Many is a great word for its ambiguity. You needn't know the actual number and you're still fine to say many. It lets you speak without knowing, or focus on the main idea without getting lost in the details.
But, still, how many is many?—especially in the construction "many of", which defines a total population, or when that total population can be assumed/is defined with a different construction (e.g. "many Americans").
Certainly it is relative. "Many people in my class" is a very different number than "many people in this country", because the total population for my class is much less than the total population of this country.
Is many proportional, then?, relative to the overall total? Is there a certain percentage where it begins to make sense?
NOAD defines the noun form as a majority,
noun (as plural noun the many) the majority of people: music for the many;
but there seems to be no need for more than half in the adjective's definition,
det., pronoun, & adjective (more, most) a large number of
But does many still need to refer to more than 30%? 10%? 1%?
Or is it relative to something else—not to some total but to a more ordinary amount?
Perhaps the underlying (or secondary?) question is whether many presumes a denominator.
It seems that when working with smaller total populations, many is proportional, but as the total population gets bigger, many can refer to a smaller percentage. As the overall total grows, the amount that qualifies as many grows at a smaller rate.
I recognize that I am perhaps attempting to define the indefinable—yet we all have a working definition of what constitutes many, and I would not be surprised if it is generally shared definition. Let's record it.