Three mathematical notes
On a proof from Analysis II
I knew what continuity meant until taking up a frame
We cut the picture cut the top
Half again then
Them out and slid them back in now
They sit in order from bottom to top yet infinitely dis-
On a proof from Algebra II
We opened a new dimension
Then closed it.
On mathematical terminology
Since my return home for Christmas break I’ve had several conversations with my little brother about mathematics and the impossibility of imagining what it is we mean by it. (Cf. this xkcd comic.) This uncertainty offers, in the end, a kind of sublimity; we teeter always on the edge of the unspeakable and yet somehow have enough of an idea what we mean to discern whether what we say is true or false. We are estranged from our words but come at last to a new understanding of them. It’s almost like poetry.
In first-year calculus you learn what “continuity” and “discontinuity” mean when speaking of a function: a discontinuity is a break in the line; continuity is when there are no such breaks, when you can draw the graph of the function without lifting up your pencil. In higher-level analysis you define “continuity” in terms of epsilons and deltas and eventually you realize that these words can be used in ways you had never before imagined. For example we can define a function from [0,1] to [0,1] discontinuous at every rational number and continuous at every irrational, a function where we can’t even put the pencil down in the first place, let alone have to pick it up again, yet which is nevertheless continuous at infinitely many more points than it is discontinuous. Would we say we changed the definition of the word? Yes, of course. But did we change its meaning? I don’t think so; I would say we clarified its meaning. But now something that previously seemed impossible, isn’t.
Analysis changed the way I saw words, but algebra changed the way I saw language itself. It offers an image of structure both like and unlike those of various linguistic philosophers (Saussure, Wittgenstein, Derrida). It both opens radical new possibilities and suggests radical limitations. It can bring us from the natural number line to the rationals, open us out onto the complex plane, fill it up, bring us closure, suggest we cannot even imagine that there could be more numbers than these–then it gives us quaternions. And then it tells us that it did not give any of this to us; that we made it ourselves, and could have done otherwise. I refer the reader to Unknown Quantity for a layman-friendly exploration of the history of algebraic mathematics.