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Regards from Comer, where our cool beautiful spring has suddenly been replaced by humidity and insects, Bob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