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This sentence is said to be correct:

Skeptical of the abilities of prophets to tell the future, a significant number of philosophers in Athens began to value reason over revealed truths.

"Skeptical" modifies "number", but "number" cannot be "Skeptical." "Skeptical" should modify "philosophers."

Can someone shed light on this sentence?

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    It's OK: a number of is understood to act as a quantifier rather than a noun (technically, a "quantificational noun"), so the actual subject of the head clause is philosophers. – StoneyB on hiatus Jun 9 '15 at 18:27
  • Welcome to ELL Stack Exchange! Nice first question. – DCShannon Jun 12 '15 at 20:14
  • @StoneyB I looked up quantificational nouns, and I'm not sure that it actually applies in this case. Maybe it's just a different way of looking at things. Would you care to elaborate, perhaps in an answer? – DCShannon Jun 12 '15 at 20:16
  • @DCShannon Take a look at this question, where my answer cites CGEL. Other common 'number-transparent quantificational nouns' that are singular in form but accept a plural oblique are lot, plenty, rest and couple. – StoneyB on hiatus Jun 12 '15 at 20:31
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Nope, a number can indeed be skeptical. Something like:

A skeptical number began to value reason.

is just fine. There's nothing wrong with this. A valid meaning of number is a few from a larger group.

Of course, without context or previous conversation/text we'd be asking number of what?. But that's explicitly stated your example, so no confusion.

Numbers as in numerals can't be skeptical, so anyone hearing/reading something like this won't be thinking that.

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Simplify for Clarity

Let's strip this sentence down:

Skeptical of the abilities of prophets to tell the future, a significant number of philosophers in Athens began to value reason over revealed truths.

The first clause adds meaning, but isn't essential to the core sentence.

A significant number of philosophers in Athens began to value reason over revealed truths.

The last clause only tells us a bit more about how much they're valuing reason, so it's not essential.

A significant number of philosophers in Athens began to value reason.

Significant is just an adjective, let's drop that.

A number of philosophers in Athens began to value reason.

Who cares where these philosophers are located?

A number of philosophers began to value reason.

"Of philosophers" is just a prepositional phrase describing what we have a number of.

A number began to value reason.

This is a perfectly valid sentence. "A number" is the subject. This is one of the most common usages of the word 'number'.

Dictionary Definitions

Some different phrasings of its definition from a few dictionaries:

Wiktionary:

  1. Quantity

Reference.com:

  1. the sum, total, count, or aggregate of a collection of units, or the like

Merriam-Webster:

1a. (1) : a sum of units : total (2) : complement 1b (3) : an indefinite usually large total (4) plural : a numerous group : many (5) : a numerical preponderance

Therefore, here are some other, mostly synonymous phrasings:

Many (philosophers) began to value reason.

A group (of philosophers) began to value reason.

A preponderance (of philosophers) began to value reason.

A Skeptical Number

So the subject of the sentence being modified is not "philosophers", it's "a number". It might be more useful to look at it as "a number of philosophers", but the phrase specifying what we have a number of may not always be present in the sentence. It could be from context:

At this time there were a great many philosophers in Athens. Skeptical of the abilities of prophets to tell the future, a significant number began to value reason over revealed truths.

So yes, 'skeptical' can modify number, because "a number" is a group of things, and the things in the group can be skeptical.

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Try thinking of "number" as "group" or "set."

You're right that skeptical modifies number, but in this case the phrase "of philosophers" defines the number as a set of people, which can be skeptical.

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