# The World of Mathematical Reality

This always blows my mind that this is true. If you take any four-sided shape at all, make it as awkward and as ridiculous as you want. Four-sided shape, any. If you take the middles of the sides… and connect them… it always makes a parallelogram. Always. In other words, these two lines are always in the same direction, and these two are always in the same direction. No matter what crazy, kooky thing you started with. That’s scary, to me! That’s a conspiracy. That’s amazing. That’s completely unexpected. I would’ve expected you make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t. It’s always a slanted box, beautifully parallel. WHY is it that? That’s not a scientific question. The scientific question would be about ink and distances roughly approximate, da da da, test tubes and all that. The mathematical question is “why?”, its always “why?”. And the only way we know how to answer such questions is to come up, from scratch, with these… narrative arguments that explain. And you can have more than one explanation. You can compare them, you can critique them, you can contrast them. Just like you can poems. And so for mathematicians our aesthetic is: your argument, your explanation, your discovery, your deduction: it needs to make sense. It can’t be wishful thinking, it can’t be pie-in-the-sky, it can’t just be what you want. It has to follow logically. Which is a tough demand. So your reasoning needs to be tight and correct. But, more importantly, really, is that your argument has to be beautiful. It has to be astounding. It has to be revelatory. It has to be charming. That’s really hard to do. So, what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod… known as problems. And, how we can possibly… craft these elegant reason poems to get at what we want to get at. And it may not be everyone’s cup of tea. Certainly I’m not suggesting, or even necessarily wanting it to be that everyone becomes a mathematician. Just like I don’t think everyone wants to be a photographer or a ballet dancer. But I think it IS important that everyone understand that there IS an art of photography, there IS an art of ballet, there IS an art of mathematics, and those things have their own challenges, frustrations, joys, etc. And that one can then choose to integrate such arts into one’s life to the extent that one wishes to. There’s nothing deeper than this mathematical act. There’s nothing that comes even close.