What do variables represent, anyway?

[HyperbolicMan]

"let x be a real number. Then x^2+1 does not equal 0."

"For all x in lR, x^2+1 does not equal 0"

As far as I know, both of these statements mean exactly the same thing. From a grammatical perspective, in the first statement, x is singular ("a real number"), while in the second, x appears to be plural (preceded by "all" which implies more than one). Is one way 'more correct' than the other, or is there some principle in mathematics that equates the two? Instinctively, I feel like the second is better because it sounds more general, but the first one is easier to visualize.

 

[mathwonk]

i think both usages are singular. i.e. "everyone here is a nerd." every is singular. it the same as for all x, to me at least.

 

[mathwonk]

your question is not about variables, but about quantifiers. get a book on quantifiers. i.e. propositional calculus, or mathematical logic. or just a book on proofs and read the part on quantifiers. you will never regret it.