Has miscommunication ever benefited mathematics? Let's list examples.

Question

I'm looking for examples of when miscommunication has lead to advancements in mathematics.

Perhaps many a theorem has been discovered by accidents of communication. For instance, perhaps a mathematician has tried to explain definition $A$ to another mathematician, who then misunderstood it as definition $B$ but went on to prove a theorem or two using definition $B$.

Note that I'm not looking for failed proofs that might be construed as misunderstandings. Much of algebraic number theory, for instance, stems from a failed attempt at proving Fermat's Last Theorem, but I don't think that classifies as a miscommunication.

Answer 1

According to his Nobel lecture, Richard Feynman came up with the path integral formalism of quantum mechanics by attempting to decipher a cryptic remark in an article written by Dirac. The circumstances and mathematical detail are given in this article, particularly § IV, but some details are given by Feynman in the aforementioned lecture (PDF version):

What Dirac said was the following: There is in quantum mechanics a very important quantity which carries the wave function from one time to another, besides the differential equation but equivalent to it, a kind of a kernel, which we might call $K(x', x)$, which carries the wave function $\pi(x)$ known at time $t$, to the wave function $\psi(x')$ at time, $t+\varepsilon$. Dirac points out that this function $K$ was analogous to the quantity in classical mechanics that you would calculate if you took the exponential of $i\varepsilon$, multiplied by the Lagrangian $L(\dot{x},x)$ imagining that these two positions $x$, $x'$ corresponded to $t$ and $t+\varepsilon$. In other words,

$$ K(x',x) \text{ is analogous to } e^{i\varepsilon L\left( \frac{x'-x}{\varepsilon} , x \right)/\hbar} $$

Professor Jehle showed me this, I read it, he explained it to me, and I said, "what does he mean, they are analogous; what does that mean, analogous? What is the use of that?" He said, "you Americans! You always want to find a use for everything!" I said, that I thought that Dirac must mean that they were equal. "No", he explained, "he doesn't mean they are equal." "Well", I said, "let's see what happens if we make them equal." [...]

(As with most of Feynman's stories, Feynman comes out of it rather well, but the bare facts are probably true.)