Why doesn't the silent "e" work on the word "infinite"?
What I mean is, why does mate have a long "a", but infinite has a short "i"?
As @tchrist has pointed out with excellent examples, the final e's ability to create a long vowel sound is a 'tendency' and not a 'rule'.
The specific reason why 'infinite' is pronounced as it is has been discussed here: Pronunciation differences between "finite" and "infinite"
It boils down to where stresses occur in English speech. The link above provides far more precise explanation than I'm capable of.
Edit: also see Brian Hooper's answer: "Finite" and "infinite": another example? for more examples of how an additional syllable can completely change pronunciation.
The apparently irregular pronunciation of infinite (that it is not
ˈɪnfənaɪt) is a confluence of two factors, both historical, I think.
The first is that the negative prefix in- (and its kin, il-, im-, ...) is only rarely stressed in English. In most cases where it is, there has either been semantic drift between the prefixed and unprefixed forms (impotent is not quite the negation of potent; likewise for infamous and famous, and invalid [
ˈɪnvəlɪd] and valid); or else the unprefixed form is simply unattested (impudent, indigent, indolent, insolent). Semantically transparent examples like infinite and finite, or indirect and direct, are in the minority. This suggests that the process that drew stress onto the in- prefix was an archaic one, possibly one that operated in the language from which English borrowed the words in question (e.g., Old French).
The second is that pronunciation of -finite as
fənət occurs, I think, only in infinite and definite (and derivatives, like indefinite). Again, this contrasts with the pronunciation
faɪnaɪt, which occurs in cofinite, subfinite, and transfinite. Given that the latter are all modern innovations, this suggests that the pronunciation
fənət, too, may be a relic from earlier phonological processes, possibly active in the source language from which English borrowed these words.
So, the pronunciation of infinite is doubly archaic—which may be the most principled account one can give of this particular oddity of English spelling.
BTW, I know that many people have a higher vowel than schwa in
-fənət, but I’m following my own (native) pronunciation here. Also, I’ve deliberately omitted secondary stress. Definite may have come to English straight from Latin (in about the 16th century), but its stress and vowel quality may have analogised with infinite.