Discussion on question by Qiaochu Yuan: What does the word "let" mean in mathematics?

I guess it depends whether the phrase using "let" occurs at the beginning of an argument or proof, or somewhere in the middle? At the beginning of a proof, the phrase "Let X be Y, then F(X) holds" should be equivalent to "If X is Y, then F(X)", or "Assuming X is Y, then F(X)", or any similar phrasings. But things may be a bit more complicated if this expression occurs in the middle of a proof...

I don't think if means much, formally. It is similar to starting a sentence "if", but applying to a lot of sentences. "If $G$ is a group, ..." but the assumption applies across sentences is, and $G$ is the same group for the rest of the text or section of text or proof. That is, we,are giving a name for one supposed entity.

@QiaochuYuan To me the answer is obvious, and as a consequence it is completely unclear to me what the problem might be. Could you provide some example usage to illustrate the problem with the obvious interpretation? Which (to me) is simply the definition of a symbol.