Well, I once knew a mathematician who could only be described as stupid (not to mention ignorant). He was not an especially good mathematician, but he did research and published it. But if you engaged him in a conversation on any other topic, you quickly realized that this was a fundamentally stupid person.
On the other hand, I have known some highly cultured mathematicians. One thing about mathematicians, though, none of them will say to me that their worst subject was math, which is a typical response when you say you’re a mathematician.
Excellence at math is difficult to fake and it is easier to establish absolute standards. This lack of subjectivity in evaluation is quite different from deciding whose poem is lovely.
As to your question about whether or not mathematical abilities correlate with intelligence: yes, they do. Take a difficult problem, for example: Land a man on the moon. The folks who did that were highly intelligent–it’s a difficult problem to solve. I’d venture to say (and lay off the “cite” crapola, wouldya?) the key figures involved in all of the really difficult problems (not counting “problems” such as managing people) were facile with math.
Intelligence is necessary to grasp math. You can paste an isolated fact into an idiot’s brain and sometimes it will even stick. You cannot paste robust mathematical ability onto a person of average intelligence.
This is not to say the ability to regurgitate lots of facts, or display linguistic facility does not also correlate with intelligence. The Pedant’s mind is a dadburn attic of never-forgotten useless facts despite a math IQ of 50.
This, of course, is emblematic of the problems with the way schools work. And the more they rely on mindless memorization and decontextualized teaching (i.e. “reading classes”), the more likely that we will continue to be a nation of people who can (possibly) say when, but will be unlikely to know why.
For sure, people can be very intelligent without any expressed mathematical ability. OTOH people who can display mathematical ability, especially in abstract or esoteric matters, are generally viewed as “intelligent”. As already mentioned this ability to think and communicate using symbols, notation, and language that are foreign to most people, yet seem relevant to the real world, gives off an aura of intelligence. But we all know that intelligence in this sense does not necessarily mean street smart or even having a lot of common sense.
I think you’re confusing mathematics with the language that people use to communicate that mathematics in. IMHO, mathematicians are seen as smart not because they use esoteric symbols, rather because mathematics itself is so abstract. Being asked to reason about something that couldn’t possibly exist, or something you have absolutely no grounding with, etc., is what drives the stereotype, I think.
I don’t know about that. If a person does not communicate his math (in speech or writing) how can you tell that he has math ability? And once he/she starts communicating that’s when the association with intelligence occurs. I do grant that the notion of mathematician = abstract thinker is a key part of the stereotypical association with intelligence.