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?

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    In your third example, maximum must be a typo for maximal. – Tsuyoshi Ito Aug 17 '11 at 12:22

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.

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    Why was this downvoted? Seems like a reasonable answer. – Clay Nichols Aug 17 '11 at 12:18
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    I don't believe this answer is correct. The usage you cite is simply a misunderstanding about 'maximum' etc., which are frequently used to mean 'very big' rather than their correct meaning which is 'the biggest'. – DJClayworth Aug 17 '11 at 14:29
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    Surely if words are frequently used to mean something then that becomes their correct meaning? Language isn't static. After all the question was about difference "in usage" not "in original meaning" – Waggers Aug 18 '11 at 7:53
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    Really? I’m with Clay and Waggers. I’ve been listening to English for only 60 years and I’ve yet to notice ‘maximum’ ever used to mean 'very big' rather than 'the biggest’ If I had from more than about age 12, I’d have been sure that in the unlikely event the speaker knew the difference, he surely didn’t care. ‘Maximum’ is the greatest; ‘maximal’ is tending towards the greatest. People who neither know nor care are welcome to make up their own language and to try to explain their own rules but please don’t let’s kow-tow to that… – Robbie Goodwin Apr 10 '17 at 20:49
  • @ DJClayworth Once it's listed without caveat in a respectable dictionary, the people saying that the usage is wrong also need to check their facts. 'Minimal' (covered in your 'etc' here) is certainly used acceptably for 'very low / negligible'. – Edwin Ashworth Sep 22 '17 at 7:49

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.

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    It seems to me that mathematical definition (a maximal value=the local maximum) effectively means the same as @Waggers "non-technical" one anyway. A maximal value would not be so called if it was easy to identify a higher one "nearby" (whatever "nearby" meant in context). – FumbleFingers Aug 17 '11 at 16:39
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    @FumbleFingers: you're right ... it's essentially a mathematical adaptation of the non-technical definition. – Peter Shor Aug 17 '11 at 19:10
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    I have seen maximal used like in Peter's answer: In my algorithms class we used maximal to mean that we would have to backtrack to get a better(larger) answer. – Brian Mar 23 '13 at 2:59
  • In this two peaks graph, is the maximal point the lower peak? I know that in math, the point near zero is call minimum, but it seems that it doesn't suit the normal meaning. – Ooker Jul 1 '15 at 9:40
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    Both the higher and lower peaks are maximal; only the higher one is maximum. – Peter Shor Jul 1 '15 at 11:57

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.

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