Superman never made any money for saving the world from Solomon Grundy

Friday, July 24, 2009

In all fairness

I remember a conversation with my philosophy teacher, mentor, and pal Joel Goldstein back when dinosaurs ruled the Earth and I was trying out college for the first time. We were discussing fairness, and how in some circumstances it might be determined by the result and in others by the process. For example, if you and I were to split a cookie, a determination of fairness would probably depend on whether we each felt we got an equal share (the result) and the process - I cut and you pick, get a trusted third party to do it, measure/weigh it - wouldn't really be important. However, in an other situation, such as a poker game or Monopoly, fairness would be determined not by an equal distribution of the resources in the end, but only by whether the correct process was followed - the rules were adhered to and no one cheated. Result vs. process.

A recent lunch with my favorite Mad Engineer Dr. Burn, I was reminded of a discussion* we had a few weeks ago about Babylonian fractions, a system whose process seems to have been affected by a particular cultural positioning regarding the fairness of different results.

Let's show this through a word problem. You have seven loaves of bread and eight people. If you were to divide up the bread evenly among the people, how much would each person get?

In most classes discussing fractions, the answer might go something like this: Well, there are 7 loaves of bread that I have to divide among 8 people, so 7 is the numerator and 8 is the denominator, and each person gets 7/8 of a loaf of bread. And that's as good as far as it goes, but it doesn't go very far.

Think about how you would actually make this happen. Presuming all the loaves are the same, you could cut a one-eighth piece from the end of each one and pile them all together. Seven people would get a less-than-full loaf, and one person would get a pile of slices.

But is this really dividing the bread evenly? Is an almost-whole loaf of bread the same as a pile of slices? (Slices that would all be heels, too!) Isn't it really the case that seven people got 7/8 of a loaf a bread and one person got stuck with seven 1/8s? We can try all sorts of variations on this method - take a slice from the middle of some loaves and the ends of others and so on - but it all comes out to same sort of result unless we work at it from a completely different perspective.

The Babylonians had this other perspective. Rather than focusing on trying to make the denominator in fractions equal one, which we really like to do (6/1 is the same as 6, right?) the Babylonians liked keeping the numerator at one.

The bread problem would be solved like this in Babylonia: First we cut each of the seven loaves in half. Now we have fourteen 1/2s. Each person gets one 1/2, leaving six. We cut those 1/2s in half, making twelve 1/4s. Each person gets one 1/4, leaving four. We cut those 1/4s in half, making eight 1/8s. Each person then gets one 1/8, and all the bread is gone. Each person gets a 1/2 and a 1/4 and a 1/8 of a loaf of bread.

(And in case you think this is just one of those math tricks, it does come out the same: 1/2 = 4/8 and 1/4 = 2/8, so 4/8 + 2/8 + 1/8 = 7/8, just like the other way.)

The difference between the Babylonian method (which I am sure I have oversimplified) and just "doing fractions" is that the approach incorporates a sense of the reality of fairness in dividing things up in its very structure. I don't think I'd feel good about getting seven bread heels for my share of the baked goods, even if you showed me how it was mathematically even. In the real world, 7/8 does not necessarily equal seven 1/8s. But 1/2 and 1/4 and 1/8 should be the same as 1/2 and 1/4 and 1/8. In this case, the result and the process work with each other, and fairness comes from their intersection, not from one or the other.

So, perhaps a purely binary view of how fairness is determined is less useful than initially thought. And that might be the most important takeaway from consideration - not a new way of doing math, but a new way of thinking about things and an understanding that as unlikely as it sounds, many coins have more than two sides. These two conversations took place over thirty years apart. My learning may be slow, but it appears to be constant.

*In calling it a discussion, I flatter myself. Actually, Dr. Burn just told me what for.


John said...

Gets more complicated if you have to split 2 loaves between 5 people.

Walaka said...

Yah, apparently they had tables of reciprocals for 2, 3, and 5 to make that easier.

Yojimbo_5 said...

Universal Solution: Cage-match