Archive for February 7, 2008

First off, the conversation I will attempt to engage in at the end of this post was inspired by the book I just finished reading. It’s called A guide to Mathematics for the Intelligent Nonmathematician by Edmund C. Berkeley. I found it while looking for some Iteration stuff in the library. It seemed like it might be fruitful, so even though it’s an old book (1966) and talks about ‘New Math’ like it’s a recent phenomenon, I took a look at it.

It’s an interesting book. Berkeley does what he says he intends and presents mathematics in a very accessible way, something that anyone can understand even without a basis in math. Though some chapters did start to shift into actual mathematics, for the most part I could read the entire thing without any memories from pre-calculus rearing their ugly heads. Berkeley goes point by point, slowly drawing the reader into a mathematical world by showing them that they’ve been dwelling in it all their lives. We do math all the time, Berkeley asserts. We just don’t usually look at what we’re doing.

That attitude reminded me of how I like to describe and teach logic. People think logically; it’s how we’re wired. Learning formal logic is really just forcing yourself to slow down and look at the steps you normally take, in order to be just a bit more careful. Berkeley seems to suggest that the same is true of math.

There were a few points interesting to my work on Iteration. When discussing approximation, and how to solve a few relatively simple problems, he suggests a variation of a guessing method. In his words, “Guess a reasonable value and try it. If it does not fit, then choose a more reasonable value and try that. And so on” (91). So make an attempt at something, then see if it works. If it doesn’t, improve and try again. This sounds like the mathematical equivalent of iterative development to me. Maybe it’s a primitive form of Iteration, but it’s right there.

Berkeley calls this method the “Method of Successive Approximations” (91), saying that while it is similar in principle to trial and error, there is thought between the trials that makes each successive trial more likely to succeed. Again, iteration.

Later on, he presents a form of iterative development at work. Discussing how correct figures are calculated at a business, Berkeley suggests that calculations are made first by one person, then passed on to the second. The second person does his/her own calculations, and marks any discrepancies between the second set and the first set. The first calculator then has to check each of the discrepancies and try to fix them, until both people agree with the calculations. Then the calculations are sent to a third person who spot checks several random figures. If he/she approves of those random figures, the calculations are taken as correct (110-112). This definitely seems like small group iterative development. Try something once, send it up the line to be tested. If it’s no good, try again. Keep doing that until the first person up the line approves, then send it to the second person. Continue the process until everyone’s happy. It’s a nice description.

Which brings me to the end of the book, where Berkeley, after discussing the way probability works (and demonstrating the proper way to look at the Gambler’s Fallacy), talks about the future. Before this section, he predicts that the likelihood that humanity will destroy itself with nuclear weapons at some point in the next 1000 years is 99.996%. Not very encouraging. But after that, he presents some predictions for what the world would look like if we got lucky and did NOT kill ourselves.

His predictions and then mine: (more…)