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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.
Waggers' answer does an excellent job of explaining the difference in the non-technical meaning. In some areas of mathematics (e.g., in maximal element, maximal matching), a maximal value (you can't use the with this meaning) is essentially a local maximum—it's a maximum value in its neighborhood; i.e., there are no small changes which will increase the value. See this Wikipedia article for a more technical description in relation to partially ordered sets.
The short answer is that, unless you are a mathematician or an economist, there is no difference. However, there is a distinction between the two terms in the context of partially ordered sets (i.e. sets in which not every pair of elements need be comparable).
An element is maximal if there is no other element greater.
An element is maximum if it is itself greater than every other element.
If the "elements" under discussion are numbers, the definitions coincide, but there are contexts in which the distinction matters.
For example, in an election one might say that candidate 1 is strictly better than candidate 2 if all potential voters prefer candidate 1 to candidate 2. Say that three candidates - Mitt, Barry, and Adolf - are running for president of a club.
The club members are divided into two contingents of equal size. One group unanimously prefers Barry to Mitt and Mitt to Adolf, while the other unanimously prefers Mitt to Barry and Barry to Adolf.
Barry and Mitt are both strictly better candidates than Adolf, as all members rank Adolf last. But since some members prefer Barry to Mitt and some prefer Mitt to Barry, neither is strictly better than the other. Thus neither can be maximum with respect to this ordering. However, because no candidates exist that are strictly better than either of them, both candidates are maximal with respect to this ordering.
Mathematicians make another distinction between the terms when considering sets that satisfy a certain property. For example, a "clique" is a set of people all of whom know each other. A clique is maximal if adding anyone else to the set destroys the clique property, that is, there is no larger clique that contains it. A clique is maximum if there is no larger clique. For example, if Alice, Bob, and Cam know each other, and Deb, Ed, Fran, and Gay know each other, but none of the first three know any of the other four, then Alice,Bob,Cam are a maximal clique but not a maximum clique. There can be many maximum cliques. Every maximum clique is maximal, but not vice-versa. The other answer about backtracking is another example of this distinction because backtracking in a search means removing an element from a set.
