Zeno to Cantor

One of the griefs of my education was a choice I had to make at the beginning of my fourth year at secondary school. Although I no longer remember the actual marks, I do recall that, in the nationwide School Certificate examination the previous year (1967) I'd scored equally well in History and Maths. In the Sixth Form, however, which I was entering, these two subjects were mutually exclusive options: you couldn't 'do' both. I tried to work out a way. My father, a former History teacher who was then Headmaster of Huntly College, though leery of showing any favouritism, also tried. There were family discussions. I was going to attempt some kind of alternative study programme during free periods, sport etc. In all of this was an unspoken assumption, which I shared, that, if one of the two subjects was going to have to go, it would be Mathematics. So it proved. And, in the way of such things, a door closed that would never really open again.

Mathematical understanding, like memory, is a muscle: it strengthens with use and atrophies without. But while, as Jean Luc Godard said, memory is the only paradise we can't be expelled from, the subject matter of mathematics, if neglected, quickly turns into unbreakable code. I am as if stalled at age fifteen when it comes to the higher arcana of numbers. This doesn't mean my fascination with some, not all, of the conundrums of the count has faded, only that these puzzles seem forever beyond my reach, not simply to solve but even to formulate. I am like someone trapped in a darkened room, unable even to see the pencil and paper with which I might work out the formula that would show me the way to the door that closed so long ago.

It's the philosophical end of the subject that interests me. And I keep worrying away at those bits I can keep in mind. Among my books there is a two volume Pelican paperback called Men of Mathematics by E. T. Bell. First published in 1937, republished by Penguin in 1953, reprinted in the 1960s, it consists of thirty odd essays that are mostly, though not exclusively, structured around the biographies of notable mathematicians. Bell writes beautifully. He is elegant, concise, witty, knowing and, if not exactly irreverent, certainly never misses an opportunity to burst a bubble or prick some pomposity.

As I read these essays (I have just reached Pascal by way of Descartes and Fermat), there always comes a point where the writing stops making sense and I have to skip forward to where it does again. These are the passages, of course, where history and/or biography recedes to make way for pure mathematics. There's something ineffably frustrating about my inability to understand the figures ... they seem to float just beyond my reach, in a zone that once looked as familiar to me as the garden of the house where I grew up, but is now as strange and threatening as the landscape of an Anselm Kiefer painting, say, or the planets of some other solar system than ours. I sigh, flick forward, and pick up the thread later.

Towards the end of the last essay, Paradise Lost?, about Georg Cantor, Bell writes: We are back once more asking the Sphinx to tell us what a number is ... thus closing the circle that began with Zeno and Pythagoras two and a half millenia before. Cantor's speculations with infinite sets opened the way for, among much else, Mandelbrot's insights. Chaos theory depends partly on what the Russian born German mathematician achieved. I was brought back to Men of Mathematics by a fascination with the so-called Cantor Set which, to my mind at least, is a variation upon Zeno's famous paradoxes of motion. I'll try, with the help of James Gleick, to give a non-mathematical explanation of this:

You take a line and remove the middle third; then remove the middle third of the remaining sections; and so on. The Cantor Set is the dust of points that remains. They are infinitely many, but their total length is zero.

Cantor was a tragic figure who ended his days in a mental asylum in 1918. Much of his later life was consumed by an attempt to prove that the Works of Shakespeare were written by Francis Bacon. Part of his depression originated in the implacable opposition German academic orthodoxy expressed against his work. They felt, in Gauss' 1831 phrase, horror of the actual infinite. Bell divides mathematicians into two classes: those who believe that mathematics is a purely human invention and those who think that it has an independent existence, containing universal truths we can discover. It was not so much horror of the infinite that sent Cantor mad, but rather the horror of the horrified for his work.

As for me, lacking the tools of understanding, I can't decide where I stand. I'm still back there in Egypt, asking the Sphinx to tell me what a number is ...


richard lopez said...

I have no facility for numbers, and so an elegant equation to me looks as strange and as riddling as what an extraterrestial's alphabet would look. even so, philosophy and music depend, or maybe its the other way around, on math. tho, I do agree with Gauss that one can go insane with faced with the actual infinite, for I believe that math is not created but exists within the natural world and equations are our own script for it. the infinite is beyond our ken, and to even glimpse at it makes one feel absolutely puny and without effect within the space of the universe. a flash of interstellar cold, when we sit under a thick canopy of stars and realize how old, how far, and how much space is between those stars, and how we, as viewers, possess such a small lifespan compared to the universe. that ain't infinity, but just a taste of vastness, and that is enough to make one shudder with that cold.

Martin Edmond said...


After reading your note I went outside. It was about 2 am, very clear and cold. I looked up into the sky for a long time, until the stars began to 'move' ... or should I say my eyes began to wander? It seems when we contemplate these things we look both ways: simultaneously into the abyss of the mind & the abyss of space. Double vertigo ... the natural world & our equations for it? Both, perhaps, infinite?

mark young said...

"How many points are there in a line segment? As Cantor's diagonal proof reveals, infinitely more than there are integers. And in the infinite plane? Just as many as there are in the line segment. Indeed, and in general, precisely as many as there are in a space of n dimensions, n is equal to or greater than 1. 'I see it,' Cantor wrote to Dedekind, 'but I don't believe it.' Is this infinite number of points, then, the highest degree of infinity available? No. Cantor also proved that for any set, a set with more members (the original set's power set, consisting of all its subsets) is constructible. Thus there is no greatest set. Hence also (Cantor's paradox) there is not a set of all sets, since such a supposed total set would at once yield a larger one."

Hence my love of Gödel. & Magritte. "Ceci n'est pas une pipe"

Martin Edmond said...

Mark, such ... clarity. In the mathematics of ambiguity.