2009-03-25

λ version succ - left or right

Japanese version

Last time, we told about a rule, function application is left-associative. First of all, why we need to think about something associative? Because it is not so useful when the answers are different even the expressions are the same. If one computes 1-2-3 as (1-2)-3 (=-4) and the other computes 1-(2-3) (=2), the values are different. Someone might think useful if a calculator gives different answers every time, however, I assume many people could agree such calculator is not so useful. We need to define associativity if the input expressions are the same, then the answer are also the same.

This is the reason why we need to think about associativity. Then the next question might be why we use left-associative? Actually this is just definition and it does not matter which one we take as long as it is defined one of them. Someone defined it as left long time ago. I think I use to use it for a long time, then it is almost natural for me. Maybe it is unnatural I fell it natural. However, this is definition and itself does not have meaning. We could have right-associative system. I assume the reason we use left-associative is because many of European language write the language from left to right, and nowadays mathematical notation is based on European mathematics.

No comments: