Dieudonne on Dedekind

[Timothy Y. Chow]

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.

Tim