Dieudonne on Dedekind
Timothy Y. Chow
tchow at math.princeton.edu
Tue Aug 10 22:24:47 EDT 2021
Colin McLarty wrote:
> He wants undergrad students to know the real numbers have the least
> upper bound property. But he does not want them taught that somehow you
> begin with sets of rational numbers and then derive the property for
> sets of reals from that. Note that historically Dedekind did base his
> account of real numbers on the least upper bound principle, while he
> explicitly refused to identify real numbers as cuts on the rationals.
It seems to me---though I might be reading him wrong---that he wants
undergrads to take for granted (and in particular, not worry about
proving) the existence of an ordered field with the least upper bound
property. That is, he's not arguing for this proof of existence versus
that proof of existence. Rather, on this view, undergrads need not worry
about the existence question at all.
It's true that this is close to how professional mathematicians tend to
view the matter. Dedekind cuts or Cauchy sequences are used to prove
existence, but once you've done that, you forget about the existence proof
and just use the axioms.
I'm not sure how I feel about this attitude. It's certainly a pragmatic
one---you can cover more material this way, and the students who don't see
the point of the existence proof are spared the torture of being dragged
through it. On the other hand, as a fan of f.o.m., I can't help but feel
some pangs of conscience about knowingly leaving a gaping hole in the
foundations of the subject, right around the time that most students are
just beginning to understand what rigorous proof is.
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