Every magic trick consists of three parts, or acts. The first part is called the pledge, the magician shows you something ordinary. The second act is called the turn, the magician takes the ordinary something and makes it into something extraordinary. But you wouldn’t clap yet, because making something disappear isn’t enough. You have to bring it BACK. That’s why every magic trick has a third act, the hardest part, the part we call “The Prestige.”
― The Prestige (2006)
Mathematics and magic behave in similar fashion. In mathematics, a function is a map from an input value x to a single output value y. Who cares what the function does. The point is, it takes an input, and gives us a single output. Functions make connections between inputs to predictable, single outputs.
Not all functions are made the same. Sometimes, one has different inputs and gets the same output. That is still a function. My computer is such a function. Inputs of a software shutdown button, holding down the physical power button, or pulling plugs from wall-sockets wildly all give me the same unique output of shutting off my computer.
The trouble with functions like my computer is that it becomes impossible to ‘go back.’ If I only know my computer was shut off, I can’t recover the knowledge of whether that was done by plug, physical button, or software button. Mathematicians would say, “The shut off function of computer is not one to one.”
A one to one function is a function which maps each unique input to a specific, unique output. A one to one function is still a function, but it allows us to undo the function when needed. It allows us to go back. It allows us to achieve the prestige.
This is vital! Understanding how to go back is everything. Remember, functions are maps. They allow us to travel to new places, magical places where multiplication of large numbers becomes addition of small numbers, where tricky long divisions become simple subtraction. But living in this dream world is just wishful thinking if we couldn’t ever get back to the real world around us. We need both a dream world as well as practical principles. Much like traveling to Narnia or Anthropos or Terabithia, we can learn on one side of the function-map, where it is safe to explore and learn and grow through hard lessons, and then safely return to our world through the opposite map, through the inverse.
I wrote all this to let people know about functions. That, of course, is always a worthy goal. But I also write to encourage you that, whether you’re planning new or improving old, you’re allowed to practice your skills in a dream world, where there aren’t politics or nay-saying or lies or hurt or pain. Create your vision, and then take it back to our world. It is in the taking things back to our world that we earn our prestige.