[FOM] When is it appropriate to treat isomorphism as identity?
Timothy Y. Chow
tchow at alum.mit.edu
Thu Apr 23 10:29:47 EDT 2009
On Wed, 22 Apr 2009, Monroe Eskew wrote:
> This characterization of mathematical existence seems wrong. The
> mathematicians I know all seem very careful with their existence
> claims. For instance it's not completely trivial to show that every
> field has a (unique up to isomorphism) algebraic closure, and I
> haven't seen an introduction to field theory that skips over the
> argument for it. The rule, "whenever we need something silly like
> that, it exists," is not only imprecise but too liberal.
Let me clarify what I meant to say in the specific case of an algebraic
closure. It's true that mathematicians are careful to make sure that the
algebraic closure exists before they use it. What they *could* do,
however, if they were concerned about "necessity" in Vaughan Pratt's
sense, would be to prove the existence of an algebraic closure separately
in each situation that they needed it. This would be parsimonious and
would require only very weak axioms. Instead, the attitude seems to be
that it's fine to assume some innocuous-looking but very powerful axiom,
namely the axiom of choice. This takes care of all algebraic closures in
one fell swoop. The fact that it also lets a bunch of dubious riff-raff
in the door was not recognized until later and came as an unpleasant
surprise. So this is a case study in liberality.
Similarly for uniqueness: Mathematicians are careful to prove uniqueness
when they assert it. However, they may be sloppy about isomorphism versus
identity when the distinction strikes them as an unimportant quibble in
the context.
> Such a stance seems to license assuming the existence of a countably
> additive, translation invariant measure on the real line, which is an
> intuitive principle with some interesting implications, but is of course
> not possible in the presence of the axiom of choice.
Joe Shipman has argued that if history had unfolded differently, this
might very well have happened (i.e., RVM accepted as standard instead of
AC). I'm not fully convinced by his argument, but I agree with the
notion that if the issue had come up tangentially, so to speak, then RVM
might very well have been accepted casually in the same way that AC was
accepted casually. Of course, now that AC is so entrenched, RVM has a
tough time ousting the incumbent.
> Also, some interesting statements about real numbers can be proved from
> large cardinals, but most mathematicians don't seem that eager to make
> large cardinal assumptions.
Mostly, this is because the statements aren't judged to be all that
interesting by most mathematicians. If the statements *do* cross the
threshold of being interesting to most mathematicians, then I would bet
that large cardinals would be quickly accepted (at least in the same way
that AC has been accepted---i.e., it is sometimes stated explicitly as a
hypothesis, but people don't hesitate to invoke it when needed).
Tim
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