Monday, 8 February 2010

Problems with infinity

Considering all the difficult problems which the concept of infinity poses, Aristotle suggested separating the idea of ‘actual infinite’ from ‘potential infinite’.
If you add all the terms in the series: ‘a half plus a quarter plus an eighth plus a sixteenth … and so on’ you get closer and closer to One the more terms you add, but you never quite get there unless you have an ‘infinite number’ of terms.
But suppose we ban the idea of an actually infinite number and put in its place a very big (but real) number (L). Then, considered in this way as a sum of fractions, the number One would be ‘imperfect’ since it could never quite be really one! Of course, all the other natural numbers would be ‘tarnished’ in the same way.
How could this very big number L be defined?
Here are two suggestions for what L could be:
L1 The total number of particles in the universe. (about 1087)
L2 The biggest number anyone has ever thought of. (apparently there are lots of ways to generate huge numbers, but at any given time, one of them would be the biggest).
If it was, L2 this would be dependant on human (or perhaps computer) interaction – is this the first time that a mathematical quantity depends on human behaviour?