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Re: [HM] Dedekind's objection to the Newtonian concept of number.
Posted:
May 27, 2006 10:19 AM
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Dear Heinz,
Thank you for this fine discussion from which I have learned a great deal.
You wrote: <<As long as you do not see the difference between "not
founded" and "invalid", you will not have understood Dedekind's remark
your are critizising. "Foundation" is Dedekind's point, not mine.>>
Whatever I may have said to suggest I could see no difference between
"not founded" and "invalid", I think I understand enough to say it is
not relevant. The issue here is not whether the understanding of
number current at the time D. wrote was well founded, or based on clear
concepts; it certainly wasn't. The issue I have tried to bring up
concerns the subsidiary objection D. raises in a footnote, namely that
this conception did not extended to complex numbers.
Strictly speaking, D.'s statement is correct, for if one allows no
extension of concepts, the Newtonian understanding of number does not
even include negative or zero.
But it seems to me, and I thought I had demonstrated it adequately, that
with extensions of the idea of magnitude to directed magnitude, one can
easily do exactly that and moreover it seems to me that, with some
effort, greater clarity could have been obtained. D. apparently didn't
care to do that, but preferred to move in an entirely different
direction. In fact, he says so: "Instead of this, I demand (fordere)
that arithmetic should develop (itself) out of itself. "
What status has such a "demand" in mathematics? Can one demand whatever
one wants? You can't do that in physics, or accounting. I suppose
you have to be a certified mathematician to make such demands.
Regards from Comer, where our cool beautiful spring has suddenly been
replaced by humidity and insects,
Bob
Robert Eldon Taylor
philologos at mindspring dot com
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