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Everything and More: a Compact History of Infinity by David Foster Wallace
How do I count thee?
Friday 27 February 2004
Infinity is the most paradoxical of all mathematical concepts. It seems immune to normal rules of logic. Mathematicians have gone mad trying to grapple with it. And almost everything we know about it seems quite irrational. So it's not an easy subject for a popular book.
But the novelist David Foster Wallace is not deterred. He adores complexity, and loves paradoxes. Unfortunately, that is not enough to make the notion of infinity accessible to ordinary readers. You have to be quite determined and mathematically savvy to plough through Wallace's "compact history". Even if you avoid all those parts clearly marked "IYI" (if you're interested), you have to be reasonably familiar with advanced mathematical notations to get anything out of this.
Wallace starts entertainingly enough, by asking the obvious questions. How big does a thing have to be before it is infinitely big? How small, before it is infinitely small? Physics and astronomy are full of numbers that are unbelievably enormous. Think of the dimensions of the universe.
But infinity, as Wallace is quick to point out, is not a number. It's a concept. There are no numbers bigger than infinity, but that does not mean that infinity is the biggest number. And some infinities are definitely bigger than other infinities. It all depends on what you are talking about.
Mathematicians have struggled with the notion of infinity for more than two-and-a-half millennia. The history of infinity is the story of a battle to tame the infinite. The idea was to show that our reasoning about infinity could be conducted like our reasoning about ordinary mathematical objects. Many inspired attempts have produced important mathematical advances. But infinity has always evaded control.
As you would expect from someone known for his intellectually ambitious fiction, Wallace tells the story well. It all began with Zeno of Elea, who propounded paradoxes. Their intent was to show that our reason is not self-contained or complete. When we scrutinise our arguments about change, or about classification, we find contradictions. The famous paradox about Achilles and the tortoise imagines a race in which the tortoise gets a head start. In one leap, Achilles halves the distance between them. In the next leap, he halves it again. But how do we describe his catching up? We cannot specify that last leap; after each leap, there is still the halved distance remaining. Trying to solve the paradox only confuses us further.
There's a lot of history after that. But the last engagement in the long battle was set off by the German mathematician Georg Cantor. His work produced some quite astounding results about infinite collections. Cantor came up with infinite sets that were countable, and infinite sets that were uncountable. His work proved the opposite of his intention. He found that the set of all "real numbers", the points on a line, cannot be counted. After Cantor, philosophers and mathematicians struggled heroically to rescue the "foundations" of mathematics. But their own tools were turned against them. The famous proof of Gödel showed that mathematics is not a logical system of ideas. We now accept that mathematical knowledge is quite arbitrary.
While Everything and More is good on narrative, it is abysmal on analysis. Wallace is too clever by half, and too obsessed with demonstrating his own mathematical dexterity. He shows that the attempts to prove Zeno wrong have enormously enriched our understanding of our concepts and, perhaps, ourselves too. But his rather pedestrian treatment of infinity fails to draw the obvious conclusions.
There are at least two important lessons to be learnt from the history of infinity. First: knowledge is not logical, but its genuineness comes from its roots in practice. Mathematics works because it is rooted in experience. Second: any abstract system of ideas will generate its own contradictions. Or, to put it another way, Zeno was right.
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