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Re: [HM] Dedekind's objection to the Newtonian concept of number.
Posted:
Aug 17, 2006 4:50 PM
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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.
William C. Waterhouse
Penn State
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