Mathematics can be thought of as many things: a language, a tool, a game. It might not seem like a game when you’re trying to do your homework or pass an exam, but for me one of the most exciting parts of doing research is when you’re just starting something new and you get to play around with some ideas for fun. It’s a bit like playing around with ingredients in the kitchen, which is more fun than trying to write down the recipe you invented in case you want to repeat it. And that is in turn more fun than trying to write down the recipe for someone else to be able to repeat.
I’m going to start by playing around with the idea of infinity a bit, to free our brains up and start exploring what we think might be true about it and what the consequences are. Mathematics is all about using logic to understand things, and we’ll find that if we’re not careful about exactly what we mean by “infinity,” then logic will take us to some very strange places that we didn’t intend to go. Mathematicians start by playing around with ideas to get a feel for what might be possible, good and bad. When Lego was first made, the designers must have played around with some prototypes first before settling on the wonderful final design.