infinite sequence of lies"….
One of my favorite 2012 math books, "Measurement" by Paul Lockhart, is now out in paperback, so in honor of that, some brief passages to ponder therefrom on this Sunday morning:
"The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying."
"Geometry, then, is not so much about shapes themselves as it is about the verbal patterns that define them. The central problem of geometry is to take these patterns and produce measurements -- numbers which themselves must necessarily be given by verbal patterns."
After explaining calculating the area of a circle from an infinite-sided polygon inscribed within, Lockhart writes (in one of my favorite bits from the entire book):
"Something really serious has just happened here. We have somehow obtained an exact description of the area of a circle using nothing but approximations. The point is that we didn't just make a few good approximations, we made infinitely many. We constructed an infinite sequence of increasingly better approximations, and there was enough of a pattern in those approximations that we could tell where they were heading. In other words, an infinite sequence of lies with a pattern can tell us the truth. It is arguable that this is the single greatest idea the human race has ever had."
And finally, this:
"Maybe the bottom line is that I don't have that much to say about the real world. Maybe part of it is that I'm not altogether entirely here a lot of the time. Maybe the point of this book is to give you a glimpse of what it is like to live a mathematical life -- to have the better part of one's mentality off in an imaginary world. At any rate, I know that I am by nature permanently isolated from reality -- my brain is alone, receiving only the (possibly illusory) sensory input that it does -- but mathematical reality is me."