So, I do maths. People say to me, what do you actually do when you are doing maths. That is not easy to answer. I study the dynamics of transcendental entire functions. Given that there is nothing actually dynamic, certainly nothing mystical, and no relation to the sort of things which are held in large rooms hired from pubs, you can already see there is a problem.
I can try to use a metaphor. It is from cooking. It is not a very good metaphor, but I am not a writer am I. So, perhaps you can deal with it.
Think of a chef.
Let suppose our chef reads that cheddar cheese on toast is a delicious meal. Well, of course, he could make cheddar cheese on toast. But that would be boring. (And spoil the metaphor). So he tries something else. Perhaps a different cheese. Or maybe beans. Perhaps a little parsley. Pushing the metaphor, perhaps he realises that cheese on toast is not only an excellent meal, but also helps cure indigestion. Or maybe provides good sustenance to dogs. There are all sorts of things our chef can conjure up. And good luck to him. Or her.
What has that got to do with what I do doing maths? Well, I start by reading what other mathematicians have done. Perhaps I read:
Theorem: If X = Cheddar Cheese + Toast, then X is a delicious meal.
So I get out a pen and paper and I try some variations on this. (I use a lot of paper). Perhaps
Theorem: If X = Cheshire Cheese + Toast, then X is a delicious meal.
Theorem: If X = Custard + Toast, then X is a delicious meal.
Not all theorems work out. Or I can try to generalise things. Perhaps
Theorem: If Y is a dairy product and X = Y + Toast, then X is a delicious meal.
It is better than cooking, because there are all sorts of things I can try. For example I can define a food as ‘good-with-toast’ if, when Toast is added, I get a delicious meal. Perhaps foods which are ‘good-with-toast’ have some interesting properties.
Theorem: If Y is good-with-toast, then Y is good-with-bread.
Well, you get the idea. So then I write it all up, and make it available to other mathematicians. Who knows, perhaps they will read it and then make their own new theorems.
I think it is pretty exciting.