Re: [HM] Good History--Correct Results and Errors

Roberto Baldino (baldino@linkway.com.br)
Sun, 04 Oct 1998 22:51:09 -0300

At 11:11 04/10/98, Gordon Fisher wrote:
> ...
> I must admit to not seeing much of a distinction between determining an
> irrational by a Dedekind cut (or pair of cuts), and then talking about cuts
> as irrationals, ontologically speaking, and defining an irrational by a
> cut. If I recall correctly, Dedekind gave a valid and satisfactory
> definition of cuts in *Was sind und was sollen die Zahlen*. Or am I being
> somehow anachronistic, or perhaps philosophically naive? Perhaps you are
> referring to the fact that set theory, as we came to know it, was not
> overtly developed until the time of Russell?
>

Dear Gordon
I quote Dedekind ("Essays on the theory of numbers", Dover 1963, p. 15).
"Whenever, then, we have to do with a cut (A1,A2) produced by no rational
number, we create a new, an _irrational_ number alpha, which we regard as
completely defined by this cut (A1,A2); we shall say that the number alpha
corresponds to this cut, or that it produces this cut."
Dedekind does not say that the cut IS the number. To say that the number is
"completely defined" by the cut is not saying that the number is the cut.
How do we proceed to "create" a new number? And how do we "regard it"
as...? Moreover, this "number", created out of thin air, has the power of
PRODUCING the cut, so that we can "say" that the creature corresponds to
the creator.
If we neglect this point as irrelevant I think that we risk to lose the
variation of the meaning of definitions in the last one hundred years.
Of course, if we are just "talking about", then I see no harm.
Your truly
Baldino