What is the difference in usage between maximum and maximal? When would you use one or the other?
Maximum can be a noun or an adjective:
This is the maximum it can be set to.
This is the maximum value.
whereas “maximal” is always an adjective:
This is the maximal value.
Is this correct? Is there more to it than that? When would you use one or the other as an adjective, or are they completely interchangeable?
There is a subtle difference; maximum and minimum relate to absolute values — there is nothing higher than the maximum and nothing lower than the minimum. Maximal and minimal, however, can be more vague.
In “I want to buy this at minimal cost” and “this action carries a minimal risk”, minimal means “very small” as opposed to “the lowest possible”; the same distinction is true of maximum and maximal.