Tuesday, December 28, 2010

A little bit of misdirection

... wherein we reduce the classical Sorites Question to a problem of Hilbert...

The question was "how many grains of sand are needed for a heap?"
I answer that four are enough, though ten thousands may not suffice.

Put another way, the question of "how many" is the wrong question. Heapness isn't about quantity but about geometry. You have a heap of sand on a table when some of the sand isn't touching the table, but only more sand. To stably hold one grain of sand off the table, a tripod of lower grains will do nicely. But if they're spaced, say each 1cm from the others, you can easily fit 10000 on a modest square table, and there will be naught of heap to the ensemble.

We suspect that other Sorites-type questions (how many voters make a democracy? How many roads must a man walk down?) have similar resolutions.


Post a Comment