Saturday, August 10, 2013

Book Review: String, Striaghtedge, and Shadow: The Story of Geometry, by Julia Diggins

This book was recommended on Living Math Forum. I can see why. The storyline is very engaging overall, and gets you thinking about the history as if it were happening while you watch. But...

My biggest issue is that it is so gendered:
"A string can usually be found in a boy's pocket..."
"Ancient men discovered the ideas and constructions of elementary geometry ..."
"Through the ages, men have searched to find the secrets of the universe."
"A long, long time ago primitive men observed the lines and curves and other forms of nature."
"It was from this inner sense - man's sensitivity to the order and harmony of the universe - that geometry really began."

I don't think writers do that so much these days. (This book was written in 1965.) Writers sometimes still say 'he' when they mean all of us, and still sometimes say 'man' to mean people, but not often. Reading this book made me think modern writers must be avoiding this construction, even if unconsciously.

Why does it matter? Research shows that we need to feel a part of  a community in order to do our best thinking. Women and girls are shut out by this sort of writing. As much as I might like the content of this book, it sets me up as an outsider (even though the author is a woman!), and that's part of how stereotype threat happens.

I haven't read the whole book, but I did discover one error, I believe. The discovery of the fact that the square root of two is irrational seems to be described incorrectly:
"Then was it a ratio of whole numbers between 1 and 2? ... They tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.
After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2."

There are two problems here. One, they couldn't have tried 'every possible ratio', because there are an infinite number of possibilities. More importantly, it wasn't about giving up. If I understand the history correctly, they actually proved that no such ratio can exist. This notion of proof is a very important foundation - it's part of what mathematics is. So her version of this story takes away some of its drama.

The Pythagoreans believed, as she says, "that the universe was ruled by whole numbers." So to prove that a length exists which cannot be described by a ratio of whole numbers was extremely unsettling to them.

How do we prove that the square root of two cannot be a ratio of whole numbers? There is more than one way to do it. You might like Kate and Justin's way more than the one I usually use. It's less dependent on being comfortable thinking with variables. Here's the way I think of it:

If the square root of 2 could be represented by a fraction, we could writes that fraction in simplest terms as a/b. Then we'd have (a/b)2=2, or a2 = 2*b2 . Since the right side of this equation is even, the left side must be, too. If a2 is even, a must itself be even. Let's call it 2c. Then our equation becomes (2c)2 = 2*b2 , or 4*c2 = 2*b2 , or 2*c2 = b2 .  Now the left side of this new equation is even, so the right side must be too. And that means we can write b as 2d. But if both a and b are even numbers, then the fraction can be simplified. We started out with what we thought was a fraction in simplest terms, and found out that it could be simplified. This is a contradiction. It happened because we tried to write the square root of 2 as a fraction - it can't be done, and this proves it.

Proof by contradiction is a bit weird. I think I might like Kate and Justin's proof better myself.

Well, you might like String, Straightedge, and Shadow, even with its flaws. I might, myself. But I have decided not to include it in my Book Picks section.

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