Zeroes and the Infinite
I recently read a book about the history of 0 (that is, the number, zero) and its place in the development of mathematics. The author talks about ancient Greek math towards the beginning of the work, and he suggests that the reason the Greeks had no zero was that they were fundamentally geometers who thought about numbers primarily as measures and mathematics primarily as a study of relative proportions. Ratios don't make much sense when you start to stick zeroes in there; the length of my big toe might be 3/2 as long as your big toe, but it's nonsense to say that anybody's toes could be 4/0 the size of somebody else's (and surely there's no need to compare two things if I've only got one of them, which is essentially what one is doing when one says that the ratio of apples to oranges is 0 to 52--I mean, there just aren't any apples to be dealt with in the first place).The most interesting part of this account is not the math and it's not the history, either (both of which are really quite rudimentary here). Rather, it's the passing mention of the fact that the Greek word for "ratio" is logos. This struck me; I've always taken logos to mean something like "word" or "discourse" or even "reason"--all with very verbal connotations. The reason I was so taken with logos as a mathematical term is because I know it primarily through John 1:1.
In the beginning was the Word, and the Word was with God, and the Word was God.
See? Even in its more philosophical uses, logos has clear cultural, Biblical connotations as a word to do with words. This it is; it is much more, as well.
Q offers a good historical account of where the above (KJV) translation came from--namely, the Latin Vulgate, which we presume to have used verbum in the place of logos. That sounds plausible to me, and is the sort of thing that Q would know.
But come on! Why on earth, given all the salient possibilities to do with rationality (in both the logical and the mathematical senses), would you think that the best translation of this little bit of the Gospel--the translation most likely to highlight God's inherent reason, order, and place in nature--is the one where we use the word "word" (or, for that matter, the relatively boring Latin "verbum")? Doesn't it make more sense if you say, "In the beginning was reason and mathematical truth, and these things were with God, and they were God?" That seems far more compelling to me.
Or does this just highlight that I don't really understand the compulsions of religion?
posted by Skay @ 11:02 PM
4 Comments:
I recently wnet to the "Art of the Mediterranean" exhibit at the Glenbow Museum here in Calgary. They showed Egytian, Grecian, and Roman relics.
One of the tings mentioned about the Romans was the "Golden Ratio" how all their architecture is formed by all these rectangles within rectangles, split to form another rectangle with the Golden Ratio. I thought that was very cool.
There's also a psychological aspect to this, not using a zero in their math because they had no use for a zero. But They also must have had some use or concept of zero. Afterall, since they were a slave owning society, and people being the way they are, they must have recognized this citizen owns 30 slaves, and this other, poorer citizen doesn't own any, thus has zero. Which I believe, would be a ratio, albeit, perhaps, a mathmatically unsupportable one.
Of course you must be right, Crag, that they had a concept of nothingness. I guess the claim is just that they didn't have a well-formed mathematical conception thereof.
If my sources are to be believed, they didn't have a mathematical conception of negative numbers, either. The ancient Babylonians knew the quadratic equation ages ago, and Euclid made it more elegant 1,000 years after that, but neither of them ever thought to include negative roots as solutions. Again, this makes sense if you're a highly practical and geometrical mathematician; there's not much sense in the idea of negative length or breadth, I guess.
It occurs to me, though, that this kind of math must entirely be missing the concept of the number line or the starting point. For us, negative numbers are useful even in the real world, because they allow us to represent direction as well as distance. Once negative length tells us that we mean "the same length, but in the opposite direction," we've gotten a lot out of them. If I travel 10 meters forward and then 10 meters back along the same line, it's definitely useful to know that I've just walked 20 meters and I'm all tired out. It might also be useful, though, to know that in the end I haven't even gotten anywhere new.
I can't really imagine a world without zero as a well-formed number (not just a place holder for some power of ten, as it is in 40, but a proper number, with a location on the number line and with a place in mathematical equations). That sounds dorky, but I don't mean that I lie awake at night and consider equations that wouldn't work if we'd never thought to include zero as a possible solution. I just mean, I think in lots of practical ways, things would be different. And weird.
Interesting observation that negative numbers proceed (recede???) from the concept of zero but I do wonder: How long after accepting the arithmetic zero did the notion of negative numbers occur? Did the zero spread before the concept of negative numbers? I speculate that it did and wonder, if so, did negatives occur independently in various places?
David Foster Wallace has a book out on the concept of infinity, _Everything and More, a Compact History of 8 [sideways]_. It turns out the infinity is very closely related to the concept of the infinitesimal, which is "almost" zero but not quite. Apparently this drove some German mathematician crazy. His name was Cantor, but he managed and some guy named Dedekind to come up with a definition of "really close" to zero (or any other transcendental number, like pi or phi) that most mathemeticians still use.
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