26 July 2007

fundamentalists and π

Fundamentalist math, at ooblog, via Vlorbik on Math Ed. The article quotes a "fundamentalist math textbook", which apparently actually exists. It says things like:
Carl Friedrich Gauss first proved the fundamental theorem of algebra. There are many fundamental theorems: of arithmetic, calculus, and so on. These are so “fundamental” that many other theorems are derived from them. In the Bible, there are also fundamentals, without which Christianity would not exist—the deity of Christ, His substitutionary atonement, and the inspiration of the Bible, to name a few.
Basically, this sentence says that "things, in various intellectual frameworks, are derived from other things". What else could you expect? Standing on the shoulders of giants, and all that. Something valuable for kids to know, surely, but why try so hard? I'd rather at least see a fundamentalist approach to mathematics that was based on saying "math is beautiful, and that shows us that God is great, because God invented mathematics." I wouldn't agree with this, but I could at least sympathize with it.
The passage on "fundamental" seems, to me, like a real stretch; mathematicians are well-known for using words in different ways than the general population. (So are fundamentalist Christians, I hear, although I don't know this for sure.) Mathematicians actually use "fundamental" in much the same way that Normal People do; Wikipedia has a list of fundamental theorems. I would suggest one more addition to this list: the "fundamental theorem of combinatorics", which is that if there are m ways to do one thing, and then there are n ways to do some other thing, then there are mn ways to first do one of the first things and then one of the second things. I will not edit Wikipedia to reflect this, though, as this usage seems non-standard; but insofar as combinatorics has a fundamental theorem, this is it.

(Oh, by the way, if someone refers to the "fundamental theorem" without saying what it's the fundamental theorem of, what would you think they meant? I'm not sure; I'd think they meant the F. T. of either algebra, arithmetic, or calculus.)

The most famous "math in the Bible" is, of course, the verse that can be interpreted as saying that π equals 3, due to some circular object which is thirty cubits around and ten cubits across. (Link goes to Good Math, Bad Math; the Bible verse in question is 1 Kings 7:23.) Surprisingly, everyone seems to assume that this means the writer actually thought π = 3, and comes up with complicated explanations for this. For example, some people assume that 30 cubits is the inner circumference and 10 cubits is the outer diameter. I think I even once saw someone who seriously suggested that in the time of the Bible, circles were actually hexagons; the length of a regular hexagon is exactly three times the distance between opposite corners. The obvious interpretation, though, is that either:
  • The object in question wasn't perfectly round. This seems rather doubtful, though, as you just need a stake in the ground and a big long rope in order to make a circle. If they wanted to make circles, they could.
  • Rounding error. What is called "30 cubits" here could be as large as 30.5 cubits; what is called "10 cubits" could actually be as small as 9.5. Thus the ratio of circumference to diameter could be as large as 30.5/9.5 = 3.21. Similarly, it could be as small as 29.5/10.5 = 2.81. So the Bible is only saying that 2.81 < π < 3.21, which is true.
Personally I think there are plenty of reasons to doubt the literal truth of the Bible (starting with Genesis 1 and 2, which are two creation stories that contradict each other!) -- but there's no need to nitpick by saying that they didn't know what π was.

And I'm trying to picture a world in which circles were hexagons. This seems to imply that the world is in fact a triangular lattice in the plane, which is preposterous.

7 comments:

Melanie said...

This post reminds me of http://www.trnty.edu/faculty/robbert/SRobbertWebFolder/ChristianityMath/

Wing said...

1 - That theorem of combinatorics I think is most commonly known as the Counting Principle.

2 - I've seen scans of a math textbook (at the high school level) that actually says something like "math is awesome because God is awesome, etc.". Things like "trig is good because it lets us build mighty engineering feats that are beautiful to behold like many of god's majestic works" and such.

Isabel said...

Wing,

if I were religious (which I'm not) I'd rephrase that as "God is awesome because math is awesome". But God would also be kind of mean, for making math hard.

John Armstrong said...

But God would also be kind of mean, for making math hard.

And all of that Old Testament stuff was sunshine and puppies? I'd rather have some hard math than be ordered to kill my only son.

Oh wait, I forgot. As good fundamentalists we're supposed to forget about everything from the OT but what supports our political positions. Carry on.

Aaron said...

Oh, by the way, if someone refers to the "fundamental theorem" without saying what it's the fundamental theorem of, what would you think they meant?

Okay, I'll bite... I'd think they meant the Standard Model Lagrangian. ;-)

Rounding error.

My sentiments exactly! It's quite heartwarming to know that even the supreme architect of the universe occasionally approximates pi as 3. ^_^

(Incidentally, my favorite pi approximation of all time has got to be the square root of 10. It's accurate to within 1%, and makes order-of-magnitude calculations a breeze!)

Isabel said...

Aaron,

I like √10 as a pi appoximation. But it turns out there's a better approximation involving square roots: (sqrt(53)-1)/2. I don't remember it this way, but rather as the positive root of x^2 + x = 13. Unfortunately, 22/7 is actually a better approximation than that, and easier to remember too.

Finally, 31^(1/3) = 3.14138065...

jackv said...

Hi! (I came here from the reading the plover blog.)

I just wanted to say thank you for saying rounding error. It seemed obvious to me, but no-one on either side of the debate seemed very satisfied. (Well, nor am I: as it happens, I'd like the problems with the bible to be so obvious one way or another.)

You comment about hexagons does remind me of one thing; in a DOuglas Hofstadter book, one question he asks, in a chapter on self-reference, is "what would the world be like if pi were three", and the response a reader gave was that, but with hexagons for the Os...