A Babel of Fractions

Once upon a time there was a mathematician who knew that One could be split into two halves. He was asked to participate in the construction of the Tower of Babel. He began teaching people how to split one thing into two equal pieces, and as more and more people learned, the Tower grew higher and sturdier.

One day, he awoke from his bed to see a swarm of people waiting outside his door. People were confused the halves that they had no longer were the same.

“I swear, 256/512 is the right way to divide one into two equal pieces!”
“No, I can bet my life that 3/6 is the right way!”
“It’s definitely 10/20 that is fundamental! Check the number of fingers on your hands!”

When he came out the door, he asked for everyone to give him a sample of every one half that they considered to be true. Then he told them, “I will take a look at these” and set out to examine them carefully.

He spent his days looking, looking, looking, looking, and there were so many of them that he died before he could find an answer.

Meanwhile, the constructors of Babel split and went their ways each thinking they had the correct number and everyone else was wrong.

One day two grandchildren of the constructors of Babel met and were sharing their family heirlooms, a/b and c/d. They were young and innocent and let each other play with their numbers, and they discovered that ad and bc were the same!

News spread to their parents and people everywhere started matching their fractions together. Family feuds were reconciled and the king decreed that  henceforth all fractions that can be reduced to the same lowest terms would be one and the same.

And the people lived happily ever after.

I wrote this story wondering to what extent the notion of a half could model  the notion of truth. It does explain how an idea can be one and infinitely many at the same time, without there being any logical contradiction.

Moreover, it illustrates how not obvious and critical the knowledge of ad = bc is to establish a single, united system of numbers that we call a half.

To understand a half we must not only understand how a fraction is different from another, but also how different fractions can be one and the same. Though we may believe we understand 1/2, we might really not have understood a half without figuring out all the ad = bc s. The task of translating many different expressions of the same idea is essential for mutual understanding, especially in our time when modernity has left us with many separated towers of traditions and expertise, inside each of which we can all lose ourselves forever.

It is funny how the notion of fractions can not only tell us how to share things but also how to share truths. Though for the battlefield of all the intricately intertwined truths out in the world, the task of establishing translations will be immensely more difficult, yet still I hope that perhaps at the end of it all, we will have a very handsome shared truth for everybody.

Hyunmoon graduated from Princeton in 2013 with a degree in mathematics. He is interested in modern cosmic ideologies and is now at Seoul National University trying to understand the structure of empty space through the mathematics of Lagrangian Floer Homology.

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