Plato's Laws V, 4: On the Division of Land

Today's reading is likewise short, this time because it contains a fun mystery as well as some general principles that need to be discussed. I'll take the mystery first. Plato is going to suggest that an ideal population will consist of 5,040 landholders, which he says can be divided by 59 quotients.

A contemporary reader will probably be a bit confused by this, because it would seem as if 5,000 were more obviously divisible into even units than 5,040. You could divide 5,000 into units of one thousand or two hundred-fifty, or one hundred, or fifty. Adding that extra 40 guys seems like it is going to cause a lot of fractions of guys, and while people can be divided into fractions conceptually, dividing them actually tends to ruin the use of the individual.

The Ancient Greeks had a completely different system of mathematics from ourselves, though, one that lacked both fractions and decimals -- that is, both of the ways we teach our young to handle uneven divisions. The Greeks used ratios, so that a number was divisible if you could give a whole number (say "6") and a ratio for the remainder (say "...and two for every three"). It turns out that is exactly what is being captured by the fraction 6 2/3, but this was done on the assumption that really you would not be dividing the two by three. You would be providing two here for every three there. So at first you might think that this aspect of our different systems was behind it.

But actually, if you work it out, 5,040 really is divisible by every number from 1-10, plus 12, and then turns out to have many more ways of evenly dividing it (fifty-nine total, according to Plato; actually 60 if you include itself and 1). So even though it seems like those extra forty guys are going to cause problems, they actually provide for even divisions in more cases. 5,040 can be divided by seven, for example, whereas 5,000 would cause a remainder. 

So it is our conceptual reliance on our base ten decimal system that is fooling us: a number divisible by ten looks like it would be the most obviously useful number.  Because the Greeks worked with practical divisions of whole numbers, they could see that 5,040 was a better choice. This is similar to the way that SAE wrenches and metric wrenches end up fooling the mind when you have to switch between them (as for example when working on a Ford truck, as I do sometimes, which has both SAE and metric fittings for no very good reason I can divine).

This quality of 5,040 was obvious to the Greeks because their quite different approach to math led them to it. The number has some other qualities, some of them more mystical, which are described herehere and here for those interested.


james said...

It is a sensible number for dividing things up.

But it's also foolish, since people and families aren't as easy to shuffle as integers.

Grim said...

You raise a good point, in that Plato's unit for the 5,040 here is 'householders.' Now one householder may have a family of four, another of three, and another of seven. Yet they're going to be assigned essentially equal plots of land on a per-household basis.

There are additional objections in the post above this one, but that's another problem. Households are themselves not equals.

james said...

And if they are fruitful and multiply--do some of the children have to go found another city?

Grim said...

Maybe! But the good news is that gives us yet another chance to found a colony perfectly. Anything that didn't go right last time, we can try to fix in the next iteration.