I am returning to this topic just to point out one sentence that I think had been misunderstood in the previous discussion.
After saying that he wants to develop numbers on a basis of arithmetic alone, Dedekind adds:
"In general it may be added that such references to non-arithmetical conceptions have given the next occasion for the extension of the number concept (however with the introduction of complex numbers this has decidedly not been the case)..."
Mr.Taylor (writing May 16) interprets this as meaning
"He accepts some validity, but says that it cannot be extended to complex numbers."
I think this is wrong and ignores the past tenses. It seems clear to me that Dedekind is not making any programmatic statement here. He is merely recording that the earliest extensions of the number concept were in fact prompted by consideration of geometric magnitudes, though that was not the case for complex numbers. Those statements are correct history, and I see no suggestion that they are meant as anything other than historical background.