After studying mathematics for some time, I am still confused.
The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the symbol “$\Rightarrow$”, which is read as “implies” or “if … then ….” It is customary and reasonable to treat “$\Rightarrow$” as the material conditional, i. e. as equivalent to saying “the antecedent is false or the consequence is true (or both)”.
Now there is some controversy about whether the material conditional really captures conditional statements because it doesn't really say anything about a logical connection between the antecedens and the consequence. This is quite often illustrated by the means of statements in natural languages such as “the moon is made out of cheese $\Rightarrow$ all hamsters are green” – since the moon isn't made out of cheese, is this statement true? This remained problematic to me.
While I came to accept the material conditional as a good way of describing implications and conditionals, I'm having a hard time to explain this usage to freshmen whenever I get asked.
My questions are: How can we best justify the interpretation of “$\Rightarrow$” as a material conditional? Why is it so well-suited for mathematics? How can we interpret or read it to understand it better? Can my confusion about it be lead back to some kind of misunderstandig or misinterpretation of something?
I have yet very poor background in mathematical logic (I sometimes browse wikipedia articles about it), but I'd have no problem with a technical answer to this question if it clarifies the situation.