This is a question about the logic of mathematical language concerning infinite series.
It's normal to say $\sum\limits_{n=0}^\infty 2^{-n}=2$. This type of equation is often given in the form of a notational introduction within the definition of "converge(nt)". But it's also normal to assert things about $\sum\limits_{n=0}^\infty 2^{-n}$ and deny the corresponding statements about $2$:
- $\sum\limits_{n=0}^\infty 2^{-n}$ converges, but $2$ does not "converge".
- $\sum\limits_{n=0}^\infty 2^{-n}$ is an infinite series, but $2$ is not.
On the surface, this looks like a violation of the basic substitutability of equals for equals. I see two possible explanations:
First, the problem result from talking about $\sum\limits_{n=0}^\infty 2^{-n}$ but implicitly referring to its form. For example, in "6/3 is a fraction but 2 is not", the idea "is a fraction" refers not to the value of 6/3 but to its form. This seems plausible and attractive, especially for saying "is an infinite series", but it seems to be a stretch for "converges". For example, to say that a nested sum $\sum\limits_{n=0}^\infty \ \sum\limits_{m=0}^\infty \ldots$ "converges" (treating it as a sum on $n$), we require that the inner sums be evaluated. This does not feel like a description of form alone.
Second, the problem might occur because the equality of $ \sum\limits_{n=0}^\infty 2^{-n}=2$ is not in fact sincere equality: It means something other than logical identity. This interpretation is strongly favored by the fact that it appears in a definition! (Presumably we would not be entitled to redefine logical identity.) Thus there is no reason to expect substitutability, and there is no problem. But this seems disingenuous: In many contexts, we freely substitute series and their sums. We also use this "$=$" symbol symmetrically and transitively, mixing it without comment with normal equality.
Have I correctly understood normal usage? Is either of these interpretations the "correct" one? Is there a "logician's solution"?