How many vowels does English have?

If you're like most English-speakers, you're an English-reader, and you think that English has five, sometimes six, vowels — a, e, i, o, u, and sometimes y — because the English alphabet has five, sometimes six, vowels.

But if you look at how dictionaries "respell" English words for pronunciation, you quickly realize that English — even if we look only at a single dialect, like General American English — has many, many vowels — including not just monophthongs, or pure vowels, but diphthongs and triphthongs composed of multiple vowel sounds.

International Phonetic Alphabet	Examples
æ	pat, lad, cat, ran
eɪ	pay, day
ɛər	care, hair, there
ɑ	father, palm
ɑr	arm
ɛ	let, head
iː	bee, see
ɪ	pit, city
aɪ	pie, by, my
ɪər	pier, near, here
ɒ	pot, not, wasp
oʊ	toe, no
ɔː	caught, paw, war
ɔɪ	noise, boy
ʊ	took, put
ʊər	tour
uː	boot, soon, through
aʊ	out, now
ʌ	cut, run, enough
ɝː	urge, term, firm, word, heard, bird
ə	about, item, edible, gallop, circus
ɚ	butter, winner
juː	pupil
ø, œ	feu, oeuf (French), schön, zwölf (German)
y	tu (French), über (German)
ɔ̃	bon (French)

Looking at all those crazy IPA spellings is enough to make you feel like a ghoti out of water.