The computing + is not associative for inexact types like floating-point. That's why it's important for the Lisp + to be consistently left-associative; the result could vary if that were left to the implementation to do however it wants.
In addition/relation to floating-point, another way in which addition is not associative in computing is if there are type conversions. (+ 1 1 0.1) is not the same as (+ 1 (+ 1 0.1)). The former will do an integer addition to produce 2, and then that is coerced to 2.0 which is added to 0.1. The latter adds 1.0 to 0.1, and then adds that to 1.0: two floating-point additions.
In languages that don't have bignums (which could include some Lisp dialects) whether overflows occur can depend on the order of operations, even when all operands are integers.
The reason we can have a n-ary + is that three or more arguments can be decimated through a binary +. The concept of + is defined as a binary operation.
Lisps have variadic functions that are blatantly non-associative, like, oh, list. (list 1 2 3) isn't (list 2 3 1).
In addition/relation to floating-point, another way in which addition is not associative in computing is if there are type conversions. (+ 1 1 0.1) is not the same as (+ 1 (+ 1 0.1)). The former will do an integer addition to produce 2, and then that is coerced to 2.0 which is added to 0.1. The latter adds 1.0 to 0.1, and then adds that to 1.0: two floating-point additions.
In languages that don't have bignums (which could include some Lisp dialects) whether overflows occur can depend on the order of operations, even when all operands are integers.
The reason we can have a n-ary + is that three or more arguments can be decimated through a binary +. The concept of + is defined as a binary operation.
Lisps have variadic functions that are blatantly non-associative, like, oh, list. (list 1 2 3) isn't (list 2 3 1).