There is more than one kind of infinity. That is what it says on the back of my copy of DFWs Everything and More: A Short History of Infinity. Stated another way, that means some infinities are larger than others.
Why is this fact included on the back of my book, prominently displayed atop several raves from respected papers? I’d wager that its there because someone thought it would sell books, and I tend to agree. Theres a certain magic to that clause, theres more than one kind of infinity. It goes against everything we know about infinity, everything weve been taught about the concept.
Putting that on the back of the book makes the best kind of promise an entertainer can make: to do something we know is impossible. DFW is promising to show us something that goes against everything weve learned, and you know what? He does it. Why is he able to do it? Because what we were taught was a lie.
Or, rather, not a lie but an approximation. Reading through Everything and More, it immediately becomes evident that in math class there was a good deal of approximation and equivocation. In fact, it goes back pretty to day one when we were taught that the number 3 stands for 3 apples. No it doesnt. Numbers are an abstract system of symbols created to be manipulated according to certain rules–some proven, some postulated. Its purely coincidental that we can use mathematical symbols and concepts to represent features of the real world.
Just as the real meaning of 3 was fudged so wed all have an easier time wrapping our minds around long division, the messy details about infinity were left out because the concept would make more sense without them, and they werent necessary to what we needed to learn.
Reading DFW explain this, one thing becomes immediately obvious: just as were made to believe 3 = 3 apples for reasons of convenience, were made to believe that apple = [red object that comes from a tree] for the same reason. Language, as Saussere pointed out decades ago, is every bit as abstracted from reality as are mathematical symbols.
If our understanding of math if faulty for this subtle manipulation, how might our understanding of language be impoverished by this fudging? What messy, interesting questions associated with language have been swept under the run for reasons of convenience? What interesting truths are we blind to?
Thats one of the many thoughts inspired by DFWs potent book. And, as these thoughts go, its a fairly banal one.
As far as we know, ancient Egypt was the first civilization to use numbers in a systematic way for real-world benefit. The ancient Greeks were the first to delve into the properties behind these numbers. They discovered a number of troublesome things, best summed up by Zenos Dichotomy, which we all will recall basically states that I cant possibly cross the street because with every move I make Im traversing half of the remaining distance. Since the distance can be infinitely divided, how can I ever get all the way across?
The Greeks had no good answer for that, so they stayed the hell away, and infinity wasnt revisited in any meaningful way until the 1600s, when mathematicians began bumping back up against it.
What they figured out, and how they definitively resolved Zenos Dichotomy is in DFWs book, and in revealing this DFW gets into what it means to have some infinities larger than others and why, logically, this makes perfect sense. What I find interesting about this was that the ancient Egyptians, the ancient Greeks, the assorted cavemen and cavewomen, when they started using numbers they never thought of infinities, or some infinities larger than others, or definitions of irrational numbers, or the real number line, or set theory, or any of that crap. But every last bit of it was implicit in those numbers they invented and the use of which they codified.
I find the whole idea very interesting, that we can invent something as simple as 1, 2, 3 and have almost unimagined complexity wrapped up in it, embedded, invisible, undetectable, waiting to be discovered.
What mind-bending rules and concepts lie embedded in the very words this review consists of?
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