a question that those who understand mathematics as an epos don't have to ask. They know why. So for those of you who were introduced to math in the wrong way, let me help you make a fresh start with a very common way to start telling an epos: A creation story.

In the beginning there was nothing but a pair of curly brackets, floating in endless space. And then, by the power of miracle, they found one another:

{ }

And then the pair of curly brackets started creating the universe. First they said: “Let there be a comma”, and there was a comma:

{ } ,

(Creation stories typically have an erotic aspect, so given what the curly brackets may resemble, I hope no one gets offended by what the comma resembles, because of the difference in size)

And then the pair of curly brackets saw that it was good, and they said, “Let's make another pair of curly brackets, and let it include a copy of all that already exists, and add another comma after that”, and so they did:

{ } , { { } , } ,

And then they started repeating this process

{ } , { { } , } , { { } , { { } , } , } ,

{ } , { { } , } , { { } , { { } , } , } , { { } , { { } , } , { { } , { { } , } , } , } ,

{ } , { { } , } , { { } , { { } , } , } , { { } , { { } , } , { { } , { { } , } , } , } , { { } , { { } , } , { { } , { { } , } , } , { { } , { { } , } , { { } , { { } , } , } , } , } ,

And so on.

And by this, all things that could be counted were created.

And because they could be counted they could be enumerated.

And because they could be enumerated they could be added and subtracted.

And because they could be added and subtracted, they could be multiplied and divided.

And there was evening and there was morning, one infinity.

This story tells us about how numbers came to be (can you see how ? If not, I hope it will become clear during the course). Numbers are the gods of the mathematical epos. Their properties, as rise from the creation story above, are enough to create all of mathematics, as far as we know (although it was not yet proven). They are also the thing that connects mathematics to the real world, trough measurements and estimations. There is, however, much more in the mathematical epos than numbers. There are other creatures and landscapes, adventurers and fun. Some of it… OK… most of it, is quite complex to understand, but the human heroes of the mathematical epos are always there to give a helping hand.

In the next lesson, the last of the part about mathematical, we will meet one of these human heroes. The Hercules of the mathematical epos - Carl Friedrich Gauss.