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'A number of philosophers believe' sounds right to me. Not only because philosophers is plural, but also number, while singular, has a plural meaning.

Also having the s - s ending in 'philosophers believes' is awkward because philosophers are plural and are the ones doing the believing.

'A number of philosophers believes' seems pedantically grammatically correct, since it is a number, so number is singular. But it sounds wrong to my ear.

What's the "right" way?

Wapiti
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    The question is, "If you were to substitute a pronoun for "A number of philosophers" would it be "it" or "they"?" If it is it, then this implies that the number is doing the believing... this is not possible - numbers can't believe anything. More simply, "a number [of nouns]" = an indefinite quantity of nouns that is more than one. – Greybeard Nov 17 '21 at 22:43
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    "A number" is what's known as a 'number transparent noun phrase'. The fact that it is singular doesn't affect the agreement of the verb. – Araucaria - Him Nov 17 '21 at 22:55
  • Compare 'A couple/score of philosophers believe ....' – Edwin Ashworth Nov 18 '21 at 14:40

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